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What is a number?

I'm doing research for college and wanted a wider opinion on what people consider a number to be. Particularly with things like infinity, zero and imaginary numbers, and whether one way of writing it is more valid than another i.e. is the decimal system more valid than binary (particularly for general use/in society). Whilst is something like 'e' actually a number or just a way to represent a number?I understand this is a lot to ask for but any opinion/comment will be useful. Thank you!

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Reply 1

Puddles the Monkey

Original post by Jemjams

I've moved this to the maths forum for you :h:

mind=blown

Original post by 29Bilal96

mind=blown

I really hope this thread kicks off, I'd love some maths-related philosophy discussion. :moon:

Reply 4

VinnicombeDmv

Never actually thought about this before today, I commend you.

I suppose I would define a number as a character that is part of a succession of other similar characters that are used in combination with each other to solve a problem.

Reply 5

around

Original post by VinnicombeDmv

By that definition things like 'a', 'b', 'x' and 'y' are numbers (as in, numbers in and of themselves, not just as variables which can hold number values).

Reply 6

tazarooni89

It's a symbol used to represent a quantity.

Reply 7

De Mortimer

Essentially, like words, they're insignias used to represent discrete amounts.

Reply 8

furryface12

Original post by 29Bilal96

mind=blown

Exactly, as soon as you start thinking about it it's just like 'what?!' :redface:

I'd guess it depends what you count as a number OP, like is it just 1 2 3 4 5 or would 0.1234981409587123874 count as one too? To me, e is a name we give to a particular value that holds some significance, same as pi, and zero I'd also count as a number purely because it's on the same scale. Infinity I wouldn't say is one though as it doesn't have a particular value, imaginary numbers I can't decide as they still represent something, we just don't have an exact value for it as our system doesn't allow it. I would say they're still numbers though, just a different type. Think everyone would answer them differently though, I'm not even sure of my own answers... Interesting question though!!

Reply 9

around

Original post by tazarooni89

It's a symbol used to represent a quantity.

Is i a number?

I think most mathematicians are happy to accept complex numbers as numbers, but quarternions very definitely aren't numbers.

I think there's probably some kind of sociological element involved as well ('a number is whatever people who do maths considers to be a number'), but I realise this is unsatisfying to some people. To which my answer is: tough.

Reply 10

Octohedral

I'd say a 'number' is most naturally defined as the underlying concept, which can be represented in various ways. For example, few people would deny that pi and 3.1415... and Circumference/ Diameter are all the same 'number'.

Numbers like i and infinity are just as valid as the natural numbers. All are built up by rules - for example, the natural numbers can be defined as the result on adding the identity to itself several times, and i can be defined as the number that squares to make -1. The fact that we can't 'see' i makes no difference to its mathematical validity.

However, we can't just invent numbers. i clearly 'exists' as a concept, because it appears in apparently unrelated contexts, such as e^(i pi)=-1. The concept exists as part of a consistent system (the field C), and so in that sense it was 'discovered', not created.

Reply 11

tazarooni89

Original post by around

Is i a number?

Not "really" :wink:

You could say it's a number though, in the sense that it still represents a quantity; even if that quantity doesn't exist on the real axis.

(edited 9 years ago)

Reply 12

iEatMuFFiNS

Original post by around

By that definition things like 'a', 'b', 'x' and 'y' are numbers (as in, numbers in and of themselves, not just as variables which can hold number values).

97, 98, 120, 121 :wink:

Reply 13

around

Original post by iEatMuFFiNS

97, 98, 120, 121 :wink:

This is actually more interesting than you might think: you're 'breaking the abstraction' behind ASCII representations of characters in computer memory. You can read more about it here.

The concept is very important in programming and mathematical foundations (does it really make sense to say something like

3 \in 17

, which is 'true' in the classic coding of numbers in ZFC)?

Reply 14

miser

I don't think they exist in a concrete abstract sense, but I also don't believe we've simply invented them. I think I'd be a numerical descriptivist and say they describe properties of (sometimes hypothetical) objects.

'e' I think can represent an irrational number just as well as the character '1' can represent the concept of oneness. As for infinity, that isn't considered to be a number.

I think alternative methods of writing numbers are just as valid, just decimal is much more easy to use than binary for day-to-day purposes. But decimal isn't the easiest of all - duodecimal (base 12) is arguably even easier.

Reply 15

xylas

I'm of the opinion that only the natural numbers, 1,2,3,4,5..., exist in any sense.

0 is not a number because it is not a finite quanitity, and because the group of integers must exclude 0 to allow for multiplication to have an inverse.

If you were to ask me what the other 'numbers' are I would reply they are vectors representing the visualisation of the natural numbers in N-dimensional space e.g integers are 1-dimensional (forwards and backwards), complex numbers are 2-dimensional (anywhere on the plane).

Reply 16

around

Original post by xylas

0 is not a number because it is not a finite quanitity, and because the group of integers must exclude 0 to allow for multiplication to have an inverse.

Are you absolutely sure you mean this?

Moreover, excluding 0 as a number means the group of integers doesn't have an identity, and it also means the monoid of natural numbers doesn't have an identity.

Reply 17

unprinted

Original post by xylas

There is no number to express how much respect I have for this answer then :smile:

Reply 18

Puddles the Monkey

I think I agree with you. :h:

It's easy to get confused between the abstract concept of a number and the signifier representing it.

Original post by miser

Reply 19

Noble.

Original post by xylas

Erm, the integers without 0 do not form a multiplicative group because 1/n is not an integer for non-zero (and excluding 1) natural n. The integers are a ring, but that includes 0 as the identity (so maybe you should believe 0 is a number?) 0 is also a finite number, arguably the most finite number :lol:

I also find it pretty odd that you consider "vectors in n-dim space" to be 'numbers' but not the length of anything circular that physically exists, which almost surely is an irrational number.