Question

Complex 'fractions'

Answer 1

can't wait to see this!

Answer 2

Since \[\frac{a}{b} = a^1 b^{-1}\] We can see that in the complex plane these exponents are 180 degrees apart.
\[1 = e^{i \pi*0}\]\[-1 = e^{i \pi * 1}\]

Similarly, does a number with exponents that are 120 degrees apart have meaning?
\[\omega = e^{i 2 \pi /3}\]

\[a^1 b^\omega c^{\omega^2}\]

Answer 3

My instinct is to say that these could represent a ratio of things like:
\[A \cdot H_2O \longrightarrow B \cdot H_2 + C \cdot O_2\]

Of course we already have linear algebra for that, although idk if it would quite be the same or not or if this is useful or interesting. Anyways just throwing this out there see if anyone knows or has any ideas.

Answer 4

A cool property you have is cancellation like in normal fractions.

\[\frac{ax}{ay} = \frac{x}{y}\]

We can see this from the complex view as:
\[(ax)^1(ay)^{-1} = a^{1-1} x^1y^{-1}=x^1y^{-1}\]

Which for a "three way fraction" gives us:

\[(ax)^1(ay)^\omega(az)^{\omega^2} = a^{1+\omega+\omega^2}x^1y^\omega z^{\omega^2}=x^1y^\omega z^{\omega^2}\]

So pretty cool.

Answer 5

Similarly with fractions when you multiply them their tops and bottoms combine, so maybe graphically it will be more interesting to write it: Also I'll draw out the thing like we had before: