1. Imaginary and complex numbers can be used to solve quadratic equations that have no real solutions. The square root of -1 is defined as i, and complex numbers have both a real and imaginary part. 2. An Argand diagram represents complex numbers graphically by plotting the real part on the x-axis and the imaginary part on a perpendicular y-axis. Common complex number operations like addition, subtraction, multiplication, and division can be performed by treating complex numbers as vectors. 3. The modulus of a complex number z = a + bi is the distance from the point (a, b) to the origin on the Argand diagram, and represents the absolute value of z. The argument of z,

1. 1
Imaginary and Complex Numbers
Solve the following equation:
𝑥!
− 4𝑥 + 13 = 0
The square root of -1 is defined as ‘i’ (for ‘imaginary’!)
−𝟏 = 𝒊
You found the question above had no real solutions. Using −1 = 𝑖 rewrite
the solutions to the quadratic above.
Notice that you have a real part and an imaginary part to your solutions.
What do you notice about your solutions?
If a real quadratic equation with Δ < 0, if a+bi is a complex root then
__________ is also a root.
Skills Practice
Solve for x:
𝑥!
− 10𝑥 + 29 = 0
2𝑥 +
1
𝑥
= 1
A complex number, z, is:

2. 2
The Number System
Define the following symbols:
ℕ
ℤ
ℚ
ℝ
Draw a Venn diagram to show the relationship between: ℕ, ℤ, ℚ, ℝ
How do you include irrational, imaginary and complex numbers in your
Venn diagram?

3. 3
Where do Imaginary Numbers belong on a number
line?
Any ‘real numbers’ can be thought of as existing on a number line,
which includes integers, fractions, square roots, π etc…
However as i is ‘imaginary’, we cannot put it anywhere on here:
Mathematicians decided to introduce an ‘imaginary number line’,
perpendicular to the standard one.
This is very similar to a set of coordinate axes (which is called ‘Cartesian’
after the mathematician Rene Descartes)
A set like this, with an imaginary number line, is known as an Argand
diagram, named after Jean-Robert Argand (although Mathematician Caspar
Wessel was actually the first to describe it)
Argand diagram
Imaginary
axis
1 2 3 4 5-5 -4 -3 -2 -1 6
i
2i
3i
4i
5i
6i
-i
-2i
-3i
-4i
0
Plot the following complex
numbers:
a) 3 + 4i
b) 5 – 2i
c) -4 - 4i
1 2 3 4 5-5 -4 -3 -2 -1 60
Real Axis

4. 4
Modulus of a Complex Number
The modulus of a complex number is the distance from (0,0) to P(x,y) (which
represents the complex number z = x+iy)
How would you find this distance 𝑧
If z = a+bi then we use the notation z* for the complex conjugate _________
Skills practice:
Imaginary axis
Real axis
z=x+iy
The modulus of a complex number is:
| 𝑧|

5. 5
Operations with Complex Numbers
Part A
Explain how you add, subtract, multiply and divide complex numbers.
Include an example in your explanations.
Adding two complex numbers Subtracting two complex numbers
Multiplying two complex numbers Dividing two complex numbers
Hint: use the notation z* for the complex
conjugate
Skills Practice

6. 6
Part B
Consider the complex numbers:
z1 = 2+i (let this be the vector u) and
z2 = 3+2i (let this be the vector v) and draw them on an Argand diagram
below. Find z1+z2 and call this w. What is the relationship between u,v and
w?
R
Im

7. 7
Argument of a Complex Number
We have seen the connection between complex numbers and vectors. Just
like vectors we can express a complex number in terms of the angle 𝜽
between the complex number and the x-axis. We call this the argument of z
or arg z.
Work out the angle 𝜽 (arg z) of the above diagram:
To avoid infinite number of solutions for the angle we restrict the domain for
arg z to – 𝜋 < 𝜃 ≤ 𝜋 in radians unless stated otherwise.
Imaginary axis
z=x+iy
Real axis
𝜽

8. 8
Working out arg z for – 𝝅 < 𝜽 ≤ 𝝅
When 𝟎 < 𝜽 ≤ 𝝅 , arg is positive
First quadrant
Arg z is positive
Second quadrant
Arg z is positive
𝜃 = 𝜋 − 𝛼
When – 𝝅 < 𝜽 < 𝟎 arg is negative
Third Quadrant
Arg z is negative
𝜃 = −(𝜋 − 𝛼)
Fourth Quadrant
Arg z is negative
Always draw a diagram when working out the arg (z)
Work out the modulus and argument for:
z1 = 3+4i, z2 =-3+4i, z3 = -5+12i and z4 = 5-12i
Use a separate diagram for each.
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Im
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Im
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Im
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Im