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 i ; 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:
• Complex numbers are often denoted by z and we can refer to the real and imaginary parts respectively usingand
• ?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

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
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• 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

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?

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:
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• 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