Cube Roots of Unity

Transcript

0:00Introduction

Hello everyone! Okay, one small challenge before we start. Take the complex number minus half plus iota root three by two, and raise it to the power three hundred and thirty four. Will you multiply it out, three hundred and thirty four times? Oh, the exam will be over by then! By the end of today, you will do it in a few lines, using one Greek letter: omega. Today's topic: the cube roots of unity. And look, JEE has asked omega questions again and again. So, let's begin.

0:34Three cube roots of one

So, first question. Which numbers, when you cube them, give one? Only one itself, right? One cubed is one, that's it. Hmm, not so fast! Let's solve it. So, z cubed equals one, which means z minus one, times z squared plus z plus one, equals zero. Two brackets, you see? First bracket: z equals one. The second is a quadratic, and the formula gives z equals minus one, plus or minus iota root three, over two. Iota is the imaginary unit, the plus minus sign gives two roots, and root three is the square root of three. So not one cube root: three! Clear? The first one gets a special name: omega, this Greek letter. Omega equals minus one plus iota root three, over two. In exponential form, it is e to the power two pi iota by three; the slash in the power means divided by, okay? And the other root, minus one minus iota root three, over two? That is omega squared, which is e to the power four pi iota by three. It is also omega bar; the bar on top means the conjugate. So the three cube roots of unity are one, omega and omega squared. Got it?

1:54On the Argand plane

Now let's see them on the Argand plane, in the picture on the left. All three points sit on the unit circle, the circle of radius one. Join them up. What shape do you get? Yes, an equilateral triangle! One, omega and omega squared are its three corners, equally spaced on the circle. Beautiful, right? The right picture, with eight points, is the n-th roots of unity for n equals eight: that is our next lesson.

2:22Properties of ω: the golden rules

Now for the properties that do all the work. Omega solves z squared plus z plus one equals zero, remember? So tell me: what is one plus omega plus omega squared? Zero, exactly! So, the two golden rules: omega cubed equals one, and one plus omega plus omega squared equals zero. Every omega question starts here. Next, omega bar equals omega squared, equals one over omega, because omega times omega squared is omega cubed, which is one. And mod omega, with the two vertical bars, is one: omega sits on the unit circle. Now the power rule, for any integer k, okay? Omega to the power three k is one; three k plus one gives omega; three k plus two gives omega squared. Look, in simple words: divide the power by three, and only the remainder matters. Clear? Alright, now take the golden rule, and move omega squared to the other side. One plus omega equals omega squared, right? No, no, careful! When omega squared crosses the equal sign, its sign changes. One plus omega equals minus omega squared. Same way, one plus omega squared is minus omega, and omega plus omega squared is minus one. Don't lose that minus! And one more to remember: one minus omega, times one minus omega squared, equals three. Multiply it out with the golden rules, and check. Example one needs it.

3:53Topper's trick: spot ω in disguise

Now a trick that toppers use. Omega often comes in disguise. Whenever you see minus one plus iota root three, over two, write omega straight away; and with minus iota root three, write omega squared. With me? But what if the sign of the one flips? One plus iota root three, over two. Is that also omega? Not omega! Take out a minus sign: it is minus of, minus one minus iota root three by two. So it is minus omega squared. Same way, one minus iota root three, over two, is minus omega. Keep that in mind. And with no two below: minus one plus iota root three is simply two omega. Remember this one; our challenge from the start needs it.

4:42Example 1: omega to big powers, and products of 1 - omega

So, Example one: three quick parts, marked with small roman numerals. Part one: omega to the power two thousand twenty five, plus omega to the power two thousand twenty six. Part two: one minus omega plus omega squared, times one plus omega minus omega squared. Part three: one minus omega, times one minus omega squared, times one minus omega to the four, times one minus omega to the eight. Pause the video, and try it. Done? Okay, let's check. Two thousand twenty five is three times six hundred seventy five; the dot means times. So omega to the two thousand twenty five is one, and the next power is just omega. One plus omega, and by our pair rule, that is minus omega squared. Part two. Look at the first bracket: one minus omega plus omega squared. Can you write it as one single term? Yes! One plus omega squared is minus omega, so the bracket is minus omega, minus omega: minus two omega. Same way, one plus omega is minus omega squared, so the second bracket is minus two omega squared. Multiply: minus two times minus two is four, and omega times omega squared is omega cubed. So four omega cubed, which is just four. Nice and clean. Part three. Four brackets, with omega to the four and omega to the eight inside. What are these two powers, really? Divide by three and keep the remainder: omega to the four is omega, and omega to the eight is omega squared. So we have the same pair twice: one minus omega, times one minus omega squared, which is three. The big square bracket is squared, so the answer is three squared: nine.

6:33Example 2: omega in disguise to powers 334, 365 (IIT 1999)

Now the harder one: our challenge from the start, a real IITJEE question from nineteen ninety nine. Evaluate four, plus five times minus half plus iota root three by two, to the power three hundred thirty four, plus three times the same base, to the power three hundred sixty five. Pause and try. Hint: who is the base? Done? Okay, let's check. First, the base: minus half plus iota root three by two. Remember the topper's trick. Who is this, in disguise? Yes, omega! Over a common two, it is minus one plus iota root three, over two: exactly omega. Look at that, the scary number became one letter! Now the powers. Divide by three. Three hundred thirty four is three times one hundred eleven, plus one. Three hundred sixty five is three times one hundred twenty one, plus two. Only the remainders matter, right? So the first power is just omega, and the second is omega squared. The value is four, plus five omega, plus three omega squared. See, those huge powers are gone, in two lines! Can we make it cleaner? How do we get rid of omega squared? Golden rule again! Omega squared equals minus one minus omega. So three omega squared becomes minus three minus three omega. Four minus three is one, and five omega minus three omega is two omega. So, one plus two omega. Last step, the trick once more: two omega is minus one plus iota root three. So one plus two omega is one, minus one, plus iota root three: just iota root three. And we never multiplied the base even once. Challenge solved!

8:27Quick revision: the golden rules and big powers of omega

So, quick revision. Omega is minus one plus iota root three, over two: omega cubed is one, and one plus omega plus omega squared is zero. Big powers: divide by three, keep the remainder. And spot omega in disguise: minus one plus iota root three is two omega. Now go and try the practice problems on this page. All the best!