# Quotient of Homogeneous Functions

## Theorem

Let $M \left({x, y}\right)$ and $N \left({x, y}\right)$ be homogeneous functions of the same degree.

Then:

$\dfrac {M \left({x, y}\right)} {N \left({x, y}\right)}$

is homogeneous of degree zero.

## Proof

Let:

$Q \left({x, y}\right) = \dfrac {M \left({x, y}\right)} {N \left({x, y}\right)}$

where $M$ and $N$ are homogeneous functions of degree $n$.

Let $t \in \R$. Then:

 $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle Q \left({tx, ty}\right)$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle \frac {M \left({tx, ty}\right)} {N \left({tx, ty}\right)}$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle \frac {t^n M \left({x, y}\right)} {t^n N \left({x, y}\right)}$$ $$\displaystyle$$ $$\displaystyle$$ as $M$ and $N$ are homogeneous of degree $n$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle t^0 \frac {M \left({x, y}\right)} {N \left({x, y}\right)}$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle t^0 Q \left({x, y}\right)$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle Q \left({x, y}\right)$$ $$\displaystyle$$ $$\displaystyle$$

The result follows from the definition.

$\blacksquare$