1. Complex numbers
Argand diagrams

2. 𝐼𝑚 3−𝑖 2 𝑅𝑒 2+𝑖 2 𝑅𝑒 3 𝑖 𝐼𝑚 4+𝑖 (2+3𝑖) 𝑅𝑒 1 1+𝑖 𝐼𝑚 1 1+𝑖
Complex numbers
KUS objectives
BAT Know how to represent and use Argand diagrams
Starter: Evaluate the following:
𝑅𝑒 2+𝑖 2
𝐼𝑚 3−𝑖 2
𝑅𝑒 3 𝑖
𝐼𝑚 4+𝑖 (2+3𝑖)
𝑅𝑒 𝑖
𝐼𝑚 𝑖

3. Real numbers can be represented on a number line
WB1 Argand diagrams
Real numbers can be represented on a number line
Complex numbers are represented on an Argand diagram
Real axis
Imaginary axis
4 + 3i
Named after Jean-Robert Argand, a Parisian mathematician and bookkeeper

4. Represent these complex numbers on an Argand diagram:
WB1 (cont) Argand diagrams
Real axis
Imaginary axis
Represent these complex numbers on an Argand diagram:
Conjugate z* is a reflection of z in the real axis
Argand diagram shows why, unlike real numbers on a number line, you cannot expect to use inequalities between complex numbers. On a number line greater than is to the right but there is no comparable relation between points on a plane
What do you notice about the position of a complex number and its conjugate?

5. You can use Pythagoras’ Theorem to find the magnitude of the distances
WB2 Represent the following complex numbers on an Argand diagram
𝑧 1 =2+5𝑖 𝑧 2 =3−4𝑖 𝑧 3 =−4+𝑖
Find the magnitude of |OA|, |OB| and |OC|, where O is the origin of the Argand diagram, and A, B and C are z1, z2 and z3 respectively
You can use Pythagoras’ Theorem to find the magnitude of the distances
y (Imaginary) z1 5i
√29 5
𝑂𝐴 = z3
√17 1
𝑂𝐴 = 29 2
x (Real) -5 4 3 5
𝑂𝐵 = 4 5
𝑂𝐵 =5 z2
𝑂𝐶 =
-5i
𝑂𝐶 = 17

6. 𝒂) 𝑧=1−2𝑖, Represent 𝑤=𝑧+(3+5𝑖) on an argand diagram
WB3 Add/subtract on an Argand diagrams
𝒂) 𝑧=1−2𝑖, Represent 𝑤=𝑧+(3+5𝑖) on an argand diagram
Real axis
Imaginary axis x x
‘Add 3 + 5i’ translates any point Z to a point W as in the diagram above
Can be confusing – both the points and the arrow represent complex numbers in different ways
Similarly the translation shown represents

7. Represent 𝑤+𝑧 on an argand diagram 𝒄) 𝑧=−2+3𝑖, 𝑤=8−3𝑖
WB3 Add/subtract on an Argand diagrams
𝒃) 𝑧=3+2𝑖, 𝑤=−4−5𝑖
Represent 𝑤+𝑧 on an argand diagram
𝒄) 𝑧=−2+3𝑖, 𝑤=8−3𝑖
Represent 𝑤−𝑧 on an argand diagram
Real axis
Imaginary axis
Can be confusing – both the points and the arrow represent complex numbers in different ways

8. Show z1, z2 and z1 + z2 on an Argand diagram
WB 𝑧 1 =4+𝑖 𝑧 2 =3+3𝑖
Show z1, z2 and z1 + z2 on an Argand diagram
y (Imaginary)
10i
𝑧 1 + 𝑧 2
= 4+𝑖 +(3+3𝑖)
=7+4𝑖
z1+z2 z2 z1
x (Real)
-10 10
-10i
Notice that vector z1 + z2 is effectively the diagonal of a parallelogram

9. Show z1, z2 and z1 - z2 on an Argand diagram
WB 𝑧 1 =2+5𝑖 𝑧 2 =4+2𝑖
Show z1, z2 and z1 - z2 on an Argand diagram
y (Imaginary) z1 5i
z1-z2
𝑧 1 − 𝑧 2 z2
2+5𝑖 −(4+2𝑖)
=−2+3𝑖
x (Real) -5 5
-z2
-5i
Vector z1 – z2 is still the diagram of a parallelogram
 One side is z1 and the other side is –z2 (shown on the diagram)

10. WB6 Argand diagram – multiplication by a real number
𝑧=2+𝑖 Represent w=3z on an argand diagram
Real axis
Imaginary axis x x
Can point out that we do not have a representation of a complex number multiplied by a complex number
This is similar to multiplication of real numbers – the relation between points Z and W is that W is 3 times as far from O as Z

11. WB7 Argand diagram – multiplication by conjugate
𝑧=2+𝑖 Represent w=z 𝑧 ∗ on an argand diagram
Real axis
Imaginary axis x x
Can point out that we do not have a representation of a complex number multiplied by a complex number
z 𝑧 ∗ = 2+𝑖 2−𝑖 = =5

12. KUS objectives
BAT Know how to represent and use Argand diagrams
self-assess
One thing learned is –
One thing to improve is –

13. END