Imaginary numbers
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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.
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Imaginary numbers
Imaginary numbers arise when a number is squared to produce a negative result. They are represented using the symbol (i), such that √-9 = 3i. Imaginary numbers can be used to solve equations involving negative results by including (i) in the calculation. Examples of imaginary numbers include √-9 = 3i, √-25 = 5i, (3+2i)(3-2i) = 13, and (3+2i) + (5-6i) = -5-12i.
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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.
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The document discusses imaginary numbers. Imaginary numbers are represented by the symbol i, where i = √-1. Standard form for imaginary numbers is a + bi. Rules for adding, subtracting, multiplying and dividing imaginary numbers are provided. When dividing, terms cannot have i in the denominator, and the expression must be multiplied by the same term with the opposite sign. The quadratic formula can produce imaginary number solutions when the expression under the square root is negative.
Imaginary numbers ppt.
The document discusses imaginary numbers. Imaginary numbers are represented by the symbol i, where i = √-1. Standard form for imaginary numbers is a + bi. Rules for adding, subtracting, multiplying, and dividing imaginary numbers are provided. When dividing, terms cannot have i in the denominator, and the expression must be multiplied by the same term with the opposite sign. The quadratic formula can produce imaginary number solutions when the expression under the square root is negative.
PEMDAS - The Proper Order of Mathematical Operations
The document discusses the proper order of mathematical operations known as PEMDAS. It covers absolute values, addition and subtraction of signed numbers, multiplication and division of signed numbers, and the order of operations using PEMDAS. Examples are provided to illustrate how to use PEMDAS to simplify expressions involving multiple operations. Exercises with answers are included to help readers practice applying these concepts.
Algebra Notes Factor and Simplify Quadratics
This document provides instructions and examples for solving "riddles" involving factoring quadratic expressions. It explains that the riddles involve two numbers whose product and sum are given. There are four possible combinations of signs for the product and sum. Examples are worked through step-by-step, showing how to determine the signs of the two numbers based on the information given, find the factors of the product, and write the factored quadratic expression that represents the solution. The goal is to help students recognize different patterns that arise when factoring quadratics and factor expressions into standard form.
Sept. 21, 2012
The document provides information about absolute values and the real number system. It includes:
- Definitions of rational numbers, integers, whole numbers, natural numbers, and irrational numbers.
- Examples of absolute value and how it measures the distance from zero, including that operations inside the absolute value signs must be done first before taking the absolute value.
- Class work assignments on opposites and absolute values problems from pages 17-18 including a mixed review.
Operations on Real Numbers
Real numbers include both positive and negative numbers. Operations can be performed on real numbers by following rules:
When adding like signs, add the absolute values and use the original sign. With different signs, find the difference of absolute values and use the greater sign.
For multiplication and division, a positive result occurs with same signs and negative result with different signs.
When raising a negative number to a power, the result is negative for odd exponents and positive for even exponents.
15minute-math-integers.ppt
This presentation reviews math skills that can help with everyday problems. It is divided into lessons that each take approximately 15 minutes, so the entire presentation can be completed in under 3 weeks. The lessons cover topics like decimals, fractions, percentages, integers, algebra, and order of operations. Looking for hints from the Professor can make working through the math problems faster and easier.
15minute-math-integers.ppt
This presentation reviews math skills that can help with everyday problems. It is divided into lessons that each take approximately 15 minutes, so the entire presentation can be completed in under 3 weeks. The lessons cover topics like decimals, fractions, percentages, integers, algebra, and order of operations. Looking for hints from the Professor can make working through the math problems faster and easier.
Synthetic Division
Synthetic division is a method for dividing polynomials that can be used when the divisor is of the form x - r or x + r, where r is a constant. It involves setting up a table with boxes and lines and systematically filling in numbers from the dividend polynomial and performing operations to arrive at the coefficients of the quotient and the remainder. The resulting expression provides the quotient polynomial and remainder over the divisor from the original problem.
ppt math6 qt3-wk3.pptx
The document provides instructions on subtracting integers. It explains:
1) To subtract integers, transform the subtraction into addition by keeping the first number and changing the second number's sign.
2) Examples are provided of subtracting integers with different signs and the same sign.
3) A multi-step word problem is worked out as an example of subtracting integers.
Integers
This document provides an overview of integers and operations on integers. It defines integers as natural numbers, 0, and negatives of counting numbers. It then discusses the properties of addition, subtraction, multiplication and division of integers, including:
- Adding two integers of the same sign by adding their values, and of opposite signs by taking the difference of their absolute values.
- Subtracting integers by adding the number to the negative of the number being subtracted.
- Multiplying integers of the same sign by multiplying their values, and of opposite signs by multiplying their values and assigning a negative sign to the product.
- Dividing integers of the same sign by dividing their values, and of opposite signs by dividing their values
W1-L1 Negative-numbers-ppt..pptx
This document provides an introduction to negative numbers for students in Year 8 maths. It includes examples of where negative numbers are used, such as temperatures below zero and bank account balances in debt. Students are introduced to concepts like adding and subtracting negative numbers using a number line. Rules for multiplying positive and negative numbers are explained, such as a positive times a negative equals a negative, and a negative times a negative equals a positive. Students are provided practice problems to solve involving addition, subtraction, and multiplication of negative numbers.
1.3 Complex Numbers
This document discusses complex numbers. It defines the imaginary unit i as the number whose square is -1. Complex numbers are expressed as a + bi, where a is the real part and bi is the imaginary part. All real number properties extend to complex numbers. Operations like addition, subtraction, multiplication, division and powers of i are demonstrated. Complex conjugates are used to simplify expressions and divide complex numbers. Examples are provided to simplify expressions and perform operations on complex numbers.
Introduction to imaginary numbers.ppt
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.
A combination of a real and an imaginary number in the form
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.
Thursday, september 26, 2013
The document provides instructions and examples for solving integer equations and working with absolute values. It discusses key concepts like:
- The rules for adding, subtracting, multiplying and dividing integers
- Classifying numbers as natural, whole, integer, rational or real
- Understanding that the absolute value of a number represents its distance from zero
- Solving problems involving absolute values and performing operations inside and outside of absolute value signs
Introduction Combined Number And Dp
This document provides information about a mathematics module taught at the foundation level. It includes details such as the module title, code, credit hours, teaching periods, level, learning objectives, assessment types and course content. The module aims to reinforce basic numeracy and algebraic manipulation through lectures, seminars and tutorials. Students will be assessed through classroom tests, examinations and coursework assignments. The course content will cover topics such as numbers, operations, place value, and classifications of real numbers.
Lesson5.1 complexnumbers
This document provides an introduction to complex numbers. It defines the imaginary number i and shows that i raised to even powers is 1 and raised to odd powers alternates between i and -i. Complex numbers combine real and imaginary parts in the form a + bi. The document demonstrates operations like addition, subtraction, multiplication and division on complex numbers. It warns that the properties of radicals do not directly apply when combining real and imaginary parts, and instead those expressions should be written using i. The final sections demonstrate working with complex numbers on a calculator.
Similar to Imaginary numbers (20)
PEMDAS - The Proper Order of Mathematical Operations
PEMDAS - The Proper Order of Mathematical Operations
A combination of a real and an imaginary number in the form
A combination of a real and an imaginary number in the form
Imaginary numbers
- 1. Imaginary Numbers By; Samantha Showalter
- 2. Two types of Numbers There are two types of numbers real numbers and imaginary numbers. Real numbers are numbers such as 1, -2, -1/2, √2, etc. Imaginary numbers are √negatives, √-1= I
- 3. Example of imaginary number √-9 = 3i √-25 = 5i
- 4. Calculator On your calculator click mode, then go down to where it says real and go over to a+bi and select it. Also to put an I on your calculator you must hit 2nd and the decimal sign.
- 5. Example problems (3+2i) + (5-6i)= 8-4i (-2-5i) – (3+7i)= -5-12i Those problems you can put in your calculator and they must have parenthesis Rule: you cannot have any thing over i^1=I Ex: i^2=-1 because nothing over i^1=I
- 6. Rule of I This pattern continues on forever.
- 7. Box problems You will need to use a box problem to work out multiplication of imaginary numbers. For problems that look like this: (3-2i)(5+9i) (4-3i)(6-4i)
- 8. Division RULE: you cannot have an I in the denominator (bottom)/ You have to use the conjugate pair.
- 9. Example (5-3i)(-3-5i) You would have to multiply both the top and the bottom by the conjugate pair of the denominator.
- 10. Quadratic formula You will plug in the numbers from the problem into the quadratic formula. And solve it like you would when doing a quadratic equation
- 11. Examples 4x^2+7x+35=) A=4B=7C=35 Plug into the quadratic formula you get: X=-(7)+or-√(7)^2-4(4)(35) X=-7+or-√-511/8 Plug the square root into the calculator.
- 12. Example cont. You will get X=-7+or- 22.6i/8
- 13. The end