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If I have a set of numbers {-1, 2, 3, 4, -8, 2, 0, 44}

and I make the statement that there are: 2 negative numbers 5 positive numbers and one signless number

Is there a mathematical concept used to name this count?

Thanks

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In the special case that your numbers are the eigenvalues of a matrix describing a quadratic form, you can say that the form has signature $(5, 1, 2)$. But in general I don't think so. –  Qiaochu Yuan Apr 26 '11 at 12:56
1  
What Qiaochu called the "signature" would also be termed the inertia in some references. –  J. M. Apr 26 '11 at 13:01

2 Answers 2

up vote 2 down vote accepted

Cardinality is the term you are looking for, e.g. :

Cardinality of the subset of positive numbers for your set is 5,

Cardinality of the subset of negative numbers for your set is 2,

Cardinality of your set is 8

Cardinality of the subset of numbers=0 in your set is 1.

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I have no idea about the answer to your question, but I would like to make a general observation which is too long for a comment.

Mathematics thrives on brevity, accurate description and minimal sets of definitions. Its beauty is in the fact that with so little we can create so much, and prove further and further.

One can define pretty much anything, the question is why would you define something like that?

In most cases I have seen (and seeing now as I take my first dip into original research) definitions arise naturally from the work being done, or from some problem being dealt with.

For example, we see that if a function between two groups preserves the group's multiplication then it can be useful for us. We name it "homomorphism", further along the way we may see that it was more useful than we'd expect.

I can say that in set theory many times you find yourself baffled at notions of large cardinals, trying to understand why and how thought about them. These notions all came to be when someone was trying to prove something and needed "these sort of properties" - so he gave them a name. Sometimes you figure out that what you define is somewhat compatible to the things that were known before, and your definition represents a stronger, weaker or equivalent object.

In a way you found way to describe some idea, known or unknown. From time to time you actually come up with something new. These new ideas are hardly ever "out of the blue", as I said; they tend to be natural from some research.

Now you might find some idea useful when you solve a question, or two, or fifteen. However you need to consider what sort of problem this notion that you have defined helps you with, it might either help you define a problem or it could be a stepping stone in the solution. If it is indeed helpful, then it is useful.

There can be of course the need in shorter notation of something that you use a lot, but giving everything a short notation rises two problems:

The first is that not everyone will be familiar with your notations, and you would have to introduce and explain them often enough at the beginning - and so unless this notation is extremely useful it is likely that it will not catch on.

The second problem is that if you add more and more notations at a certain point it will overtake your work. It would be extremely hard to follow the work that you did, which is not a good thing usually.

To conclude, my point is that definitions and notations usually arise naturally when facing a problem we want to try to solve and/or define better; we may think that some mathematical object will help us in the solution and once this object becomes useful it tends to be named.

One last thing is that it is not customary to name things after yourself.

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