Video Transcript
Let 𝜔 be the primitive cubic
root of unity. 1) Find 𝜔 plus 𝜔 squared. 2) Find 𝜔 minus 𝜔
squared. 3) What is 𝜔 plus one and how
is it related to the other roots of unity? 4) What is 𝜔 squared plus one
and how is it related to the other roots of unity?
To answer part one, we could
try writing 𝜔 and 𝜔 squared in algebraic form and finding their sum that
way. Alternatively, we recall that
𝜔 squared is the same as the conjugate of 𝜔. And this means that 𝜔 plus 𝜔
squared is equal to 𝜔 plus the conjugate of 𝜔. This has its own property. We know that the sum of a
complex number and its conjugate is two times the real part of that complex
number. So 𝜔 plus the conjugate of 𝜔
is two times the real part of 𝜔.
Well, the real part of the
primitive cubic root of unity is negative one-half. And two times negative a half
is negative one. So we can see that 𝜔 plus 𝜔
squared is equal to negative one. This also means that 𝜔 squared
plus 𝜔 plus one is equal to zero. Notice that these are the three
cubic roots of unity. And we’ve shown that their sum
is equal to zero.
Let’s repeat this process for
part two. Once again, we express 𝜔
squared as the conjugate of 𝜔. But this time, the difference
between a complex number and its conjugate is two 𝑖 times the imaginary part of
that complex number. The imaginary part of 𝜔 is
root three over two. So the difference between 𝜔
and 𝜔 squared is 𝑖 root three or root three 𝑖. And we can use what we’ve
calculated here to work out 𝜔 plus one for part three and 𝜔 squared plus one
for part four. We saw that 𝜔 squared plus 𝜔
plus one is equal to zero. So let’s subtract 𝜔 squared
from both sides of this equation. When we do, we see that 𝜔 plus
one is equal to negative 𝜔 squared. Similarly, we can also deduce
that 𝜔 squared plus one is equal to negative 𝜔.
So the cubic roots of unity
have three really important properties. We know that 𝜔 squared is
equal to the conjugate of 𝜔. We know that the sum of the
three cubic roots is equal to zero. And we know that 𝜔 minus 𝜔
squared is equal to 𝑖 root three.
