Typed and summarized IEB AP Math notes - does not include statistics notes. Includes worked examples and important concepts to note.
Received a distinction in AP Maths at the end of Grade 11 due to the basis of these notes. Content includes:
Complex Number Theory, Mathematical Induction, Radian...
(Exercise 1).
In general, complex number ℤ can
Integers (ℤ) be written in the form: ℤ = 𝑎 + 𝑏𝑖
𝑎, 𝑏 are real numbers.
✓ Examples of Non-real numbers: 𝑎 = real part of the complex
number, can be written as 𝑅𝑒(𝑧) =
√ −1 ; √ −3.
✓ Examples of irrationals: 𝑎.
𝑏 = imaginary part of the complex
√2 ; 𝜋 ; √5
number, can be written as 𝐼𝑚 (𝑧) =
The concept of non-real numbers: 𝑏.
➢ The square root of a negative Multiplication, addition and subtraction of
number is called a non-real or complex numbers:
imaginary number.
a) (5 − 3𝑖)2
Some notation: = 25 − 30𝑖 + 9𝑖 2
= 25 − 30𝑖 + 9(−1)
❖ The internationally accepted
= 25 − 30𝑖 − 9
notation for √ −1 is the letter 𝑖.
= 16 − 30𝑖
❖ We say that 𝑖 = √ −1. b) (3 + 2𝑖 )(3 − 2𝑖 ) − 𝑖 2 (𝑖 4 − 1)
❖ If we square both sides of the = 9 − 4𝑖 2 − 𝑖 6 + 𝑖 2
equality we get: 𝑖 2 = (√ −1)2 = = 9 − 4(−1) − 𝑖 4 . 𝑖 2 − 1
−1 = 9 + 4 − (1)(−1) − 1
❖ It is easy to verify that = 13
𝑖 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 4 = 1
❖ Notice that 𝑖 is a non-real solution (Exercise 2).
of the equation: 𝑥 2 + 1 = 0. If you
,Complex conjugates and division of represented on the Argand plane as
complex numbers: follows:
▪ If 𝑧 = 𝑎 + 𝑏𝑖 is a given complex
Imaginary Axis
number where a and b are real
numbers and 𝑖 = √ −1, then the 3𝑖 𝑧 = 4 + 3𝑖
complex number 𝑧 ∗= 𝑎 − 𝑏𝑖 is 𝑅𝑒(𝑧) = 4
called the complex conjugate of
𝐼𝑚 (𝑧) = 3
𝑧 = 𝑎 + 𝑏𝑖.
Examples:
Write down the complex conjugates of the
following complex numbers: Real Axis 4
a) 𝑧 = 3 + 2𝑖
𝑧 ∗= 3 − 2𝑖
o Any complex number written
b) 𝑧 = 5𝑖 − 6
in the form 𝑧 = 𝑎 + 𝑏𝑖 is said
𝑧 ∗= −5𝑖 − 6
to be in rectangular form (or
4+𝑖 Cartesian form).
Write 𝑧 = 2−𝑖 in the form 𝑧 = 𝑎 + 𝑏𝑖 and
then state 𝑅𝑒(𝑧), 𝐼𝑚(𝑧) 𝑎𝑛𝑑 𝑧 ∗. The modulus of a complex number:
4+ 𝑖 2+ 𝑖 • The modulus of a complex number
𝑧= ×
2− 𝑖 2+ 𝑖 𝑧 is is the distance of 𝑧 from the
(4 + 𝑖 )(2 + 𝑖 ) origin.
𝑧= • We write the modulus of 𝑧 as |𝑧| or
(2 − 𝑖 )(2 + 𝑖 )
simply as 𝑟.
8 + 6𝑖 + 𝑖 2 • Consider the number 𝑧 = 𝑥 + 𝑦𝑖.
𝑧=
4 − 𝑖2
7 + 6𝑖
𝑧= 𝑧 = 𝑥 + 𝑦𝑖
5
7 6 𝑟 = |𝑧|
𝑧= + 𝑖
5 5
[𝐼𝑚(𝑧)]
7 6 7 6
𝑅𝑒(𝑧) = ; 𝐼𝑚(𝑧) = ; 𝑧 ∗= − 𝑖
5 5 5 5
[𝑅𝑒(𝑧) ]
(Exercise 3).
From Pythagoras we know that:
Complex (Argand) Planes:
𝑟2 = 𝑥 2 + 𝑦 2
o Complex numbers can be
represented by points in a plane 𝑟 = √𝑥 2 + 𝑦 2
called the complex plane or argand
plane. |𝑟| = √𝑥 2 + 𝑦 2
o Consider the complex number 𝑧 =
4 + 3𝑖. This number can be Examples:
Determine |𝑧| if 𝑧 = 4 + 3𝑖
, 2
Solution: 𝑟2 = (1)2 + (√3)
𝑟=2
𝑧 = 4 + 3𝑖 3. Using any of the three
𝑟 trigonometric ratios, we can
calculate the size of 𝜃.
√3
sin 𝜃 =
2
𝑟2 = (4)2 + (3)2
𝜃 = 60°
𝑟2 = 25
∴ arg(𝑧) = 60°
𝑟=5
(Exercise 4).
|𝑧| = 5
The modulus-argument form of a complex
The argument of z: number:
1. The argument of a complex a) Consider the following diagram
number 𝑧 is the angle between the representing the complex number
real axis and the line joining 𝑧 to 𝑧 = 𝑥 + 𝑦𝑖.
the origin. We refer to the
argument of 𝑧 as arg (𝑧).
𝑧 = 𝑥 + 𝑦𝑖
𝑟
𝑦
𝜃
𝑧 = 𝑥 + 𝑦𝑖 𝑥
𝑟 = |𝑧| b) We know from Trig that:
𝑥
𝜃 = arg (𝑧) = cos 𝜃 ; ∴ 𝑥 = 𝑟 cos 𝜃
𝑟
𝑦
and = sin 𝜃 ; ∴ 𝑦 = 𝑟 sin 𝜃
𝑟
2. We can use the following c) We can now rewrite 𝑧 = 𝑥 + 𝑦𝑖 as
trigonometric ratios to calculate the
follows:
size of 𝜃:
𝑦 𝑥 𝑦 = (𝑟 cos 𝜃) + 𝑖 (𝑟 sin 𝜃)
sin 𝜃 = ; cos 𝜃 = ; tan = 𝑟 cos 𝜃 + 𝑖𝑟 sin 𝜃
𝑟 𝑟 𝑥
= 𝑟(cos 𝜃 + 𝑖 sin 𝜃)
Example: d) 𝑧 = 𝑟(cos 𝜃 + 𝑖 sin 𝜃) is called the
Calculate arg (𝑧) if 𝑧 = 1 + (√3)𝑖 modulus-argument form or the
polar form of a complex number 𝑧.
Examples:
𝑧 = 1 + (√3)𝑖
Write 𝑧 = 2(cos 60° + 𝑖 sin 60°) in
𝑟 rectangular form 𝑧 = 𝑎 + 𝑏𝑖:
√3
𝜃
1
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