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Question Video: Finding Common Roots of Unity Mathematics

How many of the 8th roots of unity are also 12th roots of unity?

02:39

Video Transcript

How many of the eighth roots of unity are also 12th roots of unity?

When we think about the eighth roots of unity, they’re the eight values for 𝑧 such that 𝑧 to the eighth power equals one. And similarly, the 12th roots of unity are all the values of 𝑧 such that 𝑧 to the 12th power equals one. To find the 𝑛th roots of unity, we have the general form cos of two πœ‹π‘˜ over 𝑛 plus 𝑖 sin of two πœ‹π‘˜ over 𝑛. This gives us the polar coordinates of each of the roots for π‘˜-values such that π‘˜ are all the integers between zero and 𝑛 minus one.

One method for solving this question would, of course, be to find all of the eighth roots of unity and all of the 12th roots of unity and then comparing the values. However, this method would give us 20 polar coordinates to compare. Again, the strategy would work, but it would be very time-consuming. Fortunately, we know something about common roots of unity. And that is this; the common roots of 𝑧 to the 𝑛 power minus one equals zero and 𝑧 to the π‘š power minus one equals zero are the roots of 𝑧 to the 𝑑 power minus one equals zero, where 𝑑 equals the greatest common divisor of π‘š and 𝑛.

We can rewrite our eighth roots of unity as 𝑧 to the eighth power minus one equals zero. And then we’ll rewrite our 12th roots of unity as 𝑧 to the 12th power minus one equals zero. From there, we can let 𝑛 equal eight and π‘š equal 12. And then the number of common roots will be 𝑑, and it will be equal to the greatest common divisor of 12 and eight. The largest factor of 12 and eight is four. So we can say that 𝑑 equals four. This answers the question, β€œhow many of the roots are common?” However, what if we wanted to find out what those common roots were? We would plug that 𝑑-value in such that 𝑧 to the fourth power minus one equals zero. And then we can say that all of the fourth roots of unity are shared between the eighth roots of unity and the 12th roots of unity.

Using the common roots of unity principle, not only are we able to find out how many shared roots two sets of roots would have, but we’re also able to identify which roots they are. In this case, we only needed to know how many common roots there were. And between the eighth roots of unity and the 12th roots of unity, they have four common roots.

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