Difference among Show as, Show with, and Show by

Question

Is there any differences among theses sentences? Which one is better?

We show the quantity with n.

We show the quantity as n.

We show the quantity by n.

Answer 1

I am making a wild guess that you mean "We use n to represent the quantity". If so, none of the three options clearly means that, though 2) and 3) could mean it with enough context.

I would say "We denote (or represent) the quantity by n".

Answer 2

There is a difference between these sentences.

1 and 3 appear to be saying that you are showing the quantity using 'n' to find said quantity.

2 means that 'n' is the quantity and you are displaying said quantity with the variable 'n'.

If you are attempting to say that you are finding the quantity using 'n' as a factor to find the value of the quantity, I would go with 1).

If you are attempting to say that 'n' is in fact the quantity and the said equation displays the quantity using 'n' as a placeholder variable, 2). would be your best bet