Complex Numbers

Imaginary number

Imaginary number (when squared give a negative result)

i is the unit of imaginary number which is square root of -1

i = √-1, because i2=-1

So, (7i)2 = -7

Definition of Complex Number

A Complex Number is a combination of a Real Number and an Imaginary Number


4 + 7i, where 4 is Real Number, and 7i is Imaginary Number

Either Part Can Be Zero

  • a Complex Number has a real part and an imaginary part.
  • Either part can be 0


  • all Real Numbers are Complex numbers
  • all Imaginary Numbers are Complex Numbers.
 Real PartImaginary PartComplex Number
Combination47i4 + 7i
Pure Real40i4
Pure Imaginary07i7i

We often use z for a complex number. And Re() for the real part and Im() for the imaginary part, like this:

z = a + bi, where Re(z) = a, Im(z) = b

Adding 2 complex numbers

To add 2 complex numbers, add each part separately

(a+bi) + (c+di) = (a+c) + (b+d)i

Multiplying 2 complex numbers

To multiply 2 complex numbers, each part of the first complex number gets multiplied by

each part of the second complex number

(a+bi)(c+di) = ac + adi + bci + bdi2

Because i2 = -1, so

(a+bi)(c+di) = ac – bd + adi + bci

Dividing 2 complex numbers

To divide 2 complex numbers, multiply both top and bottom by the conjugate of the bottom.

Note: A conjugate is where we change the sign in the middle

Complex numberconjugateMultiplying By the Conjugate
a + bia – bia2 + b2
a – bia + bia2 + b2

So, to divide a + bi by c – di:

Dividing 2 Complex Number 1

Or, to divide a + bi by c + di:

Dividing 2 Complex Number 2
z1Given Re(z1) = 4, Im(z2)=74+7i
z2Given Re(z1) = 4, Im(z2)=-74-7i
z1+z2(4+4) + (7-7)i8 + 0i = 8
z1 x z24×4 – 7x(-7) + 4x(-7)i + 4x7i44 + 0i = 44
z1 / z2Dividing 2 Complex Number Example 1Dividing 2 Complex Number Example 2

Learn More

Reciprocal of Number

Table of Factors and Multiples

Common Factor

Simplifying Square Roots