Operations with Complex Numbers (DP IB Analysis & Approaches (AA)): Revision Note

Complex addition, subtraction & multiplication

How do I add and subtract complex numbers?

• To add or subtract complex numbers, add or subtract their real and imaginary parts separately

  • e.g.

    • 

    • 

How do I multiply complex numbers?

• To multiply by you need to expand brackets

  • 

    • just remember that

• e.g.

  • gives

  • which simplifies to

How do I find powers of i?

• Because , higher powers of can be found as follows:

  • 

  • 

  • 

  • 

• The powers of form a sequence with period 4

  • 

• Use index laws to find much higher powers

  • 

  • Just remember that

    • if is even

    • if is odd

Complex conjugation & division

What is a complex conjugate?

• For the complex number , the complex conjugate of (written ) is

  • 

• If then 

• You will find that:

  • is always real

    • e.g.

  • is always imaginary

    • e.g.

  • or is always real

    • e.g.

How do I divide complex numbers?

• To divide two complex numbers, do the following:

  • STEP 1
    Write the division as a fraction

  • STEP 2
    Multiply the top and bottom of the fraction by the conjugate of the denominator

    • 

    • This is similar to rationalising a denominator with surds

  • STEP 3
    Add in brackets and multiply out the top and bottom, simplifying your answer

    • The denominator will always be real

  • STEP 4
    Give your answer in the form required

    • or in Cartesian form if not specified