Question

State the modulus and argument of each complex number. Explain how you know.

a. 4 + 0i
b. −2 + 2i

Answer 1

a)
`|z|=4`,
`arg\ z=0\ radians`

b)
`|z|=2.82`,
`arg\ z=2.35`

Explanation:

a)
`z=4+0i`

Modulus

`|z|=√(4^2+0^2)\\\Rightarrow |z|=√(16)\\\Rightarrow |z|=4`

`|z|=4`

The point is in the first quadrant

Argument

`arg\ z=tan^(-1)(0)/(4)\\\Rightarrow arg\ z=tan^(-1)0\\\Rightarrow arg\ z=0`

`arg\ z=0\ radians`

b)
`z=-2+2i`

Modulus

`|z|=√((-2)^2+2^2)\\\Rightarrow |z|=√(8)\\\Rightarrow |z|=2.82`

`|z|=2.82`

The point is in the second quadrant

Argument

`arg\ z=tan^(-1)(2)/(-2)+\pi\\\Rightarrow arg\ z=tan^(-1)-1+\pi\\\Rightarrow arg\ z=-0.7853+\pi\\\Rightarrow arg\ z=2.35`

`arg\ z=2.35`