# Complex Numbers

Complex numbers are numbers which are expressed in the form of a+ib where i is an imaginary number called iota and has the value of (√-1). The main application of these numbers is to represent periodic motions such as water waves, alternating current, light waves, etc., which relies on sine or cosine waves etc. The key concepts are highlighted in this lesson will include the following:

• Introduction
• Algebraic Operation on Complex numbers
• Formulas
• Power of Iota (i)
• Complex Number Identities
• Modulus and Conjugate of a Complex number
• Examples of Complex Numbers
• Argand Plane & Polar Representation of Complex Number

When we solve a quadratic equation of the form ax2 +bx+c = 0, the roots of the equations can be determined in three forms;

• Two Distinct Real Roots
• Similar Root
• No Real roots (Complex Roots)

## Complex Numbers Definition

The complex number is basically the combination of a real number and an imaginary number.

### Real Numbers Re()

Any number which is present in a number system such as positive, negative, zero, integer, rational, irrational, fractions, etc. are real numbers. For example: 12, -45, 0, 1/7, 2.8, √5 are all real numbers.

### Imaginary Numbers Im()

The numbers which are not real are imaginary numbers. When we square an imaginary number, it gives a negative result. Example: √-2, √-7, √-11 are all imaginary numbers.

In the 16th century, the complex numbers were introduced which made it possible to solve the equation x2  +1 = 0. The roots of the equation are of form x = ±√-1 and no real roots exist. Thus, with the introduction of complex numbers, we have Imaginary roots.

We denote √-1 with the symbol ‘i’, where i denotes Iota (Imaginary number).

An equation of the form z= a+ib, where a and b are real numbers, is defined to be a complex number. The real part is denoted by Re z = a and the imaginary part is denoted by Im z = b.

## Algebraic Operation on Complex numbers

There can be four types of algebraic operation on complex numbers which are mentioned below. Visit the linked article to know more about these algebraic operations along with solved examples. The four operations on the complex numbers include:

• Subtraction
• Multiplication
• Division

## Complex Number Formulas

(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)

• Division

(a + ib) / (c + id) = (ac+bd)/ (c2 + d2) + i(bc – ad) / (c2 + d2)

### Power of Iota (i)

Depending upon the power of “i”, it can take the following values;

i4k+1 = i . i4k+2 = -1 i4k+3 =  -i . i4k = 1

where k can have an integral value (positive or negative).

Similarly, we can find for the negative power of i, which are as follows;

i-1 = 1 / i

Multiplying and dividing the above term with i, we have;

i-1 = 1 / i  ×  i/i  × i-1  = i / i2 = i / -1 = -i / -1 = -i

Note: √-1 × √-1 = √(-1  × -1) = √1 = 1 contradicts to the fact that i2 = -1.

Therefore, for an imaginary number, √a × √b is not equal to √ab.

## Complex Number Identities

1. (z+ z2)2 = (z1)+ (z2)2 + 2 z1 × z2
2. (z1 – z2)2 = (z1)+ (z2)2 – 2 z1 × z2
3. (z1)2 – (z2)2 =  (z+ z2)(z1 – z2)
4. (z+ z2)3 = (z1)3 + 3(z1)2 z +3(z2)2 z1 + (z2 )3
5. (z– z2)3 = (z1)3 – 3(z1)2 z +3(z2)2 z1 – (z2 )3

### Modulus and Conjugate of a Complex number

Let z=a+ib be a complex number.

The Modulus of z is represented by |z|.

Mathematically, $\left | z \right |= \sqrt{a^{2}+b^{2}}$

The conjugate of “z” is denoted by $\bar{z}$.

Mathematically, $\bar{z}$= a – ib

### Argand Plane and Polar Representation of Complex Number

Similar to the XY plane, the Argand(or complex) plane is a system of rectangular coordinates in which the complex number a+ib is represented by the point whose coordinates are a and b.

We find the real and complex components in terms of r and θ, where r is the length of the vector and θ is the angle made with the real axis. Check out the detailed argand plane and polar representation of complex numbers in this article and understand this concept in a detailed way along with solved examples.

### Complex Numbers Example Problem

Example: Express the following in a+ib form.

$\frac{5+ \sqrt{3}i}{1-\sqrt{3}i}$.

And then find the Modulus and Conjugate of the complex number.

Solution:

Given $\frac{5+ \sqrt{3}i}{1-\sqrt{3}i}$

z= $\frac{5+ \sqrt{3}i}{1-\sqrt{3}i} \times \frac{1+\sqrt{3}i}{1+\sqrt{3}i} = \frac{5-3+6\sqrt{3}i}{1+3}=\frac{1}{2} + \frac{3}{2}i$

Modulus, $\bar{z} = \sqrt{\left ( \frac{1}{2}\right )^{2} + \left (\frac{3}{2}\right )^{2}}$ $\bar{z} = \sqrt{\left ( \frac{10}{4}\right )}$ $\bar{z} = \frac{\sqrt{10}}{2}$

Conjugate, $\bar{z} = \frac{1}{2} – \frac{3}{2}i$