Complex Number (Argand diagram)Hi, Here is the question: z = 4 + 2i on an Argand Diagram. Verify |2z| = 2|z|

Expert Answers
cosinusix eNotes educator| Certified Educator

The coordinates of z in the diagram are (x=real part=4, y=imaginary part=2)


Plot the point  z of coordinates (4,2)


Plot the point 2z of coordinates (8,4)


1/2 z=0.5*(4+2i)=0.5*4+0.5*2i=2+i

Plot the point (1/2 z) of coordinates (2,1)


|2z|=sqrt(real part ^2+imaginary part ^2)=sqrt(8^2+4^2)=sqrt(80)=sqrt(16*5)=4sqrt(5)






sciencesolve eNotes educator| Certified Educator

You should find the complex number `2z`  such that:

`2z = 2(4 + 2i) =gt 2z = 8 + 4i`

Hence, evaluating the absolute value `|2z|`  yields:

`|2z| = sqrt(8^2 + 4^2) =gt |2z| = sqrt(64 + 16) = sqrt80 `

`|2z| = 4sqrt5`

You should evaluate `2|z|`  to compare its value to `|2z|, ` hence:

`|z| = sqrt(4^2 + 2^2) =gt |z| = sqrt(16+4) =gt |z| = sqrt20`

`|z| = 2sqrt5`

Multiplying by 2 both sides yields:

`2|z| = 4sqrt5`

Comparing  both 2|z| and |2z| yields that they are equal, hence |2z| = 2|z|.