# Complex Numbers

## What are the complex numbers?

• A complex number is said to be a combination of a real number and an imaginary number.

• Any number in Mathematics can be known as a real number.

• Imaginary Numbers are the numbers which when squared give a negative number.

• A complex number is represented as z=a+ib, where a and b are real numbers and where i=$\sqrt{-1}$.

## Complex Number Example is –

 5i , 4i+1, 3i/4 ,  -2.8i , 0.4+2i

### Facts about a Complex Number-

As we know, a Complex Number has a real part and an imaginary part. Either part of a complex number can be 0, so we can say all Real Numbers and Imaginary Numbers are also Complex Numbers.

## Complex Number Examples-

 Complex Part Real Part Imaginary number Type 3+2i 3 2i Complex number 6 6 0 Purely Real -4i -4 i Purely Imaginary

### Real and Imaginary Parts of a Complex Number-

Now we know what complex numbers. Let’s take a complex number z=a+ib, then the real part here is a and it is denoted by Re (z) and here b is the imaginary part and is denoted by Im (z).

For example,

If z is a complex number and z = 7, then z can be written as z= 7+0i, here the real part of the complex number is Re (z)=7 and Im(z) = 0.

If z is a complex number and z = -5i, then z can be written as z= 0 + (-5)i , here the real part of the complex number is Re(z)= 0 and Im(z) = -5.

If z is a complex number and z = -3+√4i, here the real part of the complex number is Re(z)=-3 and Im(z) = $\sqrt{4}$.

## Real and Imaginary Parts of a Complex Number Examples -

 Complex number Re(z) Im (z) z=5+2i 5 2i z=7 7 0 z=-5i+2 2 -5

### What is a Purely Imaginary Number?

If in a complex number z = x+iy ,if the value of x is equal to 0 and the value of y is not equal to zero.

Therefore, z=iy and z is known as a purely imaginary number.

### What is a Real Number?

If in a complex number z = x+iy ,if the value of y is equal to 0 and the value of z is equal to x.

Therefore, z=x and z is known as a real number.

### What is a Non-Real Complex Number?

If in a complex number z = x+iy ,if the value of y is not equal to 0 and the value of z is equal to x.

## Therefore, z=x+iy is Known as a Non- Real Complex Number.

 i $\sqrt{-1}$ i2 $\sqrt{-1}$X$\sqrt{-1}$ -1 i3 $i^{2Xi}$ (-1) ×i = -i i4 $i^{2}$X$i^{2}$ (-1) ×-1= 1 i5 $i^{4Xi}$ (1) ×i = i

### Algebra of Complex Number-

Complex number formulas and complex number identities-

1. Addition of Complex Numbers- If we want to add any two complex numbers we add each part separately:

Complex Number Formulas :(x+iy) + (c+di) = (x+c) + (y+d)i

For example: If we need to add the complex numbers 5 + 3i and 6 + 2i

• We need to add the real numbers, and

• We need to add the imaginary numbers:

= (5 + 3i) + (6 + 2i)
= 5 + 6 + (3 + 2)i
= 11 + 5i

Let's try another example, lets add the complex numbers 2 + 5i and 8 − 3i

= (2 + 5i) + (8 − 3i)
= 2 + 8 + (5 − 3)i
= 10 + 2i

1. Subtraction of Complex Numbers – If we want to subtract any two complex numbers we subtract each part separately:

Complex Number Formulas : (x-iy) - (c+di) = (x-c) + (y-d)i

For example: If we need to add the complex numbers 9 +3i and 6 + 2i

• We need to subtract the real numbers, and

• We need to  subtract the imaginary numbers:

= (9+3i) - (6 + 2i)
= (9-6) + (3 -2)i
= 3+1i

1. Conjugate of a Complex Number- We will need to know about conjugates of a complex number in a minute!

A conjugate of a complex number is where the sign in the middle of a complex number changes. A conjugate of a complex number is often written with a bar over it.

For example, we take a complex number 2+4i the conjugate of the complex number is 2-4i.

1. Multiplication of Complex Numbers-

## Suppose we Have Two Complex Numbers,

 Each part of the first complex number (z1)  gets multiplied by each part of the second complex number(z2) .

## We Generally use the FOIL Rule Which Stands for "Firsts, Outers, Inners, Lasts".

 Firsts: a × cOuters: a × diInners: bi × cLasts: bi × di Complex number formulas : (a+ib)(c+id) = ac + aid+ bic + bdi2

Where the value of i2 = -1

For Example: (4 + 2i) (3 + 7i)

= (4 + 2i) (3 + 7i) = 4×3 + 4×7i + 2i×3+ 2i×7i

= 12 + 28i + 6i + 14i2

= 12 + 34i − 14   (because i2 = −1)

= −2 + 34i

### Questions to be Solved-

Question 1) Add the complex numbers 4 + 5i and 9 − 3i.

Solution) From complex number identities, we know how to add two complex numbers.

= (4+ 5i) + (9 − 3i)
= 4 + 9 + (5 − 3) i
= 13+ 2i

Question 2) Subtract the complex numbers 12 + 5i and 4 − 2i.

Solution) From complex number identities, we know how to subtract two complex numbers.

=12 + 5i – (4 − 2i)

=8 +7i

### Uses of Complex numbers -

Here are the uses of Complex numbers,

• Complex numbers are mainly used in electrical engineering techniques.

• As Fourier transforms are used in understanding oscillations and wave behavior that occur both in AC Current and in modulated signals, the concept of a complex number is widely used in Electrical engineering.