Complex Numbers

What are the 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 × c

  • Outers: a × di

  • Inners: bi × c

  • Lasts: 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.

FAQ (Frequently Asked Questions)

Question 1) Is 5 a Complex Number?

Answer) A complex number is a number in the form of x + iy , where x and y are real numbers. For example, 5 + 2i, -5 + 4i and - - i are all complex numbers. x is known as the real part of the complex number and it is known as the imaginary part of the complex number.

Question 2) Are all Numbers Complex Numbers?

Answer) A Complex Number is a combination of the real part and an imaginary part. But either part can be 0, so we can say all Real Numbers and Imaginary Numbers are also Complex Numbers.

Question 3) What are Complex Numbers Examples?

Answer) 4 + 3i is a complex number. A complex number is defined as a polynomial with real coefficients in the single indeterminate I, for which the relation i2 + 1 = 0 is imposed and the value of i2 = -1. Based on this definition, we can add and multiply complex numbers, using the addition and multiplication for polynomials.