# Do “complex percentages” exist?

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

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 am resurrecting an ancient question because I ran across it and thought of a random paper I had seen a few years ago. The original conversation participants have long since moved on, so I'm posting this for the benefit of anyone making their way here from Google.

As @joriki points out, there is not necessarily anything "god-given" about complex percentages, but maybe someone might find a useful way of viewing such a beast. (In fact, there may be multiple different definitions of complex percentages, where each might be useful in a different way.) Here is a paper that discusses a possible interpretation of something related, which is negative probability:

http://www.wilmott.com/pdfs/100609_gjs.pdf

The paper discusses an interpretation of negative probabilities as corresponding to flipping "half a coin." Specifically:

Fundamental theorem: For every generalized g.f. $f$ (of a signed probability distribution) there exist two p.d.f.’s $g$ and $h$ (of ordinary nonnegative probability distributions) such that the product $fg = h$.

Thus if $f$ is the generalized g.f. of a half coin C, a third of a die, (or any other related mystical object), then we can always find two ordinary coins, ordinary dice (ordinary random object) C1 , C2 such that if we flip C and C1 , their sum is C2 . In this sense every generalized (signed) distribution is a kind of difference (‘so-called convolution difference’) of two non-signed (ordinary) probability distributions. This result justifies the application of signed probabilities in the same sense as we use negative numbers.

So one can view a single coin flip as the succession of flipping another coin and a half-coin. The probabilities associated with flipping the half coins are negative. You're "not allowed" to stop after only flipping the a half coin, in the same way a negative balance in your bank account doesn't give you the ability to carry negative one $20 bill in your pocket. The authors also briefly allude to the wavefunction in quantum mechanics, which might better correspond to your question. One recovers a probability from the wavefunction by taking the square of the norm of the wavefunction. So the wavefunction is a probability-related thing which may be complex. At a (very) high level, the complex nature of the wavefunction allows you to do "bookkeeping" to simultaneously keep track of conjugate quantities such as position and momentum. So the wavefunction might correspond to a kind of complex probability (complex percentage), although if you want to get anything physically measurable out of this "complex probability," you end up feeding it through a process that spits out a real number, without any imaginary part. This is similar to the linked paper, where negative probabilities can be thought of as corresponding to half-coins, and the half-coin is a "bookkeeping" technique that doesn't make its way into your pocket next to single coins. - I tend to interpret % as shorthand for the fraction$\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... - +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