The Real Number System and the Complex Number System

The real number consists of both rational and irrational numbers and can be defined by infinite decimal representation, such as 5.1232445556677… - The Real Number System and the Complex Number System introduction. 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).

Figure 1: The Real Numbers and Number Line

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(Source: http://www.helpalgebra.com/onlinebook/realnumbersystem.htm)

In mathematics, a complex number z is a number of the form z = a + ib, where the symbol i = √(1) is called imaginary unit and a, b Î R.

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 Î 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)

The complex number z = x + iy, x, y ÎR can be represented by a point z = P(x, y) in xy-plane or Cartesian plane. The x-axis is called the axis of real number (real axis) whereas the y-axis is called the axis of imaginary numbers (imaginary axis).

For 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.

 

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