Basic operations

Basic Operations with Complex Numbers

Addition and Subtraction

  • Understand that to add or subtract complex numbers, simply combine like terms.
  • Recall that a + bi and c + di become (a+c) + (b+d)i when added and (a-c) + (b-d)i when subtracted.


  • Recognise that multiplication of complex numbers involves applying the distributive law and replacing i^2 with -1.
  • Review how multiplying a + bi and c + di results in ac - bd + (ad + bc)i.


  • Become proficient in dividing complex numbers by multiplying the numerator and denominator by the conjugate of the denominator, which changes it into a real number.
  • The division of a + bi and c + di will be ((ac+bd)/(c^2+d^2)) + ((bc-ad)/(c^2+d^2))i.

Conjugate of a Complex Number

  • Recognise the conjugate of a complex number a + bi as a - bi.
  • Understand the implications of the conjugate in complex number operations - the product of a complex number and its conjugate always results in a real number.

Modulus and Argument

  • Recall that the modulus of a complex number is the length or magnitude of the vector from the origin to the point representing the complex number in the Argand Diagram.
  • It is calculated as the square root of the sum of the squares of the real and the imaginary parts (√(a^2 + b^2)).
  • The argument of a complex number is the counter-clockwise angle from the positive x-axis to the vector representing the complex number.

Refresh your abilities to perform these operations with ease as they form the foundation for more advanced Complex Number topics.