If $a$ is proportional to $b$ does it imply that $b$ is proportional to $a$?

Question

If $a$ is proportional to $b$ does it imply that $b$ is proportional to $a$? This is a simple question which I have found others giving different answers.

Answer 1

I've found different people mean different things when they use the word proportional. I always take it to mean the following: $$a\propto b \Leftrightarrow a=Cb, \ \ \ \ \text{(proportional)}$$ where $C$ is something that doesn't depend on $a$ or $b$. By the double arrow I mean the logic flows both ways. Now, you can invert this equation to get $$b=\frac{1}{C}a.$$ This last equation meets the same requirements as $b$ being proportianal to $a$.

In words, proportional to me means if you change one thing by some percentage or fraction, the other thing changes by the same percentage or fraction; for example, doubling one implies doubling the other.

Note that linear relationships do not have to be proportional: $$a=Cb+D \ \ \ \ \text{($a$ and $b$ linear, not proportional)}$$ Some people do use the word proportional when they really mean linear. However, even if $a$ is linear with respect to $b$, then $b$ is indeed linear with respect to $a$ as you can show using an exercise similar to the one above.