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Summary Complex Numbers (Algebra) - Detailed Notes - Grade 12 Mathematics
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  • Course
  • Mathematics
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  • 12th Grade

This document contains detailed in-depth notes about complex numbers (algebra) for grade 12 mathematics. The following topics are dealt with: 1.1 Important Definitions and Notes for Complex Numbers 1.2 Multiplication, Addition and Subtraction 1.3 Division 1.4 Quadratic Formula 1.5 Determinin...

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  • March 29, 2024
  • 14
  • 2023/2024
  • Summary

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  • mathematics

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  • 12th Grade
  • Mathematics
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AP Mathematics - Notes
Complex Numbers


Content:
Complex Numbers:
1.1 Important Definitions and Notes for Complex Number
1.2 Multiplication, Addition and Subtraction
1.3 Division
1.4 Quadratic Formula
1.5 Determining the original quadratic equation from its roots
1.6 Solving Simultaneous Equations
1.7 Factorising
1.8 Solving Equations with Polynomials
1.9 Graphical Representation




Image from: https://www.linkedin.com/pulse/complex-numbers-window-higher-dimensional-thinking-reality-ganti


1.1 Complex Numbers
The square root of a negative number is called a non-real or imaginary number (𝑖).
Complex numbers are made up of two parts: 𝑧 = 𝑎 + 𝑏𝑖, where 𝑎 represents the real
part (𝑅𝑒(𝑧)) and 𝑏 represents the non-real/imaginary part (𝐼𝑚(𝑧)).




Pia Eklund 2024

, Some Important Notation:

𝑖 = √−1

𝑖 2 = (√−1)2 = −1

𝑖 3 = (√−1)3 = 𝑖 2 . 𝑖 = (−1). 𝑖 = −𝑖

𝑖 4 = (√−1)4 = 𝑖 2 . 𝑖 2 = (−1)(−1) = 1
∴ 𝑖 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 4 = 1


Worked Example 1
Given the complex numbers 𝑞 = −2 − 8𝑖 and 𝑠 = 5𝑖 + 9, find the following:
a) 𝑅𝑒(𝑞)
b) 𝑅𝑒(𝑠)
c) 𝐼𝑚(𝑞)
d) 𝐼𝑚(𝑠)


a) 𝑅𝑒(𝑞) = −2
b) 𝑅𝑒(𝑠) = 9
c) 𝐼𝑚(𝑞) = −8
d) 𝐼𝑚(𝑠) = 5



Worked Example 2
Determine the value of the following:
a) 𝑖 7
b) 𝑖 64
c) 𝑖 1999
d) 𝑖 2009


a) 𝑖 7
= 𝑖 4 . 𝑖 3 = (1)(−𝑖) = −𝑖

b) 𝑖 64
= 1 (recall: 𝑖 𝑎𝑛𝑦 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 4 = 1))




Pia Eklund 2024

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