Introduction to Complex Numbers (DP IB Analysis & Approaches (AA)) : Revision Note

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Cartesian Form

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”

    • 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

  • The square roots of other negative numbers can be found by rewriting them as a multiple of  square root of negative 1 end root

    • using 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

    • We refer to the real and imaginary parts respectively using Re left parenthesis z right parenthesisand Im left parenthesis z right parenthesis

  • 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

  • The set of all complex numbers is given the symbol straight complex numbers

What is Cartesian Form?

  • There are a number of different forms that complex numbers can be written in

  • The form z = a + bi is known as Cartesian Form

    • a, b ∈ straight real numbers

    • This is the first form given in the formula booklet

  • In general, for z = a + bi

    • Re(z) = a

    • Im(z) = b

  • A complex number can be easily represented geometrically when it is in Cartesian Form

  • Your GDC may call this rectangular form

    • When your GDC is set in rectangular settings it will give answers in Cartesian Form

    • If your GDC is not set in a complex mode it will not give any output in complex number form

    • Make sure you can find the settings for using complex numbers in Cartesian Form and practice inputting problems

  • Cartesian form is the easiest form for adding and subtracting complex numbers

Examiner Tips and Tricks

  • Remember that complex numbers have both a real part and an imaginary part

    • 1 is purely real (its imaginary part is zero)

    • i is purely imaginary (its real part is zero)

    • 1 + i is a complex number (both the real and imaginary parts are equal to 1)

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

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Complex Addition, Subtraction & Multiplication

How do I add and subtract complex numbers in Cartesian Form?

  • Adding and subtracting complex numbers should be done when they are in Cartesian form

  • 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

    • open parentheses a plus b straight i close parentheses minus open parentheses c plus d straight i close parentheses equals open parentheses a minus c close parentheses plus open parentheses b minus d close parentheses straight i

How do I multiply complex numbers in Cartesian Form?

  • Complex numbers can 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

  • Multiplying two complex numbers in Cartesian form is done in the same way as multiplying two linear expressions:

    • open parentheses a plus b straight i close parentheses open parentheses c plus d straight i close parentheses equals a c plus open parentheses a d plus b c close parentheses straight i plus b d straight i squared equals blank a c plus open parentheses a d plus b c close parentheses straight i minus b d

    • This is a complex number with real part a c minus b d blankand imaginary part a d plus b c

    • The most important thing when multiplying complex numbers is that

      • straight i squared equals negative 1

  • Your GDC will be able to multiply complex numbers in Cartesian form

    • Practise doing this and use it to check your answers

  • It is easy to see that multiplying more than two complex numbers together in Cartesian form becomes a lengthy process prone to errors

    • It is easier to multiply complex numbers when they are in different forms and usually it makes sense to convert them from Cartesian form to either Polar form or Euler’s form first

  • 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 Tips and Tricks

  • When revising for your exams, practice using your GDC to check any calculations you do with complex numbers by hand

    • This will speed up using your GDC in rectangular form whilst also giving you lots of practice of carrying out calculations by hand

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

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Complex Conjugation & Division

When dividing complex numbers, the complex conjugate is used to change the denominator to a real number.

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?

  • To divide two complex numbers:

    • STEP 1: Express the calculation in the form of a fraction

    • STEP 2: Multiply 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

    • STEP 3: Multiply out and simplify your answer

      • This should have a real number as the denominator

    • STEP 4: Write your answer in Cartesian form as two terms, simplifying each term if needed

      • OR convert into the required form if needed

  • Your GDC will be able to divide two complex numbers in Cartesian form

    • Practise doing this and use it to check your answers if you can

Examiner Tips and Tricks

  • 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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Amber

Author: Amber

Expertise: Maths Content Creator

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

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