Operations with Complex Numbers (Cambridge International A-Level Mathematics)

Complex Numbers – Basics

Introduction to 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 $ℂ$.

What is an imaginary number?

• 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 $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$

What is a complex number?

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

  • For example: $3+4i$

  • 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 $\mathrm{Re}(z)$and $\mathrm{Im}(z)$

• In general:

  • $z=a+bi$ 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, $3+2i$ and $3+3i$ are not equal

Worked Example

Exam Hint

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

Complex Numbers – Basic Operations

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
  • $(a+b \mathrm{i} )+(c+d \mathrm{i} )=(a+c)+(b+d) \mathrm{i}$

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

  • $k (a+b \mathrm{i} )=k a+k b \mathrm{i}$

How do I multiply complex numbers?

• 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:

  • $a(b+c)=ab+ac$
  • (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

What are the powers of i?

• Because $i^{2}=-1$ this can lead to some interesting results for higher powers of i

  • $i^{3}=i^{2}\times i=-i$
  • $i^4= ( i² ) ²=(−1)²=1$
  • $i^{5}=(i^{2})^{2}\times i=i$
  • $i^6=( i²)³=(−1)³= −1$

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

  • $i^23=( i²)^11× i=(−1)^11× i= − i$
  • Just remember that -1 raised to an even power is 1 and raised to an odd power is -1

Worked Example

Exam Hint

• 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.