Pure 1.2 - Argand diagrams

Question 1

What are Argand diagrams

Answer 1

It is possible to represent complex numbers on an Argand diagram. The 𝑥-axis on an Argand diagram is called the real axis and the 𝑦-axis is called the imaginary axis.
The complex number 𝑧=𝑥+i𝑦 can be represented on an Argand diagram by the point 𝑃(𝑥,𝑦) where 𝑥 and 𝑦 are the cartesian coordinates.
The complex number 𝑧=𝑥+i𝑦 can be represented on an Argand diagram by the vector (ˣᵧ).

Question 2

Describe the modulus of a complex number.

Answer 2

The modulus or absolute value of a complex number is the magnitude of the corresponding vector on an Argand diagram.
The modulus of a complex number, |𝑧|, is the distance from the origin to that number on an Argand diagram. For a complex number 𝑧=𝑥+i𝑦, the modulus is given as; |𝑧|=√(𝑥²+𝑦²).

Question 3

Describe the argument of a complex number.

Answer 3

The argument of a complex number is the angle its corresponding vector makes with the positive real axis.
The argument of a complex number, arg(𝑧), is the angle between the positive real axis and the line joining that number to the origin on an Argand diagram, measured in an anticlockwise direction. For a complex number 𝑧=𝑥+i𝑦 the argument, θ, satisfies tan(θ)=𝑦/𝑥.
The argument θ of any complex number is usually given in the range -π

Question 4

How can the principle argument of a complex number be calculated by the acute angle between the vector of the complex number and from the origin and the real axis?

Answer 4

Let α be the positive acute angle made with the real axis by the line joining the origin and 𝑧.
If 𝑧 lies in the first quadrant then arg(𝑧)=α
If 𝑧 lies in the second quadrant then arg(𝑧)=π-α
If 𝑧 lies in the third quadrant then arg(𝑧)=-(π-α)
If 𝑧 lies in the fourth quadrant then arg(𝑧)=-α

Question 5

Describe the modulus-argument form of complex numbers.

Answer 5

For a complex number, 𝑧, with |𝑧|=𝑟 and arg(𝑧)=θ, the modulus-argument form of 𝑧=𝑟(cos(θ)+isin(θ)).

Question 6

Describe for any 2 complex numbers, 𝑧₁ and 𝑧₂, what the values of |𝑧₁𝑧₂|, |𝑧₁/𝑧₂|, arg(𝑧₁𝑧₂) and arg(𝑧₁/𝑧₂) are.

Answer 6

|𝑧₁𝑧₂|=|𝑧₁||𝑧₂|
|𝑧₁/𝑧₂|=|𝑧₁|/|𝑧₂|
arg(𝑧₁𝑧₂)=arg(𝑧₁)+arg(𝑧₂)
arg(𝑧₁/𝑧₂)=arg(𝑧₁)-arg(𝑧₂)

Question 7

Describe how complex numbers can be used to describe the locus of points on an Argand diagram.

Answer 7

For two complex numbers 𝑧₁=𝑥₁+i𝑦₁ and 𝑧₂=𝑥₂+i𝑦₂, |𝑧₂-𝑧₁| represents the distance between the points 𝑧₂ and 𝑧₁ on an Argand diagram.
Given 𝑧₁=𝑥₁+i𝑦₁, the locus of point 𝑧 on an Argand diagram such that |𝑧-𝑧₁|=𝑟, or |𝑧-(𝑥₁+i𝑦₁)|=r, is a circle with centre (𝑥₁,𝑦₁) and radius 𝑟.
Given 𝑧₁=𝑥₁+i𝑦₁ and 𝑧₂=𝑥₂+i𝑦₂, the locus of points 𝑧 on an Argand diagram such that |𝑧-𝑧₁|=|𝑧-𝑧₂| is the perpendicular bisector of the line segment joining 𝑧₁ and 𝑧₂.
Given 𝑧₁=𝑥₁+i𝑦₁, the locus of points 𝑧 on an Argand diagram such that arg(𝑧-𝑧₁)=θ is a half-line from, but not including, the fixed point 𝑧₁ making an angle θ with a line from the fixed point 𝑧₁ parallel to the real axis.

Question 8

What is a half-line?

Answer 8

A half-line is a straight line extending from a point infinitely in one direction only.