Intro to Complex Numbers (DP IB Maths: AI HL)

Revision Note

Amber

Author

Amber

Last updated

Did this video help you?

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 Tip

  • 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

Did this video help you?

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 Tip

  • 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

Did this video help you?

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 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

You've read 0 of your 10 free revision notes

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

Did this page help you?

Amber

Author: Amber

Expertise: Maths

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.