Finding n th roots of a Complex Number Let w=cos(2 π/ n)+i sin(2 π/ n), where n is a positive in

Finding n th roots of a Complex Number Let w=cos(2 π/ n)+i...

Key Concepts

Roots of Unity

Roots of unity are the solutions to the equation z^n = 1 in the complex plane. They are evenly spaced around the unit circle, forming the vertices of a regular polygon. The concept of roots of unity is fundamental in various fields of mathematics, including number theory and abstract algebra, as they exhibit important symmetry properties and serve as a basis for understanding polynomial equations.

de Moivre's Theorem

de Moivre's Theorem states that for any real number ? and integer n, (cos ? + i sin ?)^n = cos(n?) + i sin(n?). This theorem is essential for raising complex numbers in polar form to a power and for finding the nth roots by reversing the process.

Complex Numbers

Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit satisfying i² = -1. They provide a way to extend the real numbers to include solutions to equations like x² + 1 = 0, and they play a crucial role in many areas of mathematics, particularly in the study of polynomials and their roots.

Polar Form Representation

Any nonzero complex number can be represented in polar form as r (cos ? + i sin ?), where r is the modulus (or absolute value) of the complex number and ? is the argument (or angle). This form is especially useful in multiplication, division, and finding powers or roots of complex numbers.

Euler's Formula

Euler's formula states that e^(i?) = cos ? + i sin ?. This powerful identity provides a deep connection between complex exponentials and trigonometric functions, and it is fundamental in expressing complex numbers in exponential form, which simplifies the computation of powers and roots.

Transcript

00:01 All right, so they have made a claim about a number n and about the nth roots of one.

00:07 So for now, let's just say n is 5, and we'll go through all the reasoning where n is 5, and then we'll come back to making it more abstract where n can be anything.

00:17 So they're saying, let this omega or w, whatever, let that be cosine of 2 pi over 5 plus i sine of 2 pi over 5.

00:27 And then what they're saying is that these are the nth roots of one.

00:31 1, omega, omega squared, omega cubed, and omega to the 4.

00:40 Right.

00:40 They're saying that those five numbers are the five roots of one, the five fifth roots of one, meaning that if you take any of those things and you raise them to the fifth power, it equals one.

00:53 Well, let's check that that makes sense, right? and we'll use our knowledge of complex numbers and their angles to help us.

01:02 In other words, we'll be using demwaverous theorem.

01:05 So let's start with the first one.

01:07 This one's pretty easy to check.

01:08 Is it true that 1 to the 5 is 1? well, yes, we know that one already.

01:14 So we can check that, yes, this really is one of the fifth roots.

01:18 Whoops, that didn't really show up too well.

01:20 This really is one of the fifth roots of 1.

01:22 All right, what about omega to the 5? if we take omega and we raise it to the fifth power, supposedly that's going to work out to be one as well.

01:33 Well, let's see.

01:34 Omega is all of that stuff over there, cosine of 2 pi over 5 plus isign of 2 pi per 5.

01:40 And according to dumois theorem, when we take a complex number like that in polar form and we raise it to the fifth power, that 5 just comes inside of the angle like this.

01:53 The 5 multiplies in here.

02:00 Right that's de mavra's theorem and so you'll notice that the five we just multiplied by and the five we were dividing by cancel each other out and so this is just cosine of two pi plus i sign of two pi uh and that is one because cosine of two pi is one and sine of two pi is zero so sure enough omega to the fifth power is one as well okay what about omega to the second power well first of all what is omega to the second power? well, we can use de mawver's theorem for that too.

02:44 It should be the same as omega, but where the two is multiplied in to that angle.

02:51 So that should be 4 pi over 5 plus i sign of 4 pi over 5.

03:03 All right.

03:05 So then taking that to the 5th power, what is omega squared to the 5th power? well, that should just be what we just had, cosine of 4 pi over 5, but with a 5 multiplied into the angle.

03:24 And again, that's just a moverous theorem that whenever you raise a complex number to any power, it just multiplies into the angle there.

03:37 So the same thing happens.

03:39 The 5s cancel out, and it's just cosine of 4 pi plus i sine of 4 pi.

03:46 And if you think about what 4 pi is, right, that's just instead of having gone around the circle once, you've gone around it twice.

03:54 So you're really just in the same spot.

03:56 This is once again one.

03:59 So we have verified that this is one as well.

04:03 When you raise that number to the fifth power, it equals one.

04:06 So it really is a fifth root of one.

04:09 And you can see that the same thing will happen for the other ones too.

04:12 W cubed would just be w, you know, with the angle multiplied by three.

04:18 So that would be 6 pi.