Question

How do I use the modulus and argument to square `(1+i)`?

Answer 1

You should use the de Moivre's Theorem. See explanation.

Explanation:

For any complex number `z` given in trigonometric form (having modulus `|z|` and argument `varphi`) we can calculate any natural power using the formula:

`z^n=|z|^n*(cosnvarphi+i*sinnvarphi)`

Here we get:

`|z|=sqrt(1^2+1^2)=sqrt(2)`

`cosvarphi=(Re(z))/|z|=1/sqrt(2)=sqrt(2)/2=> varphi=45^o`

Now we can calculate the square:

`z^2=|z|^2*(cos(2*45)+isin(2*45))=`

`=2*(cos90+isin90)=2*(0+1i)=2i`