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.

1. (Complex Number) 1

2. Presented by:
Hamza Sajjad
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3.  History.
 Number System.
 Complex numbers.
 Operations.
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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”
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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.
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Real
Number
Imaginary
Number
Complex
Number

7. • A complex number is a number that can b express
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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.
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9. Complex number extend the concept of one-dimensional
number line to the two-dimensional complex plan.
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10. • Horizontal axis use for real part.
• Vertical axis for the imaginary part.
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11. Complex number convert our visualization into physical
things.
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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.
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Subtraction:
Similarly, subtraction is defined
(a + b푖) - (c + d 푖 ) = (a - c) + (b - d) 푖 .

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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

14. (Complex Number) 14