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"A number is nothing but the proportion of one magnitude to another arbitrarily assumed as the unit."

My main argument against this, philosophically, is that some units are more logical than others, i.e. those which are the atoms of a system.

One person brought up the complex numbers as a counterexample.

I took the quote to mean: x = ur , where x is the unknown magnitude, u is the unit of measure, and r is a real number.

Taking 'number' in the quote above to mean any type of number, in what cases is the above quote invalid?

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Your question reads like a request for a survey of people's feelings on various notions of numbers. If you look at the FAQ you'll see that's not quite what the forum is about. – Ryan Budney Jun 11 '11 at 22:44
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My take on this: Think of a line. Pick a point, call it $0$. Pick another point, call it $1$, the rest is determined. – t.b. Jun 11 '11 at 22:47
Agreeing or disagreeing with an imprecise definition is futility, both because it's imprecise and because it is a definition. For more of my thoughts on this subject see math.stackexchange.com/questions/36289/is-infinity-a-number . – Qiaochu Yuan Jun 11 '11 at 22:48
@Qiaochu Yuan I feel that it's unfair that I've been downvoted, and this question closed. Take 'number' to mean any sort of 'number', I really don't care. I wanted to know under what definitions or allowable identities of 'number' the above is valid. I was mainly asking for counterexamples or reasons why an alternative definition is better. Yes, the above definition doesn't use symbols, but that does not mean that it is not logical. – Faustus Jun 11 '11 at 23:04
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For a nice introduction to modern mathematical "number" conceptions see the book Numbers by Ebbinghaus et al. – Gone Jun 11 '11 at 23:27
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closed as off topic by Qiaochu Yuan Jun 11 '11 at 22:47

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1 Answer

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The quote is neither right nor wrong. What it means for something to be "a number" is not a rigorous (or particularly useful) mathematical notion. Given any sort of objects in mathematics, you might want to give them the appellation "number", or you might not - it's a matter of preference, and not mathematical content.

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