Introduction to Complex Numbers (Edexcel A Level Further Maths): Revision Note

Cartesian Form of Complex Numbers

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

  • 

  • 

• 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

  • 

  • 

• 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

Operations with Complex Numbers

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

  • 

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

  • 

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:

  • 

  • 

• 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

  • 

  • 

  • 

  • 

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

  • 

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

Complex Conjugation & Division

When dividing complex numbers, we can use the complex conjugate to make the denominator a real number, which makes carrying out the division much easier.

What is a complex conjugate?

• For a given complex number , the complex conjugate of is denoted as , where 

• If then 

• You will find that:

  • is always real because 

    • For example:

  • is always imaginary because 

    • For example:

  • is always real because  (as )

    • For example:

How do I divide complex numbers?

• When we divide complex numbers, we can express the calculation in the form of a fraction, and then start by multiplying the top and bottom by the conjugate of the denominator:

  • 

• This ensures we are multiplying by 1; so not affecting the overall value

• This gives us a real number as the denominator because we have a complex number multiplied by its conjugate ()

• This process is very similar to “rationalising the denominator” with surds which you may have studied at GCSE