# The magnitude of a complex exponential

#### director

$$\displaystyle |e^{iwt}|$$ = 1

I've seen this expression a few times but without an explanation. Is the following true:

$$\displaystyle z = re^{iwt} = r(cos(wt) + isin(wt))$$
$$\displaystyle |z| = ( cos(wt)^{2} + sin(wt)^{2} )^{1/2} = (1)^{1/2} = 1$$

I don't have the $$\displaystyle i$$ included inside the square root though... Is that by convention? Because we're talking about something that's supposed to have a real value?

#### mr fantastic

MHF Hall of Fame
$$\displaystyle |e^{iwt}|$$ = 1

I've seen this expression a few times but without an explanation. Is the following true:

$$\displaystyle z = re^{iwt} = r(cos(wt) + isin(wt))$$
$$\displaystyle |z| = ( cos(wt)^{2} + sin(wt)^{2} )^{1/2} = (1)^{1/2} = 1$$ Mr F says: Yes (overlooking the fact that you have an r that disappears).

I don't have the $$\displaystyle i$$ included inside the square root though... Is that by convention? Because we're talking about something that's supposed to have a real value?

Have you studied complex numbers? If so, you should know that $$\displaystyle |a + ib| = \sqrt{a^2 + b^2}$$.

#### slovakiamaths

mod of a complex number suppose z=x+iy is equal to $$\displaystyle \sqrt{x^2+y^2}$$

so $$\displaystyle |z|=\sqrt{(rcoswt)^2+(rsinwt)^2}$$
|z|=r

#### director

Thank you for the replies.