Question

|COMPLEXE NUMBERS| What is the geometrical representation of |z| = 2? Thank you!

Hello! I don't get this because I thought you needed an angle to solve this? Obviously, that is probably not the case...

Answer 1

Circle

Explanation:

Consider
Z = x + iy
therefore |z| = `(x^2+y^2)^(1/2)`
also |z| = 2
hence `x^2+y^2=4`
therefore it represents a circle
Hope u find it helpful :)

Answer 2

`x^2+y^2=4`

circle centre `(0,0) " radius "=2`

Explanation:

`|z|=2`

`z=x+iy`

so replace `z" with "x+iy`

`:.|z|=|x+iy|`

by definition

`|x+iy|=sqrt(x^2+y^2`

`|z|=2=>sqrt(x^2+y^2)=2`

`x^2+y^2=4`

circle centre `(0,0) " radius "=2`

Answer 3

For any complex number `z`, to calculate its modulus `|z|`, the number needs to multiplied with its complex conjugate to obtain `|z|^2`.

Let `z` be of the type `(x+iy)`, where `xand y` are real numbers. Inserting in LHS of given equation we get

`|z|^2=zz^"*"` `=(x+iy)(x-iy)=x^2+y^2`

Now the given equation becomes

`sqrt(x^2+y^2)=2`

Squaring both sides we get

`x^2+y^2=4`

The equation represents a circle with center at the origin and radius `=2.`