# Correct usage of %

Wolfram|Alpha evaluates

$$2\%\%\%/2$$

to $$1×10^{-6},$$

as well as many other online (google) and offline calculators.

My question: Is it really so, and one can represent one millionth as $1\%\%\%$? (Or maybe it is kind of a bug?..)

Thank you for any opinions and suggestions!

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It's interpreting $\%$ to mean "divided by $100$. Dividing by $100$ three times is the same as dividing by one million. – Cameron Buie Oct 25 '12 at 17:33
I would consider it incorrect but understandable. – Ross Millikan Oct 25 '12 at 17:35
@RossMillikan: Funny, I just added an answer that says the opposite. :) – Charles Oct 25 '12 at 17:36
@Charles: I would argue it is incorrect syntax, but that depends upon your parser. There are many cases in English where something is incorrect but understandable. Sometimes one is a better way to say it than the "correct" version. – Ross Millikan Oct 25 '12 at 17:40
@RossMillikan: I think we're just giving the same answer from two different perspectives. It's grammatically incorrect but mathematically/syntactically correct. – Charles Oct 25 '12 at 17:45

Essentially all these interpret $\%$ as multiplication by a factor of $10^{-2}$ since $2 \% = 0.02$. Hence, $$2 \% \% = (0.02) \% = 0.0002 = 2 \times 10^{-4}$$

EDIT

I am not sure whether it is an abuse of notation. The way I interpret $\%$ is $\%: x \to \dfrac{x}{100}$. Hence, in my understanding it would be completely valid to think of $\% \%$ as a composition of the function $\%$ twice.

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Thank you Marvis, but my question is more about correctness of such notation. – Oleg Trakhman Oct 25 '12 at 17:49
EDIT makes it clear. – Oleg Trakhman Oct 26 '12 at 7:45

It is mathematically correct, but generally you shouldn't write it (for the same reason you shouldn't write "0.000002 million" in place of 2).

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We often use "$.3$ million" instead of "three hundred thousand" (for example). It's certainly goofy to use "$.000002$ million" instead of just "$2$", but it's certainly acceptable. – Cameron Buie Oct 25 '12 at 17:40

This is not correct usage. While "$2\%$ of $150$" is the same as $\frac2{100}\cdot 150=3$, this does not mean that we are allowed to abuse the percent notation like this with repeated percents.

One would also not say that $1^{\circ\circ}=\frac\pi{180}^\circ=\frac{\pi^2}{180^2}$.

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I wouldn't use your degree example, either, but like it. – Ross Millikan Oct 25 '12 at 17:41
Thank you Hagen! How do you think would it be good if I file a bug to Wolfram Alpha about that case? – Oleg Trakhman Oct 25 '12 at 17:47
@OlegTrakhman: But is it a bug? What I wrote was my personal understanding of the matter ("guts feeling"). Meanwhile I see that "even" Wikipedia does not support my view. It seems I need a refence, probably from outside math usage. ("We" mathematicians don't use percent anyway, it's just courtesy for them bankers and bookies ;) ) – Hagen von Eitzen Oct 25 '12 at 18:22