I want to show you how the Euler formula can be used to prove some nice results from complex numbers; such as a multiplication formula. When we multiply two complex numbers, like z1 and z2. z1 equals r cosine theta plus i sine theta and z2 equals s cosine phi plus i sine phi.
What we can do is first switch to exponential form, by way of the Euler formula. Our cosine theta plus i sine theta is re to the i theta. So this is the re to i theta. And z2 is se to the i phi. So multiplying these two z1 times z2, I get re to the i theta times se to the i phi.
And I can pull the s in front right, the associate of property of multiplication. Rs and then I have e to the i theta e to the i phi. And it’s a property of exponents, that when you have the same base you can combine these and add the exponents. So I get rs e to the i theta plus i phi.
Now let me factor out the i, I get r s e to the i times theta plus phi. Now I’ve gotten a form where I can actually convert back to trig form. So the modulus rs goes in front, and the e to the i theta plus phi becomes cosine theta plus phi, plus i sine theta plus phi.
And this is the result we had before. When you multiply two complex numbers, you multiply the moduli and you add their arguments. Let’s see how this works for Demoivre’s theorem.
Now, if we have the complex number z equals r cosine theta plus i sine theta, then we can easily just switch to re to the i theta. And so z to the n, becomes re to the i theta, to the nth power. Now I can separate these two. I can write these as r to the n times e to the i theta to then n. And then by using the power to the power property, I get e to the i n theta times r to the n.
Now let me use the Euler formula to convert this back into trig form. Modulus is r to the n, and the e to the i n theta becomes cosine of n theta plus i sine of, again, n theta. This is exactly the Demoivre’s theorem.
Demoivre’s theorem says z to the n is r to the n, times cosine n theta plus i sine theta. So proving this result is really easy, if you make the conversion from regular trig form, to exponential form by using the Euler formula.