# Are the letter choices in algebra meaningful?

p= F−E(S) / k

I saw someone post a math problem like the one above. I wonder why they used those letters, if they have a particular meaning or if they could be swapped out for different letters and mean the same thing?

Example:
X= A−B(C) / D

• Welcome to MSE. You should choose your tags carefully. What has this to do with linear-algebra? – José Carlos Santos Oct 19 '19 at 6:52
• It seems to me that algebra-precalculus or notation are natural choices for that question. – José Carlos Santos Oct 19 '19 at 6:54

Simple answer: yes, you could swap the letters and it would mean the same.

There is no standards body for mathematics so you can do what you want. However, there are some very common usages. If you go against these, it won't invalidate your work but it may surprise or confuse others.

Some examples:

A single unknown real number is very commonly $$x$$. A second or third is quite likely $$y$$ or $$z$$. For more than three, subscripts are common: $$x_1$$, $$x_2$$, ...

A single unknown complex number is very commonly $$z$$.

$$a$$, $$b$$, $$c$$ are common for fixed values.

Greek letters are very commonly used and some have a much more restricted use. A famous one is $$\pi$$ which almost always the circle constant but is occasionally used for other purposes. $$\Sigma$$ is very commonly used for sums. Others have typical but less fixed uses e.g. $$\theta$$ is popular for angles. Beginners sometimes get the impression that Greek letters have fixed meanings but they don't, it is just that the habits seem a little stronger with them.

Other scripts are rare but the Hebrew letters $$\aleph$$ and $$\beth$$ are used for infinite cardinal numbers.

Upper case is popular for sets e.g. a single arbitrary set is commonly $$S$$ rather than $$s$$. A group is commonly $$G$$ and $$g$$ is typically used for an element of it.

Over in physics and chemistry, the habits are much more rigid. For example, voltage is pretty much always $$V$$ or $$v$$ and current $$I$$ or $$i$$. If you chose to use $$I$$ for voltage and $$V$$ for current so that Ohm's law was $$I = VR$$ then you would cause lots of confusion and probably consternation. The $$i$$ for current habit is so strong that $$j$$ is sometimes used for $$\sqrt-1$$ rather than $$i$$ as is common in mathematics.

Context matters. In mathematics, you can take a symbol, and replace it with any other unique symbol, so long as you do it everywhere in context. For example, consider the identity $$\sin^2(x) + \cos^2(x) = 1$$ for all $$x \in \Bbb{R}$$. The context of the symbol $$x$$ is limited to the above equation only; there's no reason to think that if $$x$$ is used again (e.g. maybe the next line is $$\cos(2x) = 1 - \sin^2(x)$$) that the new $$x$$ and the old $$x$$ have to refer to the same number. We essentially used $$x$$ just to express the equation, and then was done with it. We would be absolutely fine to write, instead, $$\sin^2(f) + \cos^2(f) = 1$$ for all $$f \in \Bbb{R}$$, and it would mean the same thing.

However, remember that the context of certain symbols transcends the current proof. If we were to change, say, the symbol $$\sin$$ to the symbol $$\operatorname{ppp}$$, then writing $$\operatorname{ppp}^2(x) + \cos^2(x) = 1$$ would suddenly make much less sense. Why? Because whoever wrote the identity has made some safe assumptions that they know what symbols like $$\sin, \cos,$$ and even $$+, =, 1$$ and $$2$$ mean beforehand. That is, the context in which they are defined stretches beyond the given argument. If you wanted to introduce $$\operatorname{ppp}$$ as a replacement for $$\sin$$, you would have to define it (just as $$\sin$$ itself has its well-known definitions).

So, essentially, you are permitted to make symbol substitutions so long as you do so consistently in their entire relevant context (all the way back to the definition/quantifier that introduced the symbol). In some circumstances, the context extends beyond the current argument, in which you have to define the symbol somewhere in your argument.

Sometimes, but not usually. A variable refers to an unknown number, and occasionally we want to refer to multiple, possibly different, numbers.

Typically, any symbol will work: X-1=2 is the same problem as Y-1=2.

But sometimes we want to track multiple different things. Like if I am solving a system of equations, I want two different names for two different variables.

Finally, when our math refers to particular things in the real world, we name variables to note that. If Apple and Google both have the same linearly scaling profit function, I might write A=5x+7 and G=5x+7. Those mean the same things mathematically, but if I’m running google I care a lot more about the second one, because the G refers to google’s profits.