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(Complex Number) 2

Presented by:Hamza Sajjad

History.

Number System.

Complex numbers.

Operations.

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Complex numbers were first introduced by G. Cardano

R. Bombelli introduced the symbol 𝑖.

A. Girard called “solutions impossible”.

C. F. Gauss called “complex number”

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Real Number

Irrational Number

Rational Number

Natural Number

Whole Number

Integer

Imaginary Numbers

When we combine the real and imaginary number then complex number is form.

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Real Number

Imaginary Number

Complex Number

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• A complex number is a number that can b express in the form of "a+b𝒊".

• Where a and b are real number and 𝑖 is an imaginary.

• In this expression, a is the real part and b is the imaginary part of complex number.

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• 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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Complex number extend the concept of one-dimensional number line to the two-dimensional complex plan.

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• Horizontal axis use for real part. • Vertical axis for the imaginary part.

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Complex number convert our visualization into physical things.

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Addition :Complex number added by adding real part in real

and imaginary part in imaginary.

(a + b𝑖) + (c + d 𝑖) = (a + c) + (b + d) 𝑖.

Subtraction:Similarly, subtraction is defined

(a + b𝑖) - (c + d 𝑖 ) = (a - c) + (b - d) 𝑖 .

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Multiplication:

The multiplication of two complex number is define by the following formula:

(a + b𝑖).(c + d 𝑖 ) =(ac - bd) + (b c + ad) 𝑖Square of the imaginary unit is -1.

𝑖²=𝑖 ∗ 𝑖= -1

Division:Division can be defined as:

𝑎 + 𝑏𝑖

𝑐 + 𝑑𝑖= (

𝑎𝑐+𝑏𝑑

𝑐²+𝑑²) (

𝑏𝑐−𝑎𝑑

𝑐²+𝑑²)𝑖+

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