Math

The Real Number System and the Complex Number System (Student Name

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The real number consists of both rational and irrational numbers and can be defined by infinite decimal representation , such as 5 .77 . The rational numbers are defined by (a /b ) and consists of integers , whole numbers and natural numbers . Figure 1 shows the complete set of real numbers and the number line . Every real number corresponds to a distance on number line , which start at center (zero ?1 ) is called imaginary unit and a , b

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a is called the real part of complex number (z ) and b is called the imaginary part of complex number (z , which is written as

a Rez and b ?Imz

With this notation complex number can be written as z Rez ?iImz

The set of all complex numbers is denoted by C a ib a , b ( R

The addition , subtraction and multiplication of complex number are explained in below examples

Addition (a ?ib (c ?id (a ?c ?i (b ?d

Subtraction (a ?ib (c ?id ( a ?c ?i (b ?d

Multiplication (a ?ib (c ?id (ac - bd ?i (ad ?bc

The division for the complex number is quite different , and is explained below

If c id ? 0 , then (a ib (c id [ (ac bd i (bc - ad )] (c2 d2

When , complex numbers are added , the real part is added to real part and imaginary part is added to imaginary part as shown below (2 3i (3 4i (2 3 (3 4 )i 5 7i

As we know that i2 -1 , therefore when multiplication of complex number is performed , the both imaginary part after multiplication becomes real part and real part multiplied with imaginary part becomes imaginary part . This can be seen below (2 3i (3 4i 2 3 (2 4 3 3 )i 3 4 (i2 6 17i -12 -6 17i Figure 2 : Geometric representation of Z and its Conjugate Z (source Wikipedia

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5For any complex numbers A , B , C and D

The addition and multiplications are commutative

A B B A , AB BA

The addition and multiplications are Associative

A (B C (A B C , A (BC (AB )C

Complex number follows the Distributive Law

A (B C AB AC

Some other properties of complex number are written below

A 0 0 A A , A 1 1 A A and A (-A (-A A 0

The Complex Conjugate of complex number z a ib is defined as z a - ib

Reference

http /www .helpalgebra .com /onlinebook /realnumbersystem .htm accessed on 12 August 2007

http /en .wikipedia .org /wiki /Complex_number accessed on 12 August 2007...