The document discusses complex numbers. It explains that the equation x^2 = -1 has no real solutions, so an imaginary number i is defined where i^2 = -1. A complex number is defined as a + bi, where a is the real part and bi is the imaginary part. Complex numbers can be added or multiplied by treating i as a variable and using rules like i^2 = -1. Examples show how to solve equations and perform operations with complex numbers.
The document describes the number line and how it assigns numbers to points on a line. It explains that 0 is assigned to the center or origin of the line. Positive numbers are assigned to points to the right of 0, and negative numbers to points on the left. The number line defines the relative size of numbers based on their position, with numbers to the right being greater than those to the left. Intervals on the number line, such as -1 < x < 3, represent all the numbers between and including the bounds.
1) Complex numbers are numbers of the form a + bi, where a and b are real numbers. a is called the real part and bi is called the imaginary part.
2) To add or subtract complex numbers, treat i as a variable and combine like terms.
3) To multiply complex numbers, use FOIL and set i^2 equal to -1 to simplify the result.
The document discusses expressions and polynomials. It provides examples of algebraic expressions and operations that can be performed on polynomials, such as factoring. Factoring polynomials is useful for easier evaluation, simplifying rational expressions, and solving equations. One example factors the polynomial 64x3 + 125 into (4x + 5)(16x2 - 20x + 25). Factoring the polynomial 2x3 - 5x2 + 2x is recommended before evaluating it for specific values of x.
1. The document discusses subsets of real numbers including natural numbers, whole numbers, integers, and rational numbers.
2. Natural numbers are used for counting and start at 1. Whole numbers are formed by adding 0 to the natural numbers. Integers are formed by adding the negatives of natural numbers to whole numbers.
3. Rational numbers can be expressed as fractions a/b where a and b are integers and b is not equal to 0. Their decimal representations either terminate or repeat.
The document discusses solving linear equations. It states that to solve linear equations, each side should be simplified first by separating the x-terms from the numbers. It then introduces the change-side change-sign rule - when moving terms to the other side of the equation, their signs change. An example problem is worked through step-by-step to demonstrate. It also discusses the opposite rule - that if -x = c, then x = -c. Finally, it notes that equations involving fractions can always be restated without fractions by finding the least common denominator.
The document discusses solving word problems by translating English phrases into mathematical operations. It provides key words that represent different operations such as "+" for add and "-" for subtract. It then gives examples of word problems and shows how to represent the problems mathematically by defining variables, identifying unknown values, and relating other values to the variables. The goal is to break down word problems into clear mathematical expressions.
1 s2 addition and subtraction of signed numbersmath123a
The document discusses addition and subtraction of signed numbers. It states that adding signed numbers involves removing parentheses and combining the numbers. Examples are provided to demonstrate this process. For subtraction, the concept of opposite numbers is introduced, where the opposite of a positive number x is -x, and the opposite of a negative number -x is x. The process of finding opposites is demonstrated using examples.
55 inequalities and comparative statementsalg1testreview
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. This line is called the real number line. Two numbers are related by an inequality if their corresponding positions on the real number line have one number being further to the right than the other. Inequalities can be used to represent intervals of numbers on the real number line.
The document describes the number line and how it assigns numbers to points on a line. It explains that 0 is assigned to the center or origin of the line. Positive numbers are assigned to points to the right of 0, and negative numbers to points on the left. The number line defines the relative size of numbers based on their position, with numbers to the right being greater than those to the left. Intervals on the number line, such as -1 < x < 3, represent all the numbers between and including the bounds.
1) Complex numbers are numbers of the form a + bi, where a and b are real numbers. a is called the real part and bi is called the imaginary part.
2) To add or subtract complex numbers, treat i as a variable and combine like terms.
3) To multiply complex numbers, use FOIL and set i^2 equal to -1 to simplify the result.
The document discusses expressions and polynomials. It provides examples of algebraic expressions and operations that can be performed on polynomials, such as factoring. Factoring polynomials is useful for easier evaluation, simplifying rational expressions, and solving equations. One example factors the polynomial 64x3 + 125 into (4x + 5)(16x2 - 20x + 25). Factoring the polynomial 2x3 - 5x2 + 2x is recommended before evaluating it for specific values of x.
1. The document discusses subsets of real numbers including natural numbers, whole numbers, integers, and rational numbers.
2. Natural numbers are used for counting and start at 1. Whole numbers are formed by adding 0 to the natural numbers. Integers are formed by adding the negatives of natural numbers to whole numbers.
3. Rational numbers can be expressed as fractions a/b where a and b are integers and b is not equal to 0. Their decimal representations either terminate or repeat.
The document discusses solving linear equations. It states that to solve linear equations, each side should be simplified first by separating the x-terms from the numbers. It then introduces the change-side change-sign rule - when moving terms to the other side of the equation, their signs change. An example problem is worked through step-by-step to demonstrate. It also discusses the opposite rule - that if -x = c, then x = -c. Finally, it notes that equations involving fractions can always be restated without fractions by finding the least common denominator.
The document discusses solving word problems by translating English phrases into mathematical operations. It provides key words that represent different operations such as "+" for add and "-" for subtract. It then gives examples of word problems and shows how to represent the problems mathematically by defining variables, identifying unknown values, and relating other values to the variables. The goal is to break down word problems into clear mathematical expressions.
1 s2 addition and subtraction of signed numbersmath123a
The document discusses addition and subtraction of signed numbers. It states that adding signed numbers involves removing parentheses and combining the numbers. Examples are provided to demonstrate this process. For subtraction, the concept of opposite numbers is introduced, where the opposite of a positive number x is -x, and the opposite of a negative number -x is x. The process of finding opposites is demonstrated using examples.
55 inequalities and comparative statementsalg1testreview
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. This line is called the real number line. Two numbers are related by an inequality if their corresponding positions on the real number line have one number being further to the right than the other. Inequalities can be used to represent intervals of numbers on the real number line.
3 1 the real line and linear inequalities-xmath123b
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on the real number line, with positive numbers to the right of zero and negative numbers to the left. An inequality relates the positions of two numbers on the real number line, with the number farther to the right said to be greater than the number to its left. The document provides examples of inequalities and how to represent sets of numbers using inequalities, such as all numbers between two values a and b. It also outlines steps for solving inequalities algebraically.
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. An inequality relates the position of two numbers on the real number line, with the number farther to the right said to be greater than the number on the left. Specific inequalities are shown graphically on the number line. The properties of inequalities, such as adding or multiplying the same quantity to both sides preserving the inequality, are also covered.
1 s3 multiplication and division of signed numbersmath123a
The document discusses rules for multiplying signed numbers. It states that to multiply two signed numbers, multiply their absolute values and use rules to determine the sign of the product: two numbers with the same sign yield a positive product, while two numbers with opposite signs yield a negative product. It also discusses that in algebra, operations are often implied rather than written out. The even-odd rule determines the sign of products with multiple factors: an even number of negative factors yields a positive product, while an odd number yields a negative product.
This document contains notes on mathematics for 10th class students. It covers topics on real numbers and sets. For real numbers, it defines rational and irrational numbers, discusses properties of rational numbers like terminating vs non-terminating decimals. It also covers laws of logarithms. For sets, it defines basic set theory concepts like sets, subsets, universal set, empty set, union and intersection of sets. It includes examples, oral questions and multiple choice questions related to these topics.
The document discusses addition and subtraction. It defines addition as combining two quantities A and B to obtain a sum S, where A and B are the addends. To add two numbers, one lines them up vertically according to place value and adds the digits from right to left, carrying when necessary. Subtraction is defined as taking away or undoing addition. To subtract, one lines numbers up vertically and subtracts the digits from right to left, borrowing when needed. Real-life word problems involving addition and subtraction are translated into mathematical expressions by clarifying which quantity is being added to or subtracted from the other.
The document discusses complex numbers. It begins by explaining that the equation x^2 = -1 has no real solutions, so an imaginary number i is defined such that i^2 = -1. A complex number is then defined as a number of the form a + bi, where a is the real part and bi is the imaginary part. Rules for adding, subtracting and multiplying complex numbers by treating i as a variable and setting i^2 to -1 are provided. Examples of solving equations and performing operations with complex numbers are given.
This document discusses algebraic expressions and polynomial expressions. It provides examples of algebraic expressions and defines them as formulas constructed with variables and numbers using basic arithmetic operations. Polynomials are defined as expressions of the form anxn + an-1xn-1 + ... + a1x + a0, where the ai's are numbers. The document gives examples of factoring polynomials and evaluating polynomial expressions at given values. It also discusses using factoring to find the roots of polynomial equations.
The document discusses the process of long division for both numbers and polynomials. It demonstrates long division of numbers step-by-step using the example of 78 divided by 2. It then explains that long division of polynomials follows the same process, setting up the division with the numerator polynomial inside and denominator polynomial outside. An example problem divides the polynomial 2x^2 - 3x + 20 by the polynomial x - 4 using the long division process.
1. Natural numbers include counting numbers like 1, 2, 3, and continue indefinitely. Whole numbers include natural numbers plus zero. Integers include whole numbers and their opposites.
2. Rational numbers can be written as a fraction, like 1.5 = 3/2. Irrational numbers cannot be written as a fraction, like π.
3. The four basic operations are addition, subtraction, multiplication, and division. Addition and subtraction follow rules about sign and order. Multiplication and division rules depend on the signs of the factors or dividend and divisor.
32 multiplication and division of decimalsalg1testreview
The document discusses how to multiply multi-digit numbers by treating it as multiple single-digit multiplication problems. It shows working through an example problem step-by-step, multiplying 47 x 685. Each digit is multiplied by the bottom number and the results are placed in columns with carrying as needed. The columns are then added to obtain the final answer. The process is similar for multiplying decimal numbers, ignoring the decimal points during the multiplication and then placing the decimal point in the correct position in the final product.
The document discusses real numbers and their subsets. It defines natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It explains that rational numbers can be expressed as terminating or repeating decimals, while irrational numbers are non-terminating and non-repeating. Examples are provided of different types of numbers. Classification of numbers using Venn diagrams is demonstrated. Rounding and truncating decimals is also covered.
CBSE Class 10 Mathematics Real Numbers Topic
Real Numbers Topics discussed in this document:
Introduction
Rational numbers
Fundamental theorem of Arithmetic
Decimal representation of Rational numbers
Terminating decimal
Non-terminating repeating decimals
Irrational numbers
Surd
General form of a surd
Operations on surds
· Addition and subtraction
· Multiplication of surds
More Topics under Class 10th Real Numbers (CBSE):
Real numbers
Laws of
logarithms
Common and natural logarithms
Visit Edvie.com for more topics
1. The real number system includes natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Rational numbers can be expressed as fractions, while irrational numbers cannot.
2. Key concepts covered include number lines, intervals, unions and intersections of intervals, rules of indices, properties of surds, and laws of logarithms.
3. Examples are provided to illustrate solving equations involving indices, surds, and logarithms through appropriate transformations and applications of properties.
47 operations of 2nd degree expressions and formulasalg1testreview
The document discusses operations involving binomials and trinomials. It defines a binomial as a two-term polynomial of the form ax + b and a trinomial as a three-term polynomial of the form ax2 + bx + c. It states that the product of two binomials is a trinomial that can be found using the FOIL method: multiplying the first, outer, inner, and last terms. The FOIL method is demonstrated through examples multiplying binomial expressions. Expanding products involving negative binomials requires distributing the negative sign before using FOIL.
This document is a lecture on real numbers presented by Ms. Cherry Rose R. Estabillo. It introduces the different subsets of real numbers from natural numbers to irrational numbers. Diagrams are shown to illustrate the relationships between these number sets. Examples are provided to demonstrate classifying numbers within these sets. Properties of real numbers like closure, commutativity, associativity and distributivity are discussed. The order of operations and examples applying it are also covered.
The document discusses the real number system and its subsets. It defines natural numbers, whole numbers, integers, rational numbers, irrational numbers, terminating decimals, and repeating decimals. Rational numbers can be expressed as fractions, while irrational numbers are non-terminating and non-repeating decimals. The real number system is represented by a Venn diagram showing the relationships between its subsets. Examples are provided to demonstrate how to classify numbers as rational or irrational.
Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q is not equal to 0. Rational numbers are closed under addition, subtraction, and multiplication, but not division. They are commutative for addition and multiplication, but not for subtraction or division. Addition is associative for rational numbers, but subtraction is not.
This document provides an introduction to rational numbers. It defines integers, whole numbers, counting numbers, and rational numbers. Rational numbers can be expressed as the quotient of two integers, such as fractions. Some numbers, like pi, cannot be expressed as quotients of integers and are called irrational numbers. The combination of rational and irrational numbers make up the set of real numbers. The document then provides examples and theorems regarding properties of rational numbers, such as Euclid's division algorithm for finding the highest common factor of two numbers and the fundamental theorem of arithmetic regarding unique prime factorizations.
The document discusses the real number system and different types of numbers. It defines natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, imaginary numbers, and complex numbers. It also discusses radicals, including the radicand, index, rational vs. irrational radicals, simplifying radicals using the multiplication property of radicals, and writing radicals in simplest form and mixed radicals as entire radicals.
The document is notes from a class on imaginary numbers. It begins with examples of simplifying expressions with imaginary numbers. It then defines imaginary numbers as solutions to equations where the variable is squared and equals a negative number. Examples are provided to show how to take the square root of a negative number results in an imaginary number. Further examples demonstrate operations with imaginary numbers like addition, subtraction, multiplication and simplification.
For a long time, mathematicians tried unsuccessfully to find a number whose square is negative one. In the 1500s, some work with square roots of negative numbers began again. The first major work occurred in 1545, though the mathematician greatly disliked imaginary numbers. Later, Descartes standardized complex numbers as a + bi, but also doubted their usefulness. Today, complex numbers are used extensively in engineering, physics, and computing.
This document provides an overview of imaginary and complex numbers. It defines imaginary numbers as numbers whose squared value is a real number greater than zero. Complex numbers are defined as numbers of the form a + bi, where a and b are real numbers and i = √-1. Examples are provided of adding, subtracting, multiplying, and dividing complex numbers. Homework is assigned from the textbook on these topics.
3 1 the real line and linear inequalities-xmath123b
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on the real number line, with positive numbers to the right of zero and negative numbers to the left. An inequality relates the positions of two numbers on the real number line, with the number farther to the right said to be greater than the number to its left. The document provides examples of inequalities and how to represent sets of numbers using inequalities, such as all numbers between two values a and b. It also outlines steps for solving inequalities algebraically.
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. An inequality relates the position of two numbers on the real number line, with the number farther to the right said to be greater than the number on the left. Specific inequalities are shown graphically on the number line. The properties of inequalities, such as adding or multiplying the same quantity to both sides preserving the inequality, are also covered.
1 s3 multiplication and division of signed numbersmath123a
The document discusses rules for multiplying signed numbers. It states that to multiply two signed numbers, multiply their absolute values and use rules to determine the sign of the product: two numbers with the same sign yield a positive product, while two numbers with opposite signs yield a negative product. It also discusses that in algebra, operations are often implied rather than written out. The even-odd rule determines the sign of products with multiple factors: an even number of negative factors yields a positive product, while an odd number yields a negative product.
This document contains notes on mathematics for 10th class students. It covers topics on real numbers and sets. For real numbers, it defines rational and irrational numbers, discusses properties of rational numbers like terminating vs non-terminating decimals. It also covers laws of logarithms. For sets, it defines basic set theory concepts like sets, subsets, universal set, empty set, union and intersection of sets. It includes examples, oral questions and multiple choice questions related to these topics.
The document discusses addition and subtraction. It defines addition as combining two quantities A and B to obtain a sum S, where A and B are the addends. To add two numbers, one lines them up vertically according to place value and adds the digits from right to left, carrying when necessary. Subtraction is defined as taking away or undoing addition. To subtract, one lines numbers up vertically and subtracts the digits from right to left, borrowing when needed. Real-life word problems involving addition and subtraction are translated into mathematical expressions by clarifying which quantity is being added to or subtracted from the other.
The document discusses complex numbers. It begins by explaining that the equation x^2 = -1 has no real solutions, so an imaginary number i is defined such that i^2 = -1. A complex number is then defined as a number of the form a + bi, where a is the real part and bi is the imaginary part. Rules for adding, subtracting and multiplying complex numbers by treating i as a variable and setting i^2 to -1 are provided. Examples of solving equations and performing operations with complex numbers are given.
This document discusses algebraic expressions and polynomial expressions. It provides examples of algebraic expressions and defines them as formulas constructed with variables and numbers using basic arithmetic operations. Polynomials are defined as expressions of the form anxn + an-1xn-1 + ... + a1x + a0, where the ai's are numbers. The document gives examples of factoring polynomials and evaluating polynomial expressions at given values. It also discusses using factoring to find the roots of polynomial equations.
The document discusses the process of long division for both numbers and polynomials. It demonstrates long division of numbers step-by-step using the example of 78 divided by 2. It then explains that long division of polynomials follows the same process, setting up the division with the numerator polynomial inside and denominator polynomial outside. An example problem divides the polynomial 2x^2 - 3x + 20 by the polynomial x - 4 using the long division process.
1. Natural numbers include counting numbers like 1, 2, 3, and continue indefinitely. Whole numbers include natural numbers plus zero. Integers include whole numbers and their opposites.
2. Rational numbers can be written as a fraction, like 1.5 = 3/2. Irrational numbers cannot be written as a fraction, like π.
3. The four basic operations are addition, subtraction, multiplication, and division. Addition and subtraction follow rules about sign and order. Multiplication and division rules depend on the signs of the factors or dividend and divisor.
32 multiplication and division of decimalsalg1testreview
The document discusses how to multiply multi-digit numbers by treating it as multiple single-digit multiplication problems. It shows working through an example problem step-by-step, multiplying 47 x 685. Each digit is multiplied by the bottom number and the results are placed in columns with carrying as needed. The columns are then added to obtain the final answer. The process is similar for multiplying decimal numbers, ignoring the decimal points during the multiplication and then placing the decimal point in the correct position in the final product.
The document discusses real numbers and their subsets. It defines natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It explains that rational numbers can be expressed as terminating or repeating decimals, while irrational numbers are non-terminating and non-repeating. Examples are provided of different types of numbers. Classification of numbers using Venn diagrams is demonstrated. Rounding and truncating decimals is also covered.
CBSE Class 10 Mathematics Real Numbers Topic
Real Numbers Topics discussed in this document:
Introduction
Rational numbers
Fundamental theorem of Arithmetic
Decimal representation of Rational numbers
Terminating decimal
Non-terminating repeating decimals
Irrational numbers
Surd
General form of a surd
Operations on surds
· Addition and subtraction
· Multiplication of surds
More Topics under Class 10th Real Numbers (CBSE):
Real numbers
Laws of
logarithms
Common and natural logarithms
Visit Edvie.com for more topics
1. The real number system includes natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Rational numbers can be expressed as fractions, while irrational numbers cannot.
2. Key concepts covered include number lines, intervals, unions and intersections of intervals, rules of indices, properties of surds, and laws of logarithms.
3. Examples are provided to illustrate solving equations involving indices, surds, and logarithms through appropriate transformations and applications of properties.
47 operations of 2nd degree expressions and formulasalg1testreview
The document discusses operations involving binomials and trinomials. It defines a binomial as a two-term polynomial of the form ax + b and a trinomial as a three-term polynomial of the form ax2 + bx + c. It states that the product of two binomials is a trinomial that can be found using the FOIL method: multiplying the first, outer, inner, and last terms. The FOIL method is demonstrated through examples multiplying binomial expressions. Expanding products involving negative binomials requires distributing the negative sign before using FOIL.
This document is a lecture on real numbers presented by Ms. Cherry Rose R. Estabillo. It introduces the different subsets of real numbers from natural numbers to irrational numbers. Diagrams are shown to illustrate the relationships between these number sets. Examples are provided to demonstrate classifying numbers within these sets. Properties of real numbers like closure, commutativity, associativity and distributivity are discussed. The order of operations and examples applying it are also covered.
The document discusses the real number system and its subsets. It defines natural numbers, whole numbers, integers, rational numbers, irrational numbers, terminating decimals, and repeating decimals. Rational numbers can be expressed as fractions, while irrational numbers are non-terminating and non-repeating decimals. The real number system is represented by a Venn diagram showing the relationships between its subsets. Examples are provided to demonstrate how to classify numbers as rational or irrational.
Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q is not equal to 0. Rational numbers are closed under addition, subtraction, and multiplication, but not division. They are commutative for addition and multiplication, but not for subtraction or division. Addition is associative for rational numbers, but subtraction is not.
This document provides an introduction to rational numbers. It defines integers, whole numbers, counting numbers, and rational numbers. Rational numbers can be expressed as the quotient of two integers, such as fractions. Some numbers, like pi, cannot be expressed as quotients of integers and are called irrational numbers. The combination of rational and irrational numbers make up the set of real numbers. The document then provides examples and theorems regarding properties of rational numbers, such as Euclid's division algorithm for finding the highest common factor of two numbers and the fundamental theorem of arithmetic regarding unique prime factorizations.
The document discusses the real number system and different types of numbers. It defines natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, imaginary numbers, and complex numbers. It also discusses radicals, including the radicand, index, rational vs. irrational radicals, simplifying radicals using the multiplication property of radicals, and writing radicals in simplest form and mixed radicals as entire radicals.
The document is notes from a class on imaginary numbers. It begins with examples of simplifying expressions with imaginary numbers. It then defines imaginary numbers as solutions to equations where the variable is squared and equals a negative number. Examples are provided to show how to take the square root of a negative number results in an imaginary number. Further examples demonstrate operations with imaginary numbers like addition, subtraction, multiplication and simplification.
For a long time, mathematicians tried unsuccessfully to find a number whose square is negative one. In the 1500s, some work with square roots of negative numbers began again. The first major work occurred in 1545, though the mathematician greatly disliked imaginary numbers. Later, Descartes standardized complex numbers as a + bi, but also doubted their usefulness. Today, complex numbers are used extensively in engineering, physics, and computing.
This document provides an overview of imaginary and complex numbers. It defines imaginary numbers as numbers whose squared value is a real number greater than zero. Complex numbers are defined as numbers of the form a + bi, where a and b are real numbers and i = √-1. Examples are provided of adding, subtracting, multiplying, and dividing complex numbers. Homework is assigned from the textbook on these topics.
X2 t01 01 arithmetic of complex numbers (2013)Nigel Simmons
The document discusses complex numbers. It begins by using an imaginary number, i, to solve the quadratic equation x2 + 1 = 0, which has no real solutions. It then defines i as the number that satisfies i2 = -1. All complex numbers can be written as z = x + iy, where x is the real part and iy is the imaginary part. Basic operations on complex numbers, such as addition, subtraction, multiplication and division, are discussed. The conjugate of a complex number z, denoted z*, is defined as z* = x - iy. Some key properties of conjugates are also outlined.
Sinbad fell in love with his childhood friend Cosette in high school, but she did not feel the same way. They had an awkward period of time spent together with their families until they both moved on to college and later married other people. Decades later, after being widowed, they reunited and finally found romantic happiness with each other.
The document discusses complex numbers and their definitions. It states that all complex numbers can be written as z = x + iy, where x is the real part and y is the imaginary part. If the real part is 0, the number is pure imaginary, and if the imaginary part is 0, the number is real. Every complex number has a complex conjugate of z = x - iy.
The document discusses complex numbers and their arithmetic operations. It defines the imaginary unit i as the square root of -1. Complex numbers are expressed as a + bi, where a is the real part and b is the imaginary part. The four arithmetic operations of addition, subtraction, multiplication, and division are defined for complex numbers by combining or distributing real and imaginary parts. Complex numbers allow equations like x^2 + 1 = 0 to have solutions and are useful in many mathematical applications.
This document discusses complex numbers in MATLAB. It begins by introducing the complex plane and representing complex numbers in rectangular and polar forms. It then covers converting between the forms, Euler's formula, and basic operations like addition, subtraction, multiplication, division, and exponentiation of complex numbers. The document ends by demonstrating how to work with complex numbers in MATLAB, including entering them in polar/rectangular forms, plotting complex numbers, and performing operations like finding the real/imaginary parts and conjugate.
The document introduces complex numbers and different ways to represent them, including:
1) Imaginary numbers, represented by i, which allows for solutions to equations with "hidden roots". Complex numbers have both a real and imaginary part.
2) Polar form represents complex numbers using modulus (distance from origin) and argument (angle from positive x-axis).
3) Exponential or Euler's form uses modulus and an imaginary exponent to represent complex numbers, where the angle must be in radians.
4) Operations like addition, subtraction, multiplication and division are introduced for complex numbers, along with converting between rectangular, polar and exponential forms.
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1.[1] In this expression, a is the real part and b is the imaginary part of the complex number. If {\displaystyle z=a+bi} {\displaystyle z=a+bi}, then {\displaystyle \Re z=a,\quad \Im z=b.} {\displaystyle \Re z=a,\quad \Im z=b.}
Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers are a field extension of the ordinary real numbers, in order to solve problems that cannot be solved with real numbers alone.
1.3 Complex Numbers, Quadratic Equations In The Complex Number Systemguest620260
1) The document introduces complex numbers as a way to solve equations that involve taking the square root of a negative number.
2) It defines the imaginary unit i as the number such that i^2 = -1, and defines complex numbers as numbers of the form a + bi, where a is the real part and bi is the imaginary part.
3) It provides rules for adding, subtracting, multiplying and dividing complex numbers by treating the real and imaginary parts separately and using properties of i.
Complex numbers are numbers of the form a + bi, where a is the real part and bi is the imaginary part. Complex numbers can be added, subtracted, multiplied, and divided. When multiplying complex numbers, the real parts and imaginary parts are multiplied separately and combined. The conjugate of a + bi is a - bi. When a complex number is multiplied by its conjugate, the result is a real number equal to the modulus (magnitude) of the complex number squared. Complex numbers can also be expressed in polar form as r(cosθ + i sinθ), where r is the modulus and θ is the argument.
The document discusses complex numbers. It begins by defining pure imaginary numbers as any positive real number b multiplied by the imaginary unit i, where i^2 = -1. It then defines i as the square root of -1. The document proceeds to simplify various expressions involving complex numbers. It introduces the concept of a cycle of i where the value repeats every 4 exponents. It defines complex numbers as numbers in the form a + bi, where a and b are real numbers. The document concludes by showing how to add, subtract, and multiply complex numbers by distributing like terms.
This document provides information about quadratic equations, including:
- Methods for solving quadratic equations like factoring, completing the square, and using the quadratic formula.
- Key terms like discriminant and nature of roots. The discriminant determines if the roots are real, equal, or imaginary.
- Examples of solving quadratic equations using different methods and finding related values like discriminant and roots.
1) The document thanks Farooq Sir for providing a wonderful project to work on about quadratics.
2) It was a pleasure and wonderful experience for the author and their team to work on this project.
3) The author thanks all those who helped and motivated them to complete this project.
Osama Tahir's presentation introduces complex numbers. [1] Complex numbers consist of a real and imaginary part and can be written in the form a + bi, where i = -1. [2] Complex numbers were introduced to solve equations like x^2 = -1 that have no real number solutions. [3] Key topics covered include addition, subtraction, multiplication, and division of complex numbers, representing them in polar form using De Moivre's theorem, and applications in fields like electric circuits and root locus analysis.
This document discusses approaches to teaching complex numbers. It describes an axiomatic approach, utilitarian approach, and historical approach. The historical approach builds on prior knowledge of quadratic equations and introduces complex numbers to solve problems like finding the roots of quadratic and cubic equations. The document also covers definitions of complex numbers, addition, subtraction, multiplication, and division of complex numbers. It discusses pedagogical considerations like using multiple representations and building on students' prior knowledge.
The document describes the expansion of sin nθ and cos nθ in powers of sinθ and cosθ using De Moivre's theorem and the binomial theorem. It shows that cos nθ can be expressed as the sum of terms involving nC0cosnθ, nC2cosn-2θsin2θ, etc. and sin nθ can be expressed as the sum of terms involving nC1cosn-1θsinθ, nC3cosn-3θsin3θ, etc. The expansions are obtained by equating the real and imaginary parts of (cosθ + i sinθ)n.
Imaginary numbers are numbers that include a unit of the imaginary number i, which equals the square root of -1. There are two types of numbers, real numbers like integers and rational numbers, and imaginary numbers like 3i. Imaginary numbers are used in calculators by changing the mode to include complex numbers with real and imaginary parts. When working with imaginary numbers, you need to follow rules like not having i in the denominator and using conjugate pairs when dividing. Box diagrams are used to multiply terms with real and imaginary parts, while the quadratic formula can be used to solve quadratic equations that produce imaginary number solutions.
The document discusses the development and properties of complex numbers. Integers were originally used to count whole objects, then fractions were developed to represent portions of wholes. Real numbers were created to represent all numbers that can be written as decimals. However, some equations like x^2=-1 do not have real solutions. To solve these, an imaginary number i=√-1 was defined, where i^2=-1. A complex number is defined as a number of the form a+bi, where a is the real part and bi is the imaginary part. Complex numbers can be added or subtracted by treating i as a variable.
This document introduces complex numbers. It defines the imaginary unit i as the square root of -1, which allows quadratic equations with no real solutions, like x^2=-1, to be solved. Complex numbers have both a real part and an imaginary part in the form a + bi. They can be added, subtracted, multiplied, and divided by distributing terms and using properties of i such as i^2 = -1. Complex numbers are plotted on a plane with real numbers on the x-axis and imaginary numbers on the y-axis.
Calculus introduces complex numbers, which are numbers of the form x + iy, where x and y are real numbers and i = √-1. A complex number has two parts, a real part and an imaginary part. Operations like addition, subtraction, multiplication and division can be performed with complex numbers by treating i as a variable and using the property that i^2 = -1. Complex numbers can be represented graphically on a complex plane with real and imaginary axes.
(1) This document provides an introduction to complex numbers, including: defining complex numbers using i as the square root of -1, addition and multiplication of complex numbers, expressing complex numbers in polar form, and De Moivre's theorem.
(2) De Moivre's theorem states that for a complex number r(cosθ + i sinθ) and integer n, (r(cosθ + i sinθ))n = rn(cos(nθ) + i sin(nθ)). It allows taking complex numbers to any power and finding roots of complex numbers.
(3) The document provides examples of using De Moivre's theorem to find powers and roots of complex numbers in both
This document introduces complex numbers and how to perform basic operations with them. Complex numbers were created to account for negative numbers under a square root. The imaginary unit is i = √-1. Complex numbers are written in the form a + bi. To add or subtract complex numbers, simply add or subtract the real and imaginary parts separately. To multiply complex numbers, use FOIL and note that i^2 = -1. To divide complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator. Complex numbers arise when solving quadratic equations using the quadratic formula.
A combination of a real and an imaginary number in the formparassini
A complex number is a combination of a real number and an imaginary number in the form a + bi, where a and b are real numbers and i is the unit imaginary number which equals the square root of -1. Real numbers include integers and fractions like 1, 12.38, -0.8625, 3/4, and √2, while imaginary numbers square to a negative result. When combined, complex numbers take forms such as 1 + i, 2 - 6i, -5.2i, and 4.
The document provides an overview of complex numbers, including:
1) Complex numbers allow polynomials to always have n roots by defining the imaginary number i as the square root of -1.
2) Complex numbers are expressed as z = x + iy, where x is the real part and y is the imaginary part.
3) Arithmetic with complex numbers follows predictable rules, such as i^2 = -1 and (a + bi)(c + di) = (ac - bd) + (ad + bc)i.
The document appears to be discussing complex numbers in Urdu. It begins by stating that God is extremely merciful and compassionate. It then provides some key points about complex numbers, including:
- Complex numbers can be expressed in the form a + bi, where a and b are real numbers and i represents the imaginary unit.
- Operations like addition, subtraction, multiplication, and division can be performed with complex numbers by following specific rules.
- Complex numbers have properties like closure, commutativity, distributivity, identities, and inverses when performing operations.
- The conjugate of a complex number z = a + bi is a - bi. Conjugates have certain properties when performing operations
This document provides examples and explanations for performing operations with complex numbers. It begins with graphing complex numbers on a complex plane. It then covers determining the absolute value of complex numbers, adding and subtracting complex numbers by combining real and imaginary parts, and multiplying complex numbers using distribution and the property that i^2 = -1. It also discusses evaluating powers of i by expressing them as one of four values: i, -1, -i, or 1. The document aims to teach students how to perform basic operations like addition, subtraction, multiplication, and evaluating powers with complex numbers.
The document discusses complex numbers. It defines a complex number as having the form a + bi, where a is the real part and bi is the imaginary part, with i = √-1. It provides examples of complex numbers and their real and imaginary parts. It also discusses the concept of conjugates of complex numbers. Additionally, it covers the basic arithmetic operations of addition, subtraction, multiplication, and division that can be performed on complex numbers.
MIT Math Syllabus 10-3 Lesson 5: Complex numbersLawrence De Vera
The document discusses complex numbers. It introduces the imaginary unit i, where i2 = -1. A complex number is defined as a number of the form a + bi, where a is the real part and b is the imaginary part. The key operations of addition, subtraction, multiplication and division are defined for complex numbers. It also discusses powers of i, noting that i4 = 1, so in powers of i, the remainder of dividing the exponent by 4 determines the value of in.
Alg II Unit 4-8 Quadratic Equations and Complex Numbersjtentinger
The document discusses complex numbers, including:
- Complex numbers are based on the imaginary unit i, where i^2 = -1.
- Complex numbers can be expressed as a + bi and graphed in the complex plane.
- Operations like addition, subtraction, multiplication, and division can be performed on complex numbers by combining real and imaginary parts.
- Quadratic equations can have complex number solutions. Finding these solutions involves factoring or using the quadratic formula.
This document discusses complex numbers. It defines the imaginary unit i as having the property i^2 = -1. Complex numbers are expressed as a + bi, where a is the real part and bi is the imaginary part. Operations like addition, subtraction, multiplication and division are described for complex numbers. Examples are provided to simplify expressions and perform operations involving complex numbers. Properties of powers of i are also discussed, including the repeating pattern of i^4 = 1. Exercises are assigned for students to practice skills with complex numbers.
The document discusses complex numbers, which are combinations of real and imaginary numbers. Complex numbers can be represented in rectangular form (a + bi) or polar form (r(cosθ + i sinθ)). To multiply complex numbers in rectangular form, use the FOIL method or the formula (a + bi)(c + di) = (ac - bd) + (ad + bc)i. Multiplying complex numbers in polar form involves multiplying the magnitudes and adding the angles. De Moivre's formula generalizes this process to exponents such that (r cis θ)n = rn cis nθ.
The presentation is about Complex numbers: How they originated, what they are and how to do the operations of addition, subtraction, multiplication, and division.
By Dr. Farhana Shaheen
The document introduces imaginary numbers and the imaginary unit i, which represents the square root of -1. Some key points:
- i does not refer to a real number but allows for solutions to equations like x^2 = -1
- Powers of i follow a repeating pattern based on the remainder when dividing the exponent by 4
- Complex numbers consist of real and imaginary parts and can be written as a + bi, where a is real and b * i is imaginary
- Operations on complex numbers follow the same rules as real numbers, but i must be treated as a variable when multiplying terms.
1) The document discusses properties of real numbers including integers, rational numbers, decimals, and fractions. It covers the four fundamental operations on integers - addition, subtraction, multiplication, and division.
2) Key properties of integer addition and subtraction are discussed, including closure, commutativity, associativity, and additive identity. Addition is commutative and associative, while subtraction is not commutative or associative.
3) Examples are provided to illustrate performing the four operations on integers and evaluating expressions involving integers. Rules for multiplying and dividing positive and negative integers are also explained.
The document discusses complex numbers, including: defining complex numbers as numbers that can be written in the form a + bi, where a is the real part and b is the imaginary part; operations like addition, subtraction, and multiplication of complex numbers; complex conjugates; dividing complex numbers; and solving quadratic equations that have complex solutions. It provides examples of working through operations with complex numbers and solving a quadratic equation with complex roots.
The document introduces imaginary numbers and defines i as the square root of -1. It provides examples of how imaginary numbers can be used in equations and expressions. The key properties of imaginary numbers discussed are: i2 = -1; the pattern of powers of i repeating every 4 terms (1, i, -1, -i); and how to multiply and add/subtract complex numbers containing real and imaginary terms.
The document introduces matrices and matrix operations. Matrices are rectangular tables of numbers that are used for applications beyond solving systems of equations. Matrix notation defines a matrix with R rows and C columns as an R x C matrix. The entry in the ith row and jth column is denoted as aij. Matrices can be added or subtracted if they are the same size by adding or subtracting the corresponding entries. There are two types of matrix multiplication: scalar multiplication multiplies a matrix by a constant, and matrix multiplication involves multiplying corresponding rows and columns where the number of columns of the left matrix equals the rows of the right matrix.
35 Special Cases System of Linear Equations-x.pptxmath260
The document discusses special cases of systems of linear equations, including inconsistent/contradictory systems where the equations are impossible to satisfy simultaneously, and dependent systems where there are infinitely many solutions. An inconsistent system is shown with equations x + y = 2 and x + y = 3, which has no solution since they cannot both be true. A dependent system is shown with equations x + y = 2 and 2x + 2y = 4, which has infinitely many solutions like (2,0) and (1,1). The row-reduced echelon form (rref) of a matrix is also discussed, which puts a system of equations in a standard form to help determine if it is consistent, dependent, or has
The document discusses conic sections and ellipses. Conic sections are graphs of quadratic equations of the form Ax2 + By2 + Cx + Dy = E, where A and B are not both 0. Their graphs include circles, ellipses, parabolas and hyperbolas. Ellipses are defined as the set of all points where the sum of the distances to two fixed foci is a constant. Ellipses have a center, two axes called the semi-major and semi-minor axes, and radii along the x and y axes called the x-radius and y-radius. The standard form of an ellipse equation is presented.
This document discusses first degree functions and linear equations. It explains that most real-world mathematical functions can be composed of algebraic, trigonometric, or exponential/log formulas. Linear equations of the form Ax + By = C represent straight lines that can be graphed by finding the x- and y-intercepts. If an equation contains only one variable, it represents a vertical or horizontal line. The slope-intercept form y = mx + b is introduced, where m is the slope and b is the y-intercept. Slope is defined as the ratio of the rise over the run between two points on a line.
The document discusses notation and algebra of functions. It defines a function as a procedure that assigns a unique output to each valid input. Most mathematical functions are represented by formulas like f(x) = x^2 - 2x + 3, where f(x) is the name of the function, x is the input variable, and the formula defines the relationship between input and output. New functions can be formed using basic operations like addition, subtraction, multiplication, and division of existing functions. Examples are provided to demonstrate evaluating functions at given inputs and combining functions algebraically.
The document discusses exponents and exponent rules. It defines exponents as the number of times a base is multiplied by 1. It presents rules for multiplying, dividing, and raising exponents. Examples are provided to demonstrate applying the rules, such as using the power-multiply rule to evaluate (22*34)3. Special exponent rules are also covered, such as the 0-power rule where A0 equals 1 when A is not 0. The document provides examples of calculating fractional exponents by first extracting the root and then raising it to the numerator power.
The document discusses functions and their basic language. It defines a function as a procedure that assigns each input exactly one output. It provides examples of functions, such as a license number to name function. It explains that a function must have a domain (set of inputs) and range (set of outputs). Functions can be represented graphically, through tables of inputs and outputs, or with mathematical formulas.
The document discusses using sign charts to solve polynomial and rational inequalities. It provides examples of solving inequalities by setting one side equal to zero, factoring the expression, drawing the sign chart, and determining the solutions from the regions with the appropriate signs. Specifically, it works through examples of solving x^2 - 3x > 4, 2x^2 - x^3/(x^2 - 2x + 1) < 0, and (x - 2)/(2/(x - 1)) < 3.
The document discusses sign charts for factorable formulas. It provides examples of determining the sign (positive or negative) of expressions when evaluated at given values of x by factoring the expressions into their factored forms. The key steps to create a sign chart are: 1) solve for values where the expression is equal to 0, 2) mark these values on a number line, 3) select points in each segment to test the sign, 4) indicate the sign (positive or negative) in each segment based on the testing. Sign charts show the regions where an expression is positive, negative or equal to 0.
19 more parabolas a& hyperbolas (optional) xmath260
After dividing the general quadratic equation Ax2 + By2 + Cx + Dy = E by A, three types of conic sections can be obtained:
1) Parabolas occur when B = 0, resulting in equations of the form 1x2 + #x + #y = #.
2) Circles occur when A = B, resulting in the equation 1x2 + 1y2 = 1.
3) Hyperbolas occur when A and B have opposite signs, resulting in equations of the form 1x2 + ry2 + #x + #y = # with r < 0. Hyperbolas have two foci and asymptotes, and points on the hyperbola have
The document discusses conic sections, specifically circles and ellipses. It defines an ellipse as the set of points where the sum of the distances to two fixed foci is a constant. An ellipse has a center, two axes (semi-major and semi-minor), and can be represented by the standard form (x-h)2/a2 + (y-k)2/b2 = 1, where (h,k) is the center, a is the x-radius, and b is the y-radius. Examples are provided to demonstrate finding attributes of ellipses from their equations.
This document discusses conic sections and first degree equations. It begins by introducing conic sections as the shapes formed by slicing a cone at different angles. It then covers first degree equations, noting that their graphs are straight lines that can be written in the form of y=mx+b. Specific examples of first degree equations and their graphs are shown. The document ends by introducing the four types of conic sections - circles, ellipses, parabolas, and hyperbolas - and how graphs of second degree equations can represent these shapes.
The document discusses calculating the slope of a curve between two points (x, f(x)) and (x+h, f(x+h)) using the difference quotient formula. It defines the difference quotient as (f(x+h) - f(x))/h, where h is the difference between x and x+h. An example calculates the slope between the points (2, f(2)) and (2.2, f(2.2)) for the function f(x) = x^2 - 2x + 2, finding the slope to be 0.44.
The document discusses various transformations that can be performed on graphs of functions, including vertical translations, stretches, and compressions. Vertical translations move the entire graph up or down by adding or subtracting a constant to the function. Stretches elongate or compress the graph vertically by multiplying the function by a constant greater than or less than 1, respectively. These transformations can be represented by modifying the original function in a way that corresponds to the geometric transformation of its graph.
14 graphs of factorable rational functions xmath260
The document discusses graphs of rational functions. It defines rational functions as functions of the form R(x) = P(x)/Q(x) where P(x) and Q(x) are polynomials. It describes how vertical asymptotes occur where the denominator Q(x) is zero. The graph runs along either side of vertical asymptotes, going up or down depending on the sign chart. There are four cases for how the graph behaves at a vertical asymptote. The document uses examples to illustrate graphing rational functions and determining vertical asymptotes. It also mentions horizontal asymptotes will be discussed.
The document discusses factorable polynomials and graphing them. It defines a factorable polynomial P(x) as one that can be written as the product of linear factors P(x) = an(x - r1)(x - r2)...(x - rk), where r1, r2, etc. are the roots of P(x). It explains that for large values of |x|, the leading term of P(x) dominates so the graph resembles that of the leading term, while near the roots other terms contribute to the shape of the graph. Examples of graphs of polynomials like x^n are provided to illustrate the approach.
The document discusses quadratic functions and parabolas. It defines quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. It states that the graphs of quadratic equations are called parabolas. Parabolas are symmetric around a central line, with the vertex (highest/lowest point) located on this line. The vertex formula is given as x = -b/2a. Steps for graphing a parabola are outlined, including finding the vertex, another point, and reflections across the central line. An example graphs the parabola y = x2 - 4x - 12, finding the vertex as (2, -16) and x-intercepts as -
The document describes the rectangular coordinate system. Each point in a plane can be located using an ordered pair (x,y) where x represents the distance right or left from the origin and y represents the distance up or down. Changing the x-value moves the point right or left, and changing the y-value moves the point up or down. The plane is divided into four quadrants based on the sign of the x and y values. Reflecting a point across an axis results in another point with the same magnitude but opposite sign for the corresponding coordinate.
The document discusses first degree (linear) functions. It states that most real-world mathematical functions can be composed of formulas from three families: algebraic, trigonometric, and exponential-logarithmic. It focuses on linear functions of the form f(x)=mx+b, where m is the slope and b is the y-intercept. Examples are given of equations and how to determine the slope and y-intercept to write the equation in slope-intercept form as a linear function.
The document discusses the basic language of functions. It defines a function as a procedure that assigns each input exactly one output. Functions can be represented by formulas using typical variables like f(x) = x^2 - 2x + 3, where x is the input and f(x) is the output. Functions have a domain, which is the set of all possible inputs, and a range, which is the set of all possible outputs. Functions can be depicted graphically or via tables listing inputs and outputs.
Ivanti’s Patch Tuesday breakdown goes beyond patching your applications and brings you the intelligence and guidance needed to prioritize where to focus your attention first. Catch early analysis on our Ivanti blog, then join industry expert Chris Goettl for the Patch Tuesday Webinar Event. There we’ll do a deep dive into each of the bulletins and give guidance on the risks associated with the newly-identified vulnerabilities.
In the realm of cybersecurity, offensive security practices act as a critical shield. By simulating real-world attacks in a controlled environment, these techniques expose vulnerabilities before malicious actors can exploit them. This proactive approach allows manufacturers to identify and fix weaknesses, significantly enhancing system security.
This presentation delves into the development of a system designed to mimic Galileo's Open Service signal using software-defined radio (SDR) technology. We'll begin with a foundational overview of both Global Navigation Satellite Systems (GNSS) and the intricacies of digital signal processing.
The presentation culminates in a live demonstration. We'll showcase the manipulation of Galileo's Open Service pilot signal, simulating an attack on various software and hardware systems. This practical demonstration serves to highlight the potential consequences of unaddressed vulnerabilities, emphasizing the importance of offensive security practices in safeguarding critical infrastructure.
5th LF Energy Power Grid Model Meet-up SlidesDanBrown980551
5th Power Grid Model Meet-up
It is with great pleasure that we extend to you an invitation to the 5th Power Grid Model Meet-up, scheduled for 6th June 2024. This event will adopt a hybrid format, allowing participants to join us either through an online Mircosoft Teams session or in person at TU/e located at Den Dolech 2, Eindhoven, Netherlands. The meet-up will be hosted by Eindhoven University of Technology (TU/e), a research university specializing in engineering science & technology.
Power Grid Model
The global energy transition is placing new and unprecedented demands on Distribution System Operators (DSOs). Alongside upgrades to grid capacity, processes such as digitization, capacity optimization, and congestion management are becoming vital for delivering reliable services.
Power Grid Model is an open source project from Linux Foundation Energy and provides a calculation engine that is increasingly essential for DSOs. It offers a standards-based foundation enabling real-time power systems analysis, simulations of electrical power grids, and sophisticated what-if analysis. In addition, it enables in-depth studies and analysis of the electrical power grid’s behavior and performance. This comprehensive model incorporates essential factors such as power generation capacity, electrical losses, voltage levels, power flows, and system stability.
Power Grid Model is currently being applied in a wide variety of use cases, including grid planning, expansion, reliability, and congestion studies. It can also help in analyzing the impact of renewable energy integration, assessing the effects of disturbances or faults, and developing strategies for grid control and optimization.
What to expect
For the upcoming meetup we are organizing, we have an exciting lineup of activities planned:
-Insightful presentations covering two practical applications of the Power Grid Model.
-An update on the latest advancements in Power Grid -Model technology during the first and second quarters of 2024.
-An interactive brainstorming session to discuss and propose new feature requests.
-An opportunity to connect with fellow Power Grid Model enthusiasts and users.
Generating privacy-protected synthetic data using Secludy and MilvusZilliz
During this demo, the founders of Secludy will demonstrate how their system utilizes Milvus to store and manipulate embeddings for generating privacy-protected synthetic data. Their approach not only maintains the confidentiality of the original data but also enhances the utility and scalability of LLMs under privacy constraints. Attendees, including machine learning engineers, data scientists, and data managers, will witness first-hand how Secludy's integration with Milvus empowers organizations to harness the power of LLMs securely and efficiently.
What is an RPA CoE? Session 1 – CoE VisionDianaGray10
In the first session, we will review the organization's vision and how this has an impact on the COE Structure.
Topics covered:
• The role of a steering committee
• How do the organization’s priorities determine CoE Structure?
Speaker:
Chris Bolin, Senior Intelligent Automation Architect Anika Systems
Dandelion Hashtable: beyond billion requests per second on a commodity serverAntonios Katsarakis
This slide deck presents DLHT, a concurrent in-memory hashtable. Despite efforts to optimize hashtables, that go as far as sacrificing core functionality, state-of-the-art designs still incur multiple memory accesses per request and block request processing in three cases. First, most hashtables block while waiting for data to be retrieved from memory. Second, open-addressing designs, which represent the current state-of-the-art, either cannot free index slots on deletes or must block all requests to do so. Third, index resizes block every request until all objects are copied to the new index. Defying folklore wisdom, DLHT forgoes open-addressing and adopts a fully-featured and memory-aware closed-addressing design based on bounded cache-line-chaining. This design offers lock-free index operations and deletes that free slots instantly, (2) completes most requests with a single memory access, (3) utilizes software prefetching to hide memory latencies, and (4) employs a novel non-blocking and parallel resizing. In a commodity server and a memory-resident workload, DLHT surpasses 1.6B requests per second and provides 3.5x (12x) the throughput of the state-of-the-art closed-addressing (open-addressing) resizable hashtable on Gets (Deletes).
Programming Foundation Models with DSPy - Meetup SlidesZilliz
Prompting language models is hard, while programming language models is easy. In this talk, I will discuss the state-of-the-art framework DSPy for programming foundation models with its powerful optimizers and runtime constraint system.
How information systems are built or acquired puts information, which is what they should be about, in a secondary place. Our language adapted accordingly, and we no longer talk about information systems but applications. Applications evolved in a way to break data into diverse fragments, tightly coupled with applications and expensive to integrate. The result is technical debt, which is re-paid by taking even bigger "loans", resulting in an ever-increasing technical debt. Software engineering and procurement practices work in sync with market forces to maintain this trend. This talk demonstrates how natural this situation is. The question is: can something be done to reverse the trend?
HCL Notes und Domino Lizenzkostenreduzierung in der Welt von DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-und-domino-lizenzkostenreduzierung-in-der-welt-von-dlau/
DLAU und die Lizenzen nach dem CCB- und CCX-Modell sind für viele in der HCL-Community seit letztem Jahr ein heißes Thema. Als Notes- oder Domino-Kunde haben Sie vielleicht mit unerwartet hohen Benutzerzahlen und Lizenzgebühren zu kämpfen. Sie fragen sich vielleicht, wie diese neue Art der Lizenzierung funktioniert und welchen Nutzen sie Ihnen bringt. Vor allem wollen Sie sicherlich Ihr Budget einhalten und Kosten sparen, wo immer möglich. Das verstehen wir und wir möchten Ihnen dabei helfen!
Wir erklären Ihnen, wie Sie häufige Konfigurationsprobleme lösen können, die dazu führen können, dass mehr Benutzer gezählt werden als nötig, und wie Sie überflüssige oder ungenutzte Konten identifizieren und entfernen können, um Geld zu sparen. Es gibt auch einige Ansätze, die zu unnötigen Ausgaben führen können, z. B. wenn ein Personendokument anstelle eines Mail-Ins für geteilte Mailboxen verwendet wird. Wir zeigen Ihnen solche Fälle und deren Lösungen. Und natürlich erklären wir Ihnen das neue Lizenzmodell.
Nehmen Sie an diesem Webinar teil, bei dem HCL-Ambassador Marc Thomas und Gastredner Franz Walder Ihnen diese neue Welt näherbringen. Es vermittelt Ihnen die Tools und das Know-how, um den Überblick zu bewahren. Sie werden in der Lage sein, Ihre Kosten durch eine optimierte Domino-Konfiguration zu reduzieren und auch in Zukunft gering zu halten.
Diese Themen werden behandelt
- Reduzierung der Lizenzkosten durch Auffinden und Beheben von Fehlkonfigurationen und überflüssigen Konten
- Wie funktionieren CCB- und CCX-Lizenzen wirklich?
- Verstehen des DLAU-Tools und wie man es am besten nutzt
- Tipps für häufige Problembereiche, wie z. B. Team-Postfächer, Funktions-/Testbenutzer usw.
- Praxisbeispiele und Best Practices zum sofortigen Umsetzen
Connector Corner: Seamlessly power UiPath Apps, GenAI with prebuilt connectorsDianaGray10
Join us to learn how UiPath Apps can directly and easily interact with prebuilt connectors via Integration Service--including Salesforce, ServiceNow, Open GenAI, and more.
The best part is you can achieve this without building a custom workflow! Say goodbye to the hassle of using separate automations to call APIs. By seamlessly integrating within App Studio, you can now easily streamline your workflow, while gaining direct access to our Connector Catalog of popular applications.
We’ll discuss and demo the benefits of UiPath Apps and connectors including:
Creating a compelling user experience for any software, without the limitations of APIs.
Accelerating the app creation process, saving time and effort
Enjoying high-performance CRUD (create, read, update, delete) operations, for
seamless data management.
Speakers:
Russell Alfeche, Technology Leader, RPA at qBotic and UiPath MVP
Charlie Greenberg, host
Conversational agents, or chatbots, are increasingly used to access all sorts of services using natural language. While open-domain chatbots - like ChatGPT - can converse on any topic, task-oriented chatbots - the focus of this paper - are designed for specific tasks, like booking a flight, obtaining customer support, or setting an appointment. Like any other software, task-oriented chatbots need to be properly tested, usually by defining and executing test scenarios (i.e., sequences of user-chatbot interactions). However, there is currently a lack of methods to quantify the completeness and strength of such test scenarios, which can lead to low-quality tests, and hence to buggy chatbots.
To fill this gap, we propose adapting mutation testing (MuT) for task-oriented chatbots. To this end, we introduce a set of mutation operators that emulate faults in chatbot designs, an architecture that enables MuT on chatbots built using heterogeneous technologies, and a practical realisation as an Eclipse plugin. Moreover, we evaluate the applicability, effectiveness and efficiency of our approach on open-source chatbots, with promising results.
AppSec PNW: Android and iOS Application Security with MobSFAjin Abraham
Mobile Security Framework - MobSF is a free and open source automated mobile application security testing environment designed to help security engineers, researchers, developers, and penetration testers to identify security vulnerabilities, malicious behaviours and privacy concerns in mobile applications using static and dynamic analysis. It supports all the popular mobile application binaries and source code formats built for Android and iOS devices. In addition to automated security assessment, it also offers an interactive testing environment to build and execute scenario based test/fuzz cases against the application.
This talk covers:
Using MobSF for static analysis of mobile applications.
Interactive dynamic security assessment of Android and iOS applications.
Solving Mobile app CTF challenges.
Reverse engineering and runtime analysis of Mobile malware.
How to shift left and integrate MobSF/mobsfscan SAST and DAST in your build pipeline.
"Frontline Battles with DDoS: Best practices and Lessons Learned", Igor IvaniukFwdays
At this talk we will discuss DDoS protection tools and best practices, discuss network architectures and what AWS has to offer. Also, we will look into one of the largest DDoS attacks on Ukrainian infrastructure that happened in February 2022. We'll see, what techniques helped to keep the web resources available for Ukrainians and how AWS improved DDoS protection for all customers based on Ukraine experience
2. Complex Numbers
Because the square of any real number can't be negative,
the equation x2 = –1 does not have any real solution.
3. Complex Numbers
Because the square of any real number can't be negative,
the equation x2 = –1 does not have any real solution.
We make up a new number called an imaginary number
–1
to be a solution of this equation
4. Complex Numbers
Because the square of any real number can't be negative,
the equation x2 = –1 does not have any real solution.
We make up a new number called an imaginary number
–1 ↔ i
to be a solution of this equation and we name it “ i ”,
5. Complex Numbers
Because the square of any real number can't be negative,
the equation x2 = –1 does not have any real solution.
We make up a new number called an imaginary number
–1 ↔ i
to be a solution of this equation and we name it “ i ”, i.e.
(±i)2 = –1
6. Complex Numbers
Using i, the “solutions” of the equations of the form
Because the square of any real number can't be negative,
the equation x2 = –1 does not have any real solution.
We make up a new number called an imaginary number
–1 ↔ i
to be a solution of this equation and we name it “ i ”, i.e.
(±i)2 = –1
x2 = –r
7. Complex Numbers
Using i, the “solutions” of the equations of the form
Because the square of any real number can't be negative,
the equation x2 = –1 does not have any real solution.
We make up a new number called an imaginary number
–1 ↔ i
to be a solution of this equation and we name it “ i ”, i.e.
(±i)2 = –1
x2 = –r
are x = ± ir
8. Complex Numbers
Using i, the “solutions” of the equations of the form
Example A. Solve x2 + 49 = 0 using imaginary numbers.
Because the square of any real number can't be negative,
the equation x2 = –1 does not have any real solution.
We make up a new number called an imaginary number
–1 ↔ i
to be a solution of this equation and we name it “ i ”, i.e.
(±i)2 = –1
x2 = –r
are x = ± ir
9. Complex Numbers
Using i, the “solutions” of the equations of the form
Example A. Solve x2 + 49 = 0 using imaginary numbers.
Using the square-root method:
x2 + 49 = 0 → x2 = –49
Because the square of any real number can't be negative,
the equation x2 = –1 does not have any real solution.
We make up a new number called an imaginary number
–1 ↔ i
to be a solution of this equation and we name it “ i ”, i.e.
(±i)2 = –1
x2 = –r
are x = ± ir
10. Complex Numbers
Using i, the “solutions” of the equations of the form
Example A. Solve x2 + 49 = 0 using imaginary numbers.
Using the square-root method:
x2 + 49 = 0 → x2 = –49 so
x = ±–49
Because the square of any real number can't be negative,
the equation x2 = –1 does not have any real solution.
We make up a new number called an imaginary number
–1 ↔ i
to be a solution of this equation and we name it “ i ”, i.e.
(±i)2 = –1
x2 = –r
are x = ± ir
11. Complex Numbers
Using i, the “solutions” of the equations of the form
Example A. Solve x2 + 49 = 0 using imaginary numbers.
Using the square-root method:
x2 + 49 = 0 → x2 = –49 so
x = ±–49
x = ±49–1
x = ±7i
Because the square of any real number can't be negative,
the equation x2 = –1 does not have any real solution.
We make up a new number called an imaginary number
–1 ↔ i
to be a solution of this equation and we name it “ i ”, i.e.
(±i)2 = –1
x2 = –r
are x = ± ir
12. A complex number is a number of the form
a + bi
where a and b are real numbers,
Complex Numbers
13. A complex number is a number of the form
a + bi
where a and b are real numbers, a is called the real part
Complex Numbers
14. A complex number is a number of the form
a + bi
where a and b are real numbers, a is called the real part and
bi is called the imaginary part of the complex number.
Complex Numbers
15. A complex number is a number of the form
a + bi
where a and b are real numbers, a is called the real part and
bi is called the imaginary part of the complex number.
Complex Numbers
Example B. 5 – 3i, 6i, –17 are complex numbers.
16. A complex number is a number of the form
a + bi
where a and b are real numbers, a is called the real part and
bi is called the imaginary part of the complex number.
Complex Numbers
Example B. 5 – 3i, 6i, –17 are complex numbers.
The imaginary part of 5 – 3i is –3i.
17. A complex number is a number of the form
a + bi
where a and b are real numbers, a is called the real part and
bi is called the imaginary part of the complex number.
Complex Numbers
Example B. 5 – 3i, 6i, –17 are complex numbers.
The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.
18. A complex number is a number of the form
a + bi
where a and b are real numbers, a is called the real part and
bi is called the imaginary part of the complex number.
Complex Numbers
Example B. 5 – 3i, 6i, –17 are complex numbers.
The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.
Any real number a is also complex because a = a + 0i
hence –17 = –17 + 0i.
19. A complex number is a number of the form
a + bi
where a and b are real numbers, a is called the real part and
bi is called the imaginary part of the complex number
Example B. 5 – 3i, 6i, –17 are complex numbers.
The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.
Any real number a is also complex because a = a + 0i
hence –17 = –17 + 0i.
Complex Numbers
(Addition and subtraction of complex numbers)
Treat the "i" as a variable when adding or subtracting complex
numbers.
20. A complex number is a number of the form
a + bi
where a and b are real numbers, a is called the real part and
bi is called the imaginary part of the complex number
Complex Numbers
(Addition and subtraction of complex numbers)
Treat the "i" as a variable when adding or subtracting complex
numbers.
Example C.
(7 + 4i) + (5 – 3i)
Example B. 5 – 3i, 6i, –17 are complex numbers.
The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.
Any real number a is also complex because a = a + 0i
hence –17 = –17 + 0i.
21. A complex number is a number of the form
a + bi
where a and b are real numbers, a is called the real part and
bi is called the imaginary part of the complex number
Complex Numbers
(Addition and subtraction of complex numbers)
Treat the "i" as a variable when adding or subtracting complex
numbers.
Example C.
(7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i
Example B. 5 – 3i, 6i, –17 are complex numbers.
The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.
Any real number a is also complex because a = a + 0i
hence –17 = –17 + 0i.
22. A complex number is a number of the form
a + bi
where a and b are real numbers, a is called the real part and
bi is called the imaginary part of the complex number
Complex Numbers
(Addition and subtraction of complex numbers)
Treat the "i" as a variable when adding or subtracting complex
numbers.
Example C.
(7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i = 12 + i
Example B. 5 – 3i, 6i, –17 are complex numbers.
The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.
Any real number a is also complex because a = a + 0i
hence –17 = –17 + 0i.
23. A complex number is a number of the form
a + bi
where a and b are real numbers, a is called the real part and
bi is called the imaginary part of the complex number
Complex Numbers
(Addition and subtraction of complex numbers)
Treat the "i" as a variable when adding or subtracting complex
numbers.
Example C.
(7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i = 12 + i
(7 + 4i) – (5 – 3i)
Example B. 5 – 3i, 6i, –17 are complex numbers.
The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.
Any real number a is also complex because a = a + 0i
hence –17 = –17 + 0i.
24. A complex number is a number of the form
a + bi
where a and b are real numbers, a is called the real part and
bi is called the imaginary part of the complex number
Complex Numbers
(Addition and subtraction of complex numbers)
Treat the "i" as a variable when adding or subtracting complex
numbers.
Example C.
(7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i = 12 + i
(7 + 4i) – (5 – 3i) = 7 + 4i – 5 + 3i
Example B. 5 – 3i, 6i, –17 are complex numbers.
The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.
Any real number a is also complex because a = a + 0i
hence –17 = –17 + 0i.
25. A complex number is a number of the form
a + bi
where a and b are real numbers, a is called the real part and
bi is called the imaginary part of the complex number
Complex Numbers
(Addition and subtraction of complex numbers)
Treat the "i" as a variable when adding or subtracting complex
numbers.
Example C.
(7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i = 12 + i
(7 + 4i) – (5 – 3i) = 7 + 4i – 5 + 3i = 2 + 7i
Example B. 5 – 3i, 6i, –17 are complex numbers.
The imaginary part of 5 – 3i is –3i. The real part of 6i is 0.
Any real number a is also complex because a = a + 0i
hence –17 = –17 + 0i.
27. (Multiplication of complex numbers)
To multiply complex numbers, use FOIL, then set i2 to be (-1)
and simplify the result.
Complex Numbers
28. (Multiplication of complex numbers)
To multiply complex numbers, use FOIL, then set i2 to be (-1)
and simplify the result.
Complex Numbers
Example D. (4 – 3i)(2 + 7i)
29. (Multiplication of complex numbers)
To multiply complex numbers, use FOIL, then set i2 to be (-1)
and simplify the result.
Complex Numbers
Example D. (4 – 3i)(2 + 7i) FOIL
= 8 – 6i + 28i – 21i2
30. (Multiplication of complex numbers)
To multiply complex numbers, use FOIL, then set i2 to be (-1)
and simplify the result.
Complex Numbers
Example D. (4 – 3i)(2 + 7i) FOIL
= 8 – 6i + 28i – 21i2 set i2 = (-1)
= 8 – 6i + 28i + 21
31. (Multiplication of complex numbers)
To multiply complex numbers, use FOIL, then set i2 to be (-1)
and simplify the result.
Complex Numbers
Example D. (4 – 3i)(2 + 7i) FOIL
= 8 – 6i + 28i – 21i2 set i2 = (-1)
= 8 – 6i + 28i + 21
= 29 + 22i
32. (Multiplication of complex numbers)
To multiply complex numbers, use FOIL, then set i2 to be (-1)
and simplify the result.
Complex Numbers
Example D. (4 – 3i)(2 + 7i) FOIL
= 8 – 6i + 28i – 21i2 set i2 = (-1)
= 8 – 6i + 28i + 21
= 29 + 22i
The conjugate of (a + bi) is (a – bi) and vice–versa.
33. (Multiplication of complex numbers)
To multiply complex numbers, use FOIL, then set i2 to be (-1)
and simplify the result.
Complex Numbers
Example D. (4 – 3i)(2 + 7i) FOIL
= 8 – 6i + 28i – 21i2 set i2 = (-1)
= 8 – 6i + 28i + 21
= 29 + 22i
The conjugate of (a + bi) is (a – bi) and vice–versa.
The most important complex number multiplication formula is
the product of a pair of conjugate numbers.
34. (Multiplication of complex numbers)
To multiply complex numbers, use FOIL, then set i2 to be (-1)
and simplify the result.
Complex Numbers
Example D. (4 – 3i)(2 + 7i) FOIL
= 8 – 6i + 28i – 21i2 set i2 = (-1)
= 8 – 6i + 28i + 21
= 29 + 22i
(Conjugate Multiplication)
The conjugate of (a + bi) is (a – bi) and vice–versa.
The most important complex number multiplication formula is
the product of a pair of conjugate numbers.
35. (Multiplication of complex numbers)
To multiply complex numbers, use FOIL, then set i2 to be (-1)
and simplify the result.
Complex Numbers
Example D. (4 – 3i)(2 + 7i) FOIL
= 8 – 6i + 28i – 21i2 set i2 = (-1)
= 8 – 6i + 28i + 21
= 29 + 22i
(Conjugate Multiplication) The nonzero conjugate product is
(a + bi)(a – bi) = a2 + b2 which is always positive.
The conjugate of (a + bi) is (a – bi) and vice–versa.
The most important complex number multiplication formula is
the product of a pair of conjugate numbers.
36. (Multiplication of complex numbers)
To multiply complex numbers, use FOIL, then set i2 to be (-1)
and simplify the result.
Complex Numbers
Example D. (4 – 3i)(2 + 7i) FOIL
= 8 – 6i + 28i – 21i2 set i2 = (-1)
= 8 – 6i + 28i + 21
= 29 + 22i
Example E.
(4 – 3i)(4 + 3i)
The conjugate of (a + bi) is (a – bi) and vice–versa.
The most important complex number multiplication formula is
the product of a pair of conjugate numbers.
(Conjugate Multiplication) The nonzero conjugate product is
(a + bi)(a – bi) = a2 + b2 which is always positive.
37. (Multiplication of complex numbers)
To multiply complex numbers, use FOIL, then set i2 to be (-1)
and simplify the result.
Complex Numbers
Example D. (4 – 3i)(2 + 7i) FOIL
= 8 – 6i + 28i – 21i2 set i2 = (-1)
= 8 – 6i + 28i + 21
= 29 + 22i
Example E.
(4 – 3i)(4 + 3i) = 42 + 32 = 25
The conjugate of (a + bi) is (a – bi) and vice–versa.
The most important complex number multiplication formula is
the product of a pair of conjugate numbers.
(Conjugate Multiplication) The nonzero conjugate product is
(a + bi)(a – bi) = a2 + b2 which is always positive.
38. (Multiplication of complex numbers)
To multiply complex numbers, use FOIL, then set i2 to be (-1)
and simplify the result.
Complex Numbers
Example D. (4 – 3i)(2 + 7i) FOIL
= 8 – 6i + 28i – 21i2 set i2 = (-1)
= 8 – 6i + 28i + 21
= 29 + 22i
Example E.
(4 – 3i)(4 + 3i) = 42 + 32 = 25
(5 – 7i)(5 + 7i)
The conjugate of (a + bi) is (a – bi) and vice–versa.
The most important complex number multiplication formula is
the product of a pair of conjugate numbers.
(Conjugate Multiplication) The nonzero conjugate product is
(a + bi)(a – bi) = a2 + b2 which is always positive.
39. (Multiplication of complex numbers)
To multiply complex numbers, use FOIL, then set i2 to be (-1)
and simplify the result.
Complex Numbers
Example D. (4 – 3i)(2 + 7i) FOIL
= 8 – 6i + 28i – 21i2 set i2 = (-1)
= 8 – 6i + 28i + 21
= 29 + 22i
Example E.
(4 – 3i)(4 + 3i) = 42 + 32 = 25
(5 – 7i)(5 + 7i) = (5)2 + 72
The conjugate of (a + bi) is (a – bi) and vice–versa.
The most important complex number multiplication formula is
the product of a pair of conjugate numbers.
(Conjugate Multiplication) The nonzero conjugate product is
(a + bi)(a – bi) = a2 + b2 which is always positive.
40. (Multiplication of complex numbers)
To multiply complex numbers, use FOIL, then set i2 to be (-1)
and simplify the result.
Complex Numbers
Example D. (4 – 3i)(2 + 7i) FOIL
= 8 – 6i + 28i – 21i2 set i2 = (-1)
= 8 – 6i + 28i + 21
= 29 + 22i
Example E.
(4 – 3i)(4 + 3i) = 42 + 32 = 25
(5 – 7i)(5 + 7i) = (5)2 + 72 = 54
The conjugate of (a + bi) is (a – bi) and vice–versa.
The most important complex number multiplication formula is
the product of a pair of conjugate numbers.
(Conjugate Multiplication) The nonzero conjugate product is
(a + bi)(a – bi) = a2 + b2 which is always positive.
42. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
43. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2i
4 + 3i
Example F. Simplify
44. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2i
4 + 3i
Example F. Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i)
(4 + 3i)
45. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2i
4 + 3i
Example F. Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i)
(4 + 3i)
=
(4 – 3i)
(4 – 3i)*
46. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2i
4 + 3i
Example F. Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i)
(4 + 3i)
=
(4 – 3i)
(4 – 3i)* 42 + 32
47. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2iExample F. Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i)
(4 + 3i)
=
(4 – 3i)
(4 – 3i)* 42 + 32 =
25
4 + 3i
48. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2i
4 + 3i
Example F. Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i)
(4 + 3i)
=
(4 – 3i)
(4 – 3i)* 42 + 32 =
25
12 – 8i – 9i + 6i2
49. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2i
4 + 3i
Example F. Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i)
(4 + 3i)
=
(4 – 3i)
(4 – 3i)* 42 + 32 =
25
12 – 8i – 9i + 6i2
–6
50. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2i
4 + 3i
Example F. Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i)
(4 + 3i)
=
(4 – 3i)
(4 – 3i)* 42 + 32 =
25
12 – 8i – 9i + 6i2
–6
6 – 17i
=
25
6
25
17i
–
51. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2i
4 + 3i
Example F. Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i)
(4 + 3i)
=
(4 – 3i)
(4 – 3i)* 42 + 32 =
25
12 – 8i – 9i + 6i2
–6
6 – 17i
=
25
6
25
17i
–
Using the quadratic formula, we can solve all 2nd degree
equations and obtain their complex number solutions.
52. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2i
4 + 3i
Example F. Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i)
(4 + 3i)
=
(4 – 3i)
(4 – 3i)* 42 + 32 =
25
12 – 8i – 9i + 6i2
–6
6 – 17i
=
25
6
25
17i
–
Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.
Using the quadratic formula, we can solve all 2nd degree
equations and obtain their complex number solutions.
53. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2i
4 + 3i
Example F. Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i)
(4 + 3i)
=
(4 – 3i)
(4 – 3i)* 42 + 32 =
25
12 – 8i – 9i + 6i2
–6
6 – 17i
=
25
6
25
17i
–
Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.
To find b2 – 4ac first: a = 2, b = –2, c = 3,
Using the quadratic formula, we can solve all 2nd degree
equations and obtain their complex number solutions.
54. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2i
4 + 3i
Example F. Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i)
(4 + 3i)
=
(4 – 3i)
(4 – 3i)* 42 + 32 =
25
12 – 8i – 9i + 6i2
–6
6 – 17i
=
25
6
25
17i
–
Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.
To find b2 – 4ac first: a = 2, b = –2, c = 3, so b2 – 4ac = –20.
Using the quadratic formula, we can solve all 2nd degree
equations and obtain their complex number solutions.
55. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2i
4 + 3i
Example F. Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i)
(4 + 3i)
=
(4 – 3i)
(4 – 3i)* 42 + 32 =
25
12 – 8i – 9i + 6i2
–6
6 – 17i
=
25
6
25
17i
–
Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.
To find b2 – 4ac first: a = 2, b = –2, c = 3, so b2 – 4ac = –20.
x =
2 ± –20
4
Using the quadratic formula, we can solve all 2nd degree
equations and obtain their complex number solutions.
Hence
56. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2i
4 + 3i
Example F. Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i)
(4 + 3i)
=
(4 – 3i)
(4 – 3i)* 42 + 32 =
25
12 – 8i – 9i + 6i2
–6
6 – 17i
=
25
6
25
17i
–
Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.
To find b2 – 4ac first: a = 2, b = –2, c = 3, so b2 – 4ac = –20.
x =
2 ± –20
4 =
2 ± 2–5
4
Using the quadratic formula, we can solve all 2nd degree
equations and obtain their complex number solutions.
Hence
57. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2i
4 + 3i
Example F. Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i)
(4 + 3i)
=
(4 – 3i)
(4 – 3i)* 42 + 32 =
25
12 – 8i – 9i + 6i2
–6
6 – 17i
=
25
6
25
17i
–
Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.
To find b2 – 4ac first: a = 2, b = –2, c = 3, so b2 – 4ac = –20.
x =
2 ± –20
4 =
2 ± 2–5
4 =
2(1 ± i5)
4 =
1 ± i5
2
Using the quadratic formula, we can solve all 2nd degree
equations and obtain their complex number solutions.
Hence
59. The powers of i go in a cycle as shown below:
i
-1 = i2
Powers of i
60. The powers of i go in a cycle as shown below:
i
-1 = i2
-i = i3
Powers of i
61. The powers of i go in a cycle as shown below:
i
-1 = i2
-i = i3
1 = i4
Powers of i
62. The powers of i go in a cycle as shown below:
i = i5
-1 = i2
-i = i3
1 = i4
Powers of i
63. Complex Numbers
Exercise D. Divide by rationalizing the denominators.
2 + 3i
i24.
3 – 4i
i25.
3 – 4i
i26.
1 + i
1 – i27. 2 – i
3 – i28. 3 – 2i
2 + i29.
2 + 3i
2 – 3i
30.
3 – 4i
3 – 2i
31.
3 – 4i
2 + 5i
32.
33. Is there a difference between √4i and 2i?
64. The powers of i go in a cycle as shown below:
i = i5
-1 = i2 = i6 ..
-i = i3
1 = i4
Powers of i
65. The powers of i go in a cycle as shown below:
i = i5
-1 = i2 = i6 ..
-i = i3 = i7 ..
1 = i4
Powers of i
66. The powers of i go in a cycle as shown below:
i = i5
-1 = i2 = i6 ..
-i = i3 = i7 ..
1 = i4 = i8 ..
Powers of i
67. The powers of i go in a cycle as shown below:
i = i5 = i9 ..
-1 = i2 = i6 ..
-i = i3 = i7 ..
1 = i4 = i8 ..
Powers of i
68. The powers of i go in a cycle as shown below:
i = i5 = i9 ..
-1 = i2 = i6 ..
-i = i3 = i7 ..
1 = i4 = i8 ..
Example H. Simplify i59
Powers of i
69. The powers of i go in a cycle as shown below:
i = i5 = i9 ..
-1 = i2 = i6 ..
-i = i3 = i7 ..
1 = i4 = i8 ..
Example H. Simplify i59
59 = 4*14 + 3,
Powers of i
70. The powers of i go in a cycle as shown below:
i = i5 = i9 ..
-1 = i2 = i6 ..
-i = i3 = i7 ..
1 = i4 = i8 ..
Example H. Simplify i59
59 = 4*14 + 3,
hence i59 = i4*14+3
Powers of i
71. The powers of i go in a cycle as shown below:
i = i5 = i9 ..
-1 = i2 = i6 ..
-i = i3 = i7 ..
1 = i4 = i8 ..
Example H. Simplify i59
59 = 4*14 + 3,
hence i59 = i4*14+3 = i4*14+3
Powers of i
72. The powers of i go in a cycle as shown below:
i = i5 = i9 ..
-1 = i2 = i6 ..
-i = i3 = i7 ..
1 = i4 = i8 ..
Example H. Simplify i59
59 = 4*14 + 3,
hence i59 = i4*14+3 = i4*14+3 = (i4)14 i3
Powers of i
73. The powers of i go in a cycle as shown below:
i = i5 = i9 ..
-1 = i2 = i6 ..
-i = i3 = i7 ..
1 = i4 = i8 ..
Example H. Simplify i59
59 = 4*14 + 3,
hence i59 = i4*14+3 = i4*14+3 = (i4)14 i3 = 114 i3
Powers of i
74. The powers of i go in a cycle as shown below:
i = i5 = i9 ..
-1 = i2 = i6 ..
-i = i3 = i7 ..
1 = i4 = i8 ..
Example H. Simplify i59
59 = 4*14 + 3,
hence i59 = i4*14+3 = i4*14+3 = (i4)14 i3 = 114 i3 = i3 = -i
Powers of i
75. Quadratic Formula and Complex Numbers
and
From example G, the solutions of 2x2 – 2x + 3 = 0 are
x =
1 + i5
2
x = 1 – i5
2
because b2 – 4ac = –20 < 0.
76. Quadratic Formula and Complex Numbers
and
From example G, the solutions of 2x2 – 2x + 3 = 0 are
x =
1 + i5
2
x = 1 – i5
2
because b2 – 4ac = –20 < 0.
Therefore we have a complex conjugate pair as solutions.
77. Quadratic Formula and Complex Numbers
and
From example G, the solutions of 2x2 – 2x + 3 = 0 are
x =
1 + i5
2
x = 1 – i5
2
In general, for ax2 + bx + c = 0 with a, b, and c real numbers,
and b2 – 4ac < 0,
because b2 – 4ac = –20 < 0.
Therefore we have a complex conjugate pair as solutions.
78. Quadratic Formula and Complex Numbers
x =
–b +b2 – 4ac
2a
x =
–b –b2 – 4ac
2a
and
From example G, the solutions of 2x2 – 2x + 3 = 0 are
x =
1 + i5
2
x = 1 – i5
2
and
In general, for ax2 + bx + c = 0 with a, b, and c real numbers,
and b2 – 4ac < 0, then its two roots:
because b2 – 4ac = –20 < 0.
are of the form A + Bi and A – Bi, a conjugate pair.
Therefore we have a complex conjugate pair as solutions.
79. Quadratic Formula and Complex Numbers
x =
–b +b2 – 4ac
2a
x =
–b –b2 – 4ac
2a
and
If a, b, and c are real numbers, then the complex roots* for
ax2 + bx + c = 0
are a complex conjugates pair. ( * if b2 – 4ac < 0).
From example G, the solutions of 2x2 – 2x + 3 = 0 are
x =
1 + i5
2
x = 1 – i5
2
and
In general, for ax2 + bx + c = 0 with a, b, and c real numbers,
and b2 – 4ac < 0, then its two roots:
because b2 – 4ac = –20 < 0.
are of the form A + Bi and A – Bi, a conjugate pair.
Therefore we have a complex conjugate pair as solutions.
80. Quadratic Formula and Complex Numbers
x =
–b +b2 – 4ac
2a
x =
–b –b2 – 4ac
2a
and
If a, b, and c are real numbers, then the complex roots* for
ax2 + bx + c = 0
are a complex conjugates pair. ( * if b2 – 4ac < 0).
From example G, the solutions of 2x2 – 2x + 3 = 0 are
x =
1 + i5
2
For example, if x = i is a solution of #x2 + # x + # = 0,
then automatically x = – i is the other root (# real numbers.)
x = 1 – i5
2
and
In general, for ax2 + bx + c = 0 with a, b, and c real numbers,
and b2 – 4ac < 0, then its two roots:
because b2 – 4ac = –20 < 0.
are of the form A + Bi and A – Bi, a conjugate pair.
Therefore we have a complex conjugate pair as solutions.
81. Complex Numbers
In what sense are the complex numbers, numbers?
Real numbers are physically measurable quantities (or the
lack of such quantities in the case of the negative numbers).
Theoretically, we can forever improve upon the construction of
a stick with length exactly 2. But how do we make a stick of
length 3i, or a cookie that weighs 3i oz? Well, we can’t.
Imaginary numbers and complex numbers in general are not
physically measurable in the traditional sense. Only the real
numbers, which are a part of the complex numbers, are
tangible in the traditional sense.
Complex numbers are directional measurements.
They keep track of measurements and directions,
i.e. how much and in what direction (hence the two–
component form of the complex numbers).
Google the terms “complex numbers, 2D vectors” for further
information.
83. Complex Numbers
Exercise D. Divide by rationalizing the denominators.
2 + 3i
i24.
3 – 4i
i25.
3 + 4i
i26.
1 + i
1 – i27. 2 – i
3 – i28. 3 – 2i
2 + i29.
2 + 3i
2 – 3i
30.
3 – 4i
3 – 2i
31.
3 – 4i
2 + 5i
32.
Simplify
33. i92
38. Find a and b if (a + bi) 2 = i.
34. i –25 36. i 205
37. i –102
39. Is there a difference between √4i and 2i?
84. Complex Numbers
(Answers to odd problems) Exercise A.
1. 5 3. 7 + (2 – √5) i
Exercise B.
9. 10 11. 5
Exercise C.
17. 11 – 3i 19. – 25i
5. –5 – i 7. –1/4 + 5/3 i
13. 54 15. 72
21. – 2i 23. 21 + 20i
Exercise D.
– 4 – 3i25. 27.
33. 1
i 29.
1
5
(4 – 7i) 31.
1
13
(17 – 6i)
37. – 1 39. There is no difference