One formula that is used frequently to rewrite a complex number is the Euler Formula. The Euler Formula can be used to convert a complex number from exponential form to rectangular form and back. The Euler Formula is closely tied to DeMoivre's Theorem, and can be used in many proofs and derivations such as the double angle identity in trigonometry.
I want to talk about a really important formula in Mathematics called the Euler formula. And it is pronounced euler like Floyd. r e to the i theta equals r times cosine theta plus i sine theta. You probably recognize this as the trig form of a complex number. It turns out that the trig form can also be written this way, and it's much much more compact. So use the Euler formular to evaluate these numbers. So e to the i pi. According to the Euler formular this would be like 1 times e to the i pi. So r would be 1 and I'd get 1 times cosine of pi, plus i sine pi. Cosine of pi is -1 and sine of pi is 0. So this is just -1. e to the i pi is negative 1. That's actually a very famous identity. Now here is another one. 8 times e to the i pi over 2. This will be 8, right that's the r value. Turns out it's the modulus of the complex number. Cosine of pi over 2 plus i sine of pi over 2. Cosine of pi over 2 0 and sine of pi over 2 is 1. So this is 8 times 0 plus i times 1, 8i. So 8e to the i pi over 2 is 8i. Let's try another. 10e to the minus i pi over 3. This will be 10 times cosine of pi- of negative pi over 3, sorry about that. Negative pi over 3, plus i sine of negative pi over 3. That's going to equal 10 times the cosine of negative pi over 3 is the same as the cosine of pi over 3 which is a half. So one half plus i times and the sine of negative pi over 3 is the opposite of the sine of the sine of pi over 3. So negative root 3 over 2. And so this becomes 10 times a half, 5 plus 10 times root negative root 3 over 2 is -5 root 3. So minus i times 5 root 3. So 10e to the minus i pi over 3 is 5-i times 5 over 3. What about this one? r is root 2 here. We get cosine of 3 pi over 4 plus i sine of 3 pi over 4. Now cosine of 3 pi over 4 is negative root 2 over 2. And the sine of 3 pi over 4 is positive root 2 over 2. So this is root 2 times negative root 2 over 2 plus i times root 2 over 2. Then we get -2 over 2, -1 plus root 2 times root 2 over 2 is 2 over 2 which is 1 times i. So this number is the same as -1 plus i, and so what you may have gathered here is that all complex numbers can be written in the form of z=r e to the i theta. This is equivalent to the trig form z=r cosine theta plus i sine theta. And you'll find if you if you take advance Math classes that this is the preffered way to express comlex numbers. Very very short and so synched, easy to raise the powers easy to multiply and divide. This is a very nice form for complex numbers.