-1

As far as I have learnt at the school level the meaning of $\omega$ is the complex cube root of unity that is ($\omega$)³=1, i.e. ($\omega$-1)[($\omega$)²+$\omega$+1]=0. But are there any other definitions of $\omega$ possible in mathematics? Please help me out.

Shivam Kumar
  • 381
  • 1
    it would be more correct to ask "is there any other use of the symbol $\omega$". It's often used to denote an infinite ordinal. – Brevan Ellefsen Oct 16 '23 at 04:14
  • 1
    $\omega$ is used in physics (and partial differential equations) as angular frequency, so waves can be expressed as $\cos(\omega t)$ – David Raveh Oct 16 '23 at 04:17
  • By your definition, Shivam, $\omega$ couold be $1$. Better to say $\omega^2+\omega+1=0$, though even that allows for two possibilities. Alternatives are $\omega=e^{2\pi i/3}$ and $\omega=-(1/2)+i(\sqrt3/2)$. – Gerry Myerson Oct 16 '23 at 04:36
  • I posted an answer a few days ago, Shivam. Any thoughts about it? – Gerry Myerson Oct 26 '23 at 10:39

1 Answers1

2

The question is not clear. If you are asking whether there is any way to express a cube root of unity other than as a solution of $x^3=1$, then here are a couple: $e^{2\pi i/3}$, and $-(1/2)+i\sqrt3/2$, where $i$ is a square root of minus one.

If you are asking whether the symbol $\omega$ is ever used to mean something other than a cube root of unity, then several answers have been given in the comments. Another one comes from Number Theory, where $\omega(n)$ is used for the number of distinct prime divisors of $n$. For example, $\omega(72)=2$, since there are exactly two different primes dividing $72$, namely, $2$ and $3$. See https://en.wikipedia.org/wiki/Prime_omega_function

Gerry Myerson
  • 179,216