## Paranoia About Starting Equations With a Negative

Am I the only one annoyed by the tendancy to avoid negatives in front of an equation?

Alternate Form of Law of Cosines (from my textbook)

$cos A = \frac{b^2+c^2-a^2}{2bc}$
$cos B = \frac{a^2+c^2-b^2}{2ac}$
$cos C = \frac{a^2+b^2-c^2}{2ab}$

I’d much rather have

$cos A = \frac{-a^2+b^2+c^2}{2bc}$
$cos B = \frac{a^2-b^2+c^2}{2ac}$
$cos C = \frac{a^2+b^2-c^2}{2ab}$

But the first in particular seems to be absolutely unthinkable, because of the -a^2 up in front.

### 7 Responses

1. I agree with your preference.

I think a programmer would be more likely to lay things out as you suggest because there’s a strong tradition in programming for vertically aligning repetitive terms, more so than in math.

2. My original background is CS, so I understand how you feel.

I already had a rant over at Invisible Math about how mathematicians (unlike programmers) seem to give no thought to how they name their variables.

3. Hey, the same goes for fractions; people prefer 2x^2+2x+1=0 to x^2+x+\frac{1}{2}=0, don’t you think? But what about a huge preference for \frac{\sqrt{2}}{2} and not \frac{1}{\sqrt{2}}?

And the last comment is unfair. Please, read Halmos’s “How to write mathematics”, which is really a great reading.

4. mbork: I think the preference for rational denominators was motivated by manual computational effort. For example, once you’ve looked up sqrt(2) in a table, it’s easier to divide the result by 2 than to take its reciprocal. Of course that doesn’t matter anymore, and I think the rational denominator convention isn’t followed as strictly as it once was.

5. That is *really* interesting. I never thought about it this way! I always assumed that the reason is aesthetic: this way, you really see with the naked eye, that all numbers of the form, say, a+b\sqrt{2}, where a and b are rational, form a field and not only a ring.

BTW: how to embed (La)TeX code in the comments?

6. This bothers me less, though I dislike the inconsistency: $-\frac{a^2 - b^2 - c^2}{2bc}$

Jonathan