What is complex number
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This document discusses the history and properties of complex numbers. It begins by outlining the key topics to be covered: history, number systems, complex numbers themselves, and operations. It then provides details on who originally introduced complex numbers and the symbols used. Complex numbers combine real and imaginary numbers and can be expressed as a + bi, where a is the real part and b is the imaginary part. The document also describes how complex numbers extend the number line to the complex plane and defines the basic operations of addition, subtraction, multiplication, and division of complex numbers.
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Complex number
This document contains information about Md. Arifuzzaman, a lecturer in the Department of Natural Sciences at the Faculty of Science and Information Technology, Daffodil International University. It includes his employee ID, designation, department, faculty, personal webpage, email, and phone number. The document also provides an overview of complex numbers, including their history, the number system, definitions of complex numbers, operations like addition and multiplication of complex numbers, and applications of complex numbers.
Complex number
This document provides an overview of complex numbers. It defines complex numbers as numbers consisting of a real part and imaginary part written in the form a + bi. It discusses the subsets of complex numbers including real and imaginary numbers. It also covers topics such as the complex conjugate, modulus, addition, subtraction, multiplication, and division of complex numbers. Finally, it mentions applications of complex numbers in science, mathematics, engineering, and statistics.
1 complex numbers
This document discusses complex numbers including:
1. Defining complex numbers and their algebraic properties such as addition, subtraction, multiplication and division.
2. Geometrically representing complex numbers in Cartesian and polar forms.
3. Key concepts such as the absolute value, distance between complex numbers, and the interpretation of multiplication in polar form.
4. De Moivre's theorem and its expansion along with examples of evaluating complex numbers and finding roots of complex numbers using this theorem.
5. Exponential and logarithmic forms of representing complex numbers.
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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.
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The document discusses complex numbers and their properties. It defines i as the square root of -1 and shows that complex numbers can be written in the form a + bi, where a and b are real numbers. It describes how to add, subtract, multiply and divide complex numbers. It also discusses conjugates, moduli, and solving quadratic equations with complex number solutions.
Basics of Integration and Derivatives
Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
Imaginary numbers
Imaginary numbers are square roots of negative real numbers that were introduced to allow taking the square root of negatives. An example of an imaginary number is 2i√6, where i represents the imaginary unit. Imaginary numbers can be added, subtracted, multiplied, and divided like real numbers. When squared, imaginary numbers follow a repeating sequence of i, -1, -i, 1. Though sometimes complex, imaginary numbers serve an important purpose and are generally easy to work with through practice and logical thinking.
Complex Numbers
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.
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Complex number
This document contains information about Md. Arifuzzaman, a lecturer in the Department of Natural Sciences at the Faculty of Science and Information Technology, Daffodil International University. It includes his employee ID, designation, department, faculty, personal webpage, email, and phone number. The document also provides an overview of complex numbers, including their history, the number system, definitions of complex numbers, operations like addition and multiplication of complex numbers, and applications of complex numbers.
Complex number
This document provides an overview of complex numbers. It defines complex numbers as numbers consisting of a real part and imaginary part written in the form a + bi. It discusses the subsets of complex numbers including real and imaginary numbers. It also covers topics such as the complex conjugate, modulus, addition, subtraction, multiplication, and division of complex numbers. Finally, it mentions applications of complex numbers in science, mathematics, engineering, and statistics.
1 complex numbers
This document discusses complex numbers including:
1. Defining complex numbers and their algebraic properties such as addition, subtraction, multiplication and division.
2. Geometrically representing complex numbers in Cartesian and polar forms.
3. Key concepts such as the absolute value, distance between complex numbers, and the interpretation of multiplication in polar form.
4. De Moivre's theorem and its expansion along with examples of evaluating complex numbers and finding roots of complex numbers using this theorem.
5. Exponential and logarithmic forms of representing complex numbers.
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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.
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The document discusses complex numbers and their properties. It defines i as the square root of -1 and shows that complex numbers can be written in the form a + bi, where a and b are real numbers. It describes how to add, subtract, multiply and divide complex numbers. It also discusses conjugates, moduli, and solving quadratic equations with complex number solutions.
Basics of Integration and Derivatives
Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
Imaginary numbers
Imaginary numbers are square roots of negative real numbers that were introduced to allow taking the square root of negatives. An example of an imaginary number is 2i√6, where i represents the imaginary unit. Imaginary numbers can be added, subtracted, multiplied, and divided like real numbers. When squared, imaginary numbers follow a repeating sequence of i, -1, -i, 1. Though sometimes complex, imaginary numbers serve an important purpose and are generally easy to work with through practice and logical thinking.
Complex Numbers
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.
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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.
complex numbers
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.
complex numbers
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.
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complex number
1. A complex number is an ordered pair of real numbers written in the form a + ib, where a is the real part and b is the imaginary part.
2. Complex numbers combine real and imaginary numbers, extending the real number line to the two-dimensional complex plane with the horizontal axis for real parts and vertical axis for imaginary parts.
3. The fundamental operations of addition, subtraction, multiplication, and division are defined for complex numbers by applying the operations to the real and imaginary parts separately.
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This Our presentation about complex number.
This is very easy and you can explore it to all within a short time .
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iTutor provides information on complex numbers. Complex numbers consist of real and imaginary parts and can be written as a + bi, where a is the real part and b is the imaginary part. The imaginary unit i = √-1. Properties of complex numbers include: the square of i is -1; complex conjugates are obtained by changing the sign of the imaginary part; and the basic arithmetic operations of addition, subtraction, and multiplication follow predictable rules when applied to complex numbers. Complex numbers allow representing solutions, like the square root of a negative number, that are not possible with real numbers alone.
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This document discusses rational and irrational numbers. It defines rational numbers as numbers that can be written as p/q where p and q are integers and q is not equal to 0. Rational numbers include fractions, integers, and natural numbers. The document describes the different types of rational numbers such as positive, negative, and in standard form. It also discusses how to perform operations like addition, subtraction, multiplication, and division on rational numbers. Irrational numbers are defined as real numbers that cannot be expressed as a ratio of integers like the roots of prime numbers or pi.
Complex numbers org.ppt
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.
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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.
complex numbers
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.
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This document provides information about the Engineering Mathematics 1 course taught by Dr. Ir. Arman Djohan Diponegoro. It includes details about the class code, credits, schedule, topics to be covered including complex variables and vector analysis. It also outlines the grading breakdown and test schedule. Several concepts from complex numbers are then defined and explained, such as the complex plane, polar coordinates, trigonometric form, and operations like addition, subtraction and multiplication of complex numbers. Examples are provided to demonstrate converting between rectangular and trigonometric forms.
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complex number
1. A complex number is an ordered pair of real numbers written in the form a + ib, where a is the real part and b is the imaginary part.
2. Complex numbers combine real and imaginary numbers, extending the real number line to the two-dimensional complex plane with the horizontal axis for real parts and vertical axis for imaginary parts.
3. The fundamental operations of addition, subtraction, multiplication, and division are defined for complex numbers by applying the operations to the real and imaginary parts separately.
Complex Number Updated
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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
Complex Analysis
Complex analysis and its application
2.Contents,Complex number
Different forms of complex number
Types of complex number
Argand Diagram
Addition, subtraction, Multiplication & Division
Conjugate of Complex number
Complex variable
Function of complex variable
Continuity
Differentiability
Analytic Function
Harmonic Function
Application of complex Function
3.Complex Number,For most human tasks, real numbers (or even rational numbers) offer an adequate description of data. Fractions such as 2/3 and 1/8 are meaningless to a person counting stones, but essential to a person comparing the sizes of different collections of stones. Negative numbers such as -3 and-5 are meaningless when measuring the mass of an object, but essential when keeping track of monetary debits and credits.
All numbers are imaginary (even "zero“ was contentious once). Introducing the square root(s) of minus one is convenient because
all n-degree polynomials with real coefficients then haven roots, making algebra "complete";
it saves using matrix representations for objects that square to-1 (such objects representing an important part of the structure of linear equations which appear in quantum mechanics ,heat,diffusion,optics,etc) .The hottest contenders for numbers without purpose are probably the p-adic numbers (an extension of the rationales),and perhaps the expiry dates on army ration packs.
4.Complex Number is defined as an ordered pair of real number X & Y and is denoted by (X,Y)
It is also written as 𝒛=𝒙,𝒚=𝒙+𝒊𝒚,where 𝑖^2=−1
𝑥 is called Real Part of z and written as Re(z)
Y is called imaginary part of z and written as Im(z).
-If R(z) = 0 then 𝑧=𝑖𝑦, is called Purely Imaginary Number.
-If I(z) = 0 then 𝑧=𝑥, is called Purely Real Number.
-Here 𝑖can be written as (0, 1) = 0 ±1𝑖
Note:-−𝒂= 𝑎−1=𝑖𝑎
-If 𝑧=𝑥+𝑖𝑦is complex number then its conjugate or complex conjugate is defined as 𝒛=𝒙−𝒊𝒚.
5.DIFFERENT FORMS OF COMPLEX NUMBER
Cartesian or Rectangular Form :-𝑧=𝑥+𝑖𝑦
Polar Form :-𝑧=𝑟(cos𝜃+𝑖sin𝜃) 𝑜𝑟 𝑧=𝑟∠𝜃
Exponential Form :-𝑧=𝑟𝑒^𝑖𝜃
MODULUS & ARGUMENT OF COMPLEX NUMBER
Modulus of complex number (|z|) OR mod(z) OR 𝑟=√(𝑋^2+𝑌^2 )
Argument OR Amplitude of complex number (𝜃) OR arg (𝑧) OR amp(z)=tan^(−1)(𝑥/𝑦)
6.Argand Diagram
Mathematician Argand represent a complex number in a diagram known as Argand diagram. A complex number x+iy can be represented by a point P whose co–ordinate are (x,y).The axis of x is called the real axis and the axis of y the imaginary axis. The distance OP is the modulus and the angle, OP makes with the x-axis, is the argument of x+iy.
7.Addition of Complex Numbers
Let a+ib and c+id be two numbers, then
(a+ib)+(c+id)=(a+c)+i(b+d)
Procedure: In addition of complex numbers we add real parts with real parts and imaginary parts with imaginary parts.
8.Subtraction of Complex Numbers
Let a+ib and c+id be two numbers, then
(a+ib)-(c+id)=(a-c)+i(b-d)
Procedure: In subtraction of complex numbers we subtract real parts w
Complex nos demo 2
(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.
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5.4 Complex Numbers
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.
Lesson 1 imaginary and complex numbers p1 8
1. Imaginary and complex numbers can be used to solve quadratic equations that have no real solutions. The square root of -1 is defined as i, and complex numbers have both a real and imaginary part.
2. An Argand diagram represents complex numbers graphically by plotting the real part on the x-axis and the imaginary part on a perpendicular y-axis. Common complex number operations like addition, subtraction, multiplication, and division can be performed by treating complex numbers as vectors.
3. The modulus of a complex number z = a + bi is the distance from the point (a, b) to the origin on the Argand diagram, and represents the absolute value of z. The argument of z,
An introdcution to complex numbers jcw
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.
Complex numbers
This document introduces complex numbers by representing real numbers as vectors on a number line. It defines the imaginary unit i as the operator that rotates a vector 90 degrees, making it possible to take the square root of negative numbers. Complex numbers are then represented as points in a plane, allowing them to be written as z = x + iy, where x is the real component and y is the imaginary component. Key properties of complex numbers like addition and multiplication are explained geometrically and through polar coordinates, with the unit circle playing an important role.
1.real numbers
1. The document discusses properties of real numbers including: Euclid's division lemma, the fundamental theorem of arithmetic, unique prime factorizations of composite numbers, properties of prime numbers, laws of logarithms, and multiple choice questions testing knowledge of real number concepts.
2. Key real number concepts covered include: every positive integer can be uniquely expressed as a product of prime factors; there are infinitely many prime numbers; rational numbers may have terminating or repeating decimal expansions based on their prime factorizations; laws of logarithms include log(ab) = loga + logb.
3. The multiple choice questions test understanding of concepts like: determining if a number is rational or irrational; finding highest common factors and
นำเสนอจำนวนจริงเพิ่มเติม
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.
Intro Num Int Asmd
The document discusses the history and development of the Arabic numeral system. It explains that the numerals originated from the Phoenicians but were popularized by Arabs. It then provides a theory about how the shapes of the numerals may have been derived from representing different numbers of angles, showing examples. The document also briefly outlines some key developments in the history of algebra.
Freecomplexnumbers
1. Complex numbers can be expressed as a + bi, where a is the real part and b is the imaginary part. They can be visualized on a complex plane.
2. The absolute value (modulus) of a complex number z = |z| represents the distance from the origin, while the argument (angle) is denoted arg(z).
3. When multiplying complex numbers, their absolute values are multiplied and their arguments are added based on geometric representations.
Complex number
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.
Alg II Unit 4-8 Quadratic Equations and Complex Numbers
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.
Lesson 1 1 properties of real numbers
1) The document provides an overview of properties and operations of real numbers including identifying different types of real numbers like integers, rational numbers, and irrational numbers.
2) It discusses ordering real numbers and using symbols like <, >, ≤, ≥ to compare them. Properties of addition, multiplication and other operations are also covered.
3) Examples are provided to illustrate concepts like using properties of real numbers to evaluate expressions and convert between units like miles and kilometers.
complex numbers 1
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.
An applied approach to calculas
This chapter reviews real numbers including:
[1] Classifying numbers as natural numbers, integers, rational numbers, irrational numbers, and real numbers. Rational numbers can be written as fractions while irrational numbers cannot.
[2] Approximating irrational numbers like π as decimals to a given number of decimal places by rounding or truncating.
[3] How calculators handle decimals by either truncating or rounding values based on their display capabilities. A scientific or graphing calculator is recommended for this course.
Complex number multiplication
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.
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What 5G is?
This document summarizes the key technologies and architecture of 5G wireless networks. It discusses technologies like millimeter wave, small cells, massive MIMO, and beamforming that enable 5G's high speeds and low latency. It then outlines the 5G service-based architecture, including functions like the protocol data unit session, access and mobility management, and policy control function. It notes that 5G utilizes a virtualized network function approach with network slicing to provide a flexible, cost-effective infrastructure.
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Resistor and conductor
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Sonet (synchronous optical networking )
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How our body emit radiation
Human bodies emit thermal radiation due to their temperature. Thermal radiation includes radio waves and infrared waves. Humans emit infrared radiation from their bodies, which has a longer wavelength and lower frequency than visible light. This makes the infrared radiation emitted by the human body invisible to the human eye. The amount of thermal radiation emitted depends on surface temperature, area, and other characteristics, but does not indicate other attributes like weight. Radiation protection is important because radiation is present in many aspects of life and excessive exposure can contribute to health issues.
Types of ram
There are three main types of RAM: Dynamic RAM (DRAM), Static RAM (SRAM), and Magnetoresistive RAM (MRAM). DRAM stores each bit in a separate capacitor and transistor, while SRAM uses multiple transistors per cell without capacitors. MRAM uses magnetic charges instead of electrical charges, providing greater storage capacity, lower power consumption, and faster access times than other RAM types. RAM technology has advanced from SDRAM to DDR1, DDR2, DDR3, and now DDR4, with increasing speeds and capacities over time.
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What is complex number
- 2. Presented by: Hamza Sajjad (Complex Number) 2
- 3. History. Number System. Complex numbers. Operations. (Complex Number) 3
- 4. Complex numbers were first introduced by G. Cardano R. Bombelli introduced the symbol 푖. A. Girard called “solutions impossible”. C. F. Gauss called “complex number” (Complex Number) 4
- 5. (Complex Number) 5 Real Number Irrational Number Rational Number Natural Number Whole Number Integer Imaginary Numbers
- 6. When we combine the real and imaginary number then complex number is form. (Complex Number) 6 Real Number Imaginary Number Complex Number
- 7. • A complex number is a number that can b express (Complex Number) 7 in the form of "a+b풊". • Where a and b are real number and 푖 is an imaginary. • In this expression, a is the real part and b is the imaginary part of complex number.
- 8. • A complex number has a real part and an imaginary part, But either part can be 0 . • So, all real number and Imaginary number are also complex number. (Complex Number) 8
- 9. Complex number extend the concept of one-dimensional number line to the two-dimensional complex plan. (Complex Number) 9
- 10. • Horizontal axis use for real part. • Vertical axis for the imaginary part. (Complex Number) 10
- 11. Complex number convert our visualization into physical things. (Complex Number) 11
- 12. Addition : Complex number added by adding real part in real and imaginary part in imaginary. (a + b푖) + (c + d 푖) = (a + c) + (b + d) 푖. (Complex Number) 12 Subtraction: Similarly, subtraction is defined (a + b푖) - (c + d 푖 ) = (a - c) + (b - d) 푖 .
- 13. (Complex Number) 13 Multiplication: The multiplication of two complex number is define by the following formula: (a + b푖).(c + d 푖 ) =(ac - bd) + (b c + ad) 푖 Square of the imaginary unit is -1. 푖²=푖 ∗ 푖= -1 Division: Division can be defined as: 푎 + 푏푖 푐 + 푑푖 = ( 푎푐+푏푑 푐²+푑² ) ( 푏푐−푎푑 푐²+푑² + )푖