Complex numbers are a set of numbers which contain both a real part and an imaginary part.
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:
- Complex numbers are often denoted by and we can refer to the real and imaginary parts respectively using and
- In general:
- 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
- Be careful in your notation of complex and imaginary numbers.
- For example:
could also be written as , but if you wrote this could easily be confused with .
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
- Most calculators used at A-Level can work with complex numbers and you can use these to check your working.
- You should still show your full working though to ensure you get all marks though.