Introduction to Complex Numbers (Edexcel A Level Further Maths: Core Pure)

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Jamie W

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Jamie W

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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 straight complex numbers.

What is an imaginary number?

  • Up until now, when we have encountered an equation such as x to the power of 2 space end exponent equals space minus 1 we would have stated that there are “no real solutions” as the solutions are x equals plus-or-minus square root of negative 1 end root which are not real numbers
  • To solve this issue, mathematicians have defined one of the square roots of negative one as straight i; an imaginary number
    • square root of negative 1 end root equals straight i
    • straight i squared equals negative 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 square root of negative 1 end root using the fact that square root of a b end root equals square root of a cross times square root of b

What is a complex number?

  • Complex numbers have both a real part and an imaginary part
    • For example: 3 plus 4 straight i
    • The real part is 3 and the imaginary part is 4
      • Note that the imaginary part does not include the 'straight i'
  • Complex numbers are often denoted by z and we can refer to the real and imaginary parts respectively using Re left parenthesis z right parenthesisand Im left parenthesis z right parenthesis
  •  In general:
    • z equals a plus b straight i This is the Cartesian form of z
    • Re open parentheses z close parentheses equals a
    • Im open parentheses z close parentheses equals 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 plus 2 straight i and 3 plus 3 straight i are not equal

Examiner Tip

  • Be careful in your notation of complex and imaginary numbers.
  • For example:
    open parentheses 3 square root of 5 close parentheses i could also be written as 3 i square root of 5, but if you wrote 3 square root of 5 i this could easily be confused with  3 square root of 5 i end root.

Worked example

a)
Solve the equation x squared equals negative 9

 1-8-1-ib-hl-aa-cartesian-form-we-a

b)
Solve the equation open parentheses x plus 7 close parentheses squared equals negative 16, giving your answers in Cartesian form.

1-8-1-ib-hl-aa-cartesian-form-we-b

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
    • open parentheses a plus b straight i close parentheses plus open parentheses c plus d straight i close parentheses equals open parentheses a plus c close parentheses plus open parentheses b plus d close parentheses straight i
  • Complex numbers can also be multiplied by a constant in the same way as algebraic expressions:
    • k open parentheses a plus b straight i close parentheses equals k a plus k b straight i

How do I multiply complex numbers?

  • The most important thing to bear in mind when multiplying complex numbers is that straight i squared equals negative 1
  • We can still apply our usual rules for multiplying algebraic terms:
    • a left parenthesis b plus c right parenthesis equals a b plus a c
    • open parentheses a plus b close parentheses open parentheses c plus d close parentheses equals a c plus a d plus b c plus b d
  • Sometimes when a question describes multiple complex numbers, the notation z subscript 1 comma blank z subscript 2 comma blank horizontal ellipsis is used to represent each complex number

How do I deal with higher powers of i?

  • Because straight i squared equals negative 1 this can lead to some interesting results for higher powers of i
    • bold i cubed equals bold i squared cross times bold i equals blank minus bold i
    • bold i to the power of 4 equals left parenthesis bold i squared right parenthesis squared equals open parentheses negative 1 close parentheses squared equals 1
    • bold i to the power of 5 equals left parenthesis bold i squared right parenthesis squared blank cross times bold i equals bold i
    • bold i to the power of 6 equals open parentheses bold i squared close parentheses cubed equals open parentheses negative 1 close parentheses cubed equals blank minus 1
  • We can use this same approach of using i2 to deal with much higher powers
    • bold i to the power of 23 equals open parentheses bold i squared close parentheses to the power of 11 cross times bold i equals open parentheses negative 1 close parentheses to the power of 11 cross times bold i equals blank minus bold i
    • Just remember that -1 raised to an even power is 1 and raised to an odd power is -1

Examiner Tip

  • 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.

Worked example

a)
Simplify the expression 2 open parentheses 8 minus 6 straight i close parentheses minus 5 open parentheses 3 plus 4 straight i close parentheses.

1-8-1-ib-hl-aa-adding-subtracting-mulitplying-we-a

b)
Given two complex numbers z subscript 1 equals 3 plus 4 straight i and z subscript 2 equals 6 plus 7 straight i, find z subscript 1 cross times blank z subscript 2.

1-8-1-ib-hl-aa-adding-subtracting-mulitplying-we-b

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 z equals a plus b straight i, the complex conjugate of z is denoted as z to the power of asterisk times, where z to the power of asterisk times equals a minus b straight i
  • If z equals a minus b straight i then z to the power of asterisk times equals a plus b straight i
  • You will find that:
    • z plus z to the power of asterisk times is always real because left parenthesis a plus b straight i right parenthesis plus left parenthesis a minus b straight i right parenthesis equals 2 a
      • For example: left parenthesis 6 plus 5 straight i right parenthesis space plus space left parenthesis 6 minus 5 straight i right parenthesis space equals space 6 plus 6 plus 5 straight i minus 5 straight i space equals space 12
    • z minus z to the power of asterisk times is always imaginary because open parentheses a plus b straight i close parentheses minus left parenthesis a minus b straight i right parenthesis equals 2 b straight i
      • For example: left parenthesis 6 plus 5 straight i right parenthesis space minus space left parenthesis 6 minus 5 straight i right parenthesis space equals space 6 minus 6 plus 5 straight i minus left parenthesis negative 5 straight i right parenthesis space equals space 10 straight i
    • z cross times z to the power of asterisk times is always real because open parentheses a plus b straight i close parentheses open parentheses a minus b straight i close parentheses equals a squared plus a b straight i minus a b straight i minus b squared straight i squared equals a squared plus b squared (as straight i squared equals negative 1)
      • For example: left parenthesis 6 plus 5 straight i right parenthesis left parenthesis 6 minus 5 straight i right parenthesis space equals space 36 space plus 30 straight i space – space 30 straight i space minus 25 straight i squared space equals space 36 space – space 25 left parenthesis negative 1 right parenthesis space equals space 61

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:
    • fraction numerator a plus b straight i over denominator c plus d straight i end fraction equals blank fraction numerator a plus b straight i over denominator c plus d straight i end fraction blank cross times blank fraction numerator c minus d straight i over denominator c minus d straight i end fraction
  • 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 (z z to the power of asterisk times)
  • This process is very similar to “rationalising the denominator” with surds which you may have studied at GCSE

Examiner Tip

  • We can speed up the process for finding z z asterisk timesby using the basic pattern of open parentheses x plus a close parentheses open parentheses x minus a close parentheses equals x squared minus a squared
  • We can apply this to complex numbers: open parentheses a plus b straight i close parentheses open parentheses a minus b straight i close parentheses equals a squared minus b squared straight i squared equals a squared plus b squared
    (using the fact that straight i squared equals negative 1)
  • So 3 plus 4 straight i multiplied by its conjugate would be 3 squared plus 4 squared equals 25

Worked example

Find the value of open parentheses 1 plus 7 straight i close parentheses divided by left parenthesis 3 minus straight i right parenthesis.

1-8-1-ib-hl-aa-dividing-we-a

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Jamie W

Author: Jamie W

Expertise: Maths

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.