# How to signal to the reader the difference between a function and a multiplication?

The following is alpha of t times x:

$$\alpha(t)x$$

The following is alpha times t times x:

$$\alpha(t)x$$

My instructor had one interpretation, I used the other. ;)

Is there an easy or standard way to signify that we're talking about a function rather than a variable multiplication here?

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Just don't have both a function and value $\alpha$. – Raphael Nov 18 '11 at 19:37
@Raphael: If it was my choice, I'd never use greek letters for these things at all. It's not though. – Billy ONeal Nov 18 '11 at 20:33
Doesn't matter which symbol you use as long as you use if for at most one purpose. – Raphael Nov 18 '11 at 23:14
@Raphael: I think you're misunderstanding me. I'm talking about reading here. Someone else wrote it, and they did not define which they meant. – Billy ONeal Nov 18 '11 at 23:16
If they did not define $\alpha$, you better ask them. How should we know what they meant? (Although in this case, one possibility is certainly more likely than the other.) – Raphael Nov 18 '11 at 23:34

## 3 Answers

Just a suggestion: we can simply write $\alpha t x$ to mean the product of the three variables. Usually it is clear from the context what the meaning is: if $\alpha$ is a function, then $\alpha (t)$ would be the function evaluated at $t$.

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I would do this -- but I didn't write the question :) But +1 because it's a good idea in general. – Billy ONeal Mar 2 '11 at 17:31
In computer science, this is called "typing". :) – Raphael Nov 18 '11 at 19:35
@Raphael: What do you mean? – Billy ONeal Nov 18 '11 at 20:33
If $\alpha$ is defined to be function it can only be used as such. Just treat maths like a programming language, you will discover lots of notation abuses. Some things can't be done in typed systems,though. – Raphael Nov 18 '11 at 23:17
@Raphael: It wasn't defined. – Billy ONeal Nov 19 '11 at 3:53

For "alpha times t times x" I would suggest writing $$\alpha\cdot t\cdot x$$ and adding brackets if appropriate, like $(\alpha\cdot t)\cdot x$. You could also leave the centered dots.

The bracket in $\alpha(t) x$ does not make much sense.

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I have noticed that misinterpretations of notation are becoming more common. For instance, many students have difficulty when differentiating something such as y = sin(tan(x)), mistakenly interpreting it as a product as opposed to a composition. Jasper makes a good point when he emphasizes that the context is usually a good guide as to how to interpret. In my example, the function sine has to act on something, which would be tan(x). As a rule, if you are unsure how to interpret, ask for clarification. If the situation does not allow that, or the instructor declines to comment, closely examine the context.

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The difference there is that the notation is unambiguous there. If a student has a problem there, that student is simply incorrect. But in the example I show above, both interpretations are reasonable -- the syntax is ambiguous. – Billy ONeal Mar 2 '11 at 21:36
It is only less ambiguous than your example if you accept $\sin$ to be a function. Maybe $s, i, n$ are natural numbers? Or $\sin$ is one? – Raphael Nov 18 '11 at 23:35