# Is adding a plus before a positive number a mistake?

I had a simple problem to find out the function equation from a sinusoidal graph.

$$F(x)= -2\sin(2x)$$

Would it be considered a "mistake" to write it down the following way?

$$F(x)= -2\sin(+2x)$$

Or, would it simply be an unnecessary additional math symbol (not treated as a mistake). As in: $$+∞ = ∞$$

or $$5+5+(5+5)=20$$

?

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By whom? ${}{}{}$ – Qiaochu Yuan Mar 14 '13 at 20:56
Where are the unnecessary symbols in 5+5+(5+5)=20? – user39280 Mar 14 '13 at 21:04
It's unnecessary, but so is preferring (x-c)(x+c) to $x^2-c^2$, so whatever helps you think about it. – Loki Clock Mar 14 '13 at 21:06
@dado: The parentheses, I'm guessing. – mjqxxxx Mar 14 '13 at 21:14
If you treat $+$ as the binary operator indicating the sum of two variables, it is a mistake. However, if you treat $+$ as the unary operator indicating the sign of the following term, it is correct. – Daryl Mar 14 '13 at 22:02

## 3 Answers

It is a stylistic error to write unnecessary symbols, even if they are not strictly wrong. Occasionally, redundant symbols may be considered useful because they emphasise a pattern or structure in a formula, but I think your teacher's advice is good to follow in general: simpler is better.

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I would consider it a mistake for one and only one reason: it will fail to communicate your meaning. Many (most?) people who read it will either think you made a typo, be confused, or think you are confused or don't know what you're doing. Most of the rest will still be forced to pause and think about it.

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In pre-high-school mathematics, it seems to be a common mistake (thanks to Hurkyl for this example!) to misinterpret $+x$ as "the positive value of $x$", so if $x=-5$, it "follows" that $+x=5$.

$+x$ is not alternative notation for $|x|$; that's what we got the absolute value for, but this may very well be the cause of your teacher being against it.

Once you know the rules of precedence, the plus before $2x$ is pretty harmless, since it "does nothing": $+x = x$. For example, you could also write the quadratic formula which I have been taught as $$\frac{-b \pm (\sqrt{b^2-4ac})}{2a}$$ in the form

$$\frac{\pm (\sqrt{b^2-4ac}) -b}{2a}$$

In the new form, the interpretation of the $\pm$ sign changes from "use both the binary $+$ and the binary $-$" (i.e. add and subtract) to "use both the unary $+$ and the unary $-$" (i.e. do nothing and take the negative). I would also find it very acceptable to solve $x^2=25$ by writing $x=\pm 5$; just a lowered minus sign with nothing above it just doesn't look right to me. Your teacher may not agree, unfortunately.

On the other hand, in more advanced mathematics, a notation like $+x$ would have me looking through the book, trying to find out why on earth the author introduced such a silly notation: it looks like there's nothing going on, but then why did he write it down like that - why not something more visually distinct like $x^{+}$ or whatever?

Concluding, I would say you didn't make a mistake, but your teacher also has very good reasons to insist on her way of doing things. There's usually one nicest way of writing something down; would you go with $x = 5 + 0 + 7 + 0 - 3 - 4$ as a solution to $2x =10$?

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I'm not so sure it's harmless -- I think it has a good chance of tricking a student into making the mistake of substituting $x = -5$ into $+2x$ and getting $10$. – Hurkyl Mar 14 '13 at 23:52
@Hurkyl: In that case I would expect $+2-5 = -3$ to come out! Seriously, I'm very much interested in the potential reasoning mistakes in your example (I'm thinking about getting into education, IT is getting a bit boring). Have you seen people come up with 10 from $+2x$ and $x=-5$? If so, you've bowled a strike! – yatima2975 Mar 15 '13 at 0:03
I have seen people plug $x = -5$ into $-x$ to get $-5$ (you wanted the negative value, right?). So I expect to see the same thing happen with plus... especially since the notation is uncommon enough that they will have to guess at the meaning. – Hurkyl Mar 15 '13 at 0:45
@Hurkyl: Fair enough, and thanks! I'll go and edit this potential cause of confusion right in. – yatima2975 Mar 15 '13 at 0:46