Operations with Complex Numbers (Cambridge (CIE) A Level Maths): Revision Note

Complex Numbers – Basics

Complex numbers are a set of numbers which contain both a real part and an imaginary part. The set of complex numbers is denoted as .

What is an imaginary number?

• Up until now, when we have encountered an equation such as we would have stated that there are “no real solutions” as the solutions are  which are not real numbers

• To solve this issue, mathematicians have defined one of the square roots of negative one as ; an imaginary number

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• We can use the rules for manipulating surds to manipulate imaginary numbers.

• We can do this by rewriting surds to be a multiple of  using the fact that 

What is a complex number?

• Complex numbers have both a real part and an imaginary part
  

  • For example: 

  • The real part is 3 and the imaginary part is 4

    • Note that the imaginary part does not include the ''

• Complex numbers are often denoted by  and we can refer to the real and imaginary parts respectively using and

•  In general:

  • This is the Cartesian form of z

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• It is important to note that two complex numbers are equal if, and only if, both the real and imaginary parts are identical.

  • For example,  and  are not equal

Complex Numbers – Basic Operations

How do I add and subtract complex numbers?

• When adding and subtracting complex numbers, simplify the real and imaginary parts separately

  • Just like you would when collecting like terms in algebra and surds, or dealing with different components in vectors

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• Complex numbers can also be multiplied by a constant in the same way as algebraic expressions:

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How do I multiply complex numbers?

• The most important thing to bear in mind when multiplying complex numbers is that 

• We can still apply our usual rules for multiplying algebraic terms:

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• Sometimes when a question describes multiple complex numbers, the notation  is used to represent each complex number

How do I deal with higher powers of i?

• Because  this can lead to some interesting results for higher powers of i

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• We can use this same approach of using i² to deal with much higher powers

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  • Just remember that -1 raised to an even power is 1 and raised to an odd power is -1