Intro to Complex Numbers (CIE A Level Maths: Pure 3)

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14 December 2023

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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 $ℂ$ .

Up until now, when we have encountered an equation such as $x^{2} = - 1$ we would have stated that there are “no real solutions” as the solutions are $x = \pm \sqrt{- 1}$ 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
- $\sqrt{- 1} = i$
- $i^{2} = - 1$
We can use the rules for manipulating surds to manipulate imaginary numbers.
We can do this by rewriting surds to be a multiple of $\sqrt{- 1}$ using the fact that $\sqrt{a b} = \sqrt{a} \times \sqrt{b}$

Complex numbers have both a real part and an imaginary part
- For example: $3 + 4 i$
- The real part is 3 and the imaginary part is 4
  - Note that the imaginary part does not include the ' $i$ '
Complex numbers are often denoted by $z$ and we can refer to the real and imaginary parts respectively using $Re (z)$ and $Im (z)$
In general:
- $z = a + b i$ This is the Cartesian form of z
- $Re (z) = a$
- $Im (z) = b$
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, $3 + 2 i$ and $3 + 3 i$ are not equal

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Be careful in your notation of complex and imaginary numbers.
For example:
$(3 \sqrt{5}) i$ could also be written as $3 i \sqrt{5}$ , but if you wrote $3 \sqrt{5} i$ this could easily be confused with $3 \sqrt{5 i}$ .

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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
- $(a + b i) + (c + d i) = (a + c) + (b + d) i$
Complex numbers can also be multiplied by a constant in the same way as algebraic expressions:
- $k (a + b i) = k a + k b i$

The most important thing to bear in mind when multiplying complex numbers is that $i^{2} = - 1$
We can still apply our usual rules for multiplying algebraic terms:
- $a (b + c) = a b + a c$
- $(a + b) (c + d) = a c + a d + b c + b d$
Sometimes when a question describes multiple complex numbers, the notation $z_{1}, z_{2}, \dots$ is used to represent each complex number

Because $i^{2} = - 1$ this can lead to some interesting results for higher powers of i
- $i^{3} = i^{2} \times i = - i$
- $i^{4} = {(i^{2})}^{2} = {(- 1)}^{2} = 1$
- $i^{5} = {(i^{2})}^{2} \times i = i$
- $i^{6} = {(i^{2})}^{3} = {(- 1)}^{3} = - 1$
We can use this same approach of using i² to deal with much higher powers
- $i^{23} = {(i^{2})}^{11} \times i = {(- 1)}^{11} \times i = - i$
- Just remember that -1 raised to an even power is 1 and raised to an odd power is -1

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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.

the (exam) results speak for themselves:

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