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Well, the origin of this question is a little bit strange. I dreamed - with a book called "Percentages and complex numbers. When I woke up, I thought: "Is this real?" So I started thinking:

1% of 100 = 1
3% of 100 = 3

And more:

i% of 100 = i?

That's my question. Is it right - does it even make any sense? Calculating a percentage is, basically, to multiply a fraction (denominator = 100) to a number. If it's right, we can, for example, calculate

8 + 4i% of 2 - i.

So, what you can tell me? Is it real or just a dream - have you ever seen this? And more: does it respect the definition of percentage?

Thank you.

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"$x$ per cent" literally means $x$ out of every hundred. That's why to compute x% you multiply by $\frac{x}{100}$: multiplying by $\frac{1}{100}$ tells you how many "hundreds" you have, multiplying by $x$ gives you the total amount if each hundred corresponds to $x$. So you can certainly talk about "$i$%", as the result of multiplying by $\frac{i}{100}$. But the real question is: are they good for anything? – Arturo Magidin Oct 19 '11 at 18:28
@ArturoMagidin That's exactly what I thought. I don't really see any function, but I was curious about the existence of them. – Ian Mateus Oct 19 '11 at 18:31
up vote 2 down vote accepted

Whether this is "real" or "exists" is a philosophical, not a mathematical question. My personal opinion on such questions is that they are meaningless, that is, there is no way to distinguish a state of affairs in which this concept is "real" from one in which it is not "real".

Concerning your question "have you ever seen this?": No, I haven't, and I would hazard a guess that if anyone has ever thought of this, they haven't found a use for it yet. However, that doesn't keep you from doing it. The fun thing about mathematics is that you can define whatever you want, as long as it's consistent, and see what it leads to, without worrying about whether it's "real".

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I tend to interpret % as shorthand for the fraction $\frac1{100}$; $i\%$ would just be $\frac{i}{100}$, and $(8+4i)\%$ would be $\frac2{25}+\frac{i}{25}$. As to whether these things are useful...

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+1 I remember being called out in high school for using shorthand notation like $3 \cdot 50\% = 1.5$, so in the next exam I actually wrote "Let $\% = 0.01$..." – Ilmari Karonen Oct 19 '11 at 21:08
As far as I'm concerned % is just as much of a constant as $\pi$, e, $^\circ$, etc. – Charles Oct 19 '11 at 21:32

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