what is a number?

[Anonymous No.16108052]

in mathematics, the term "number" refers to a variety of concepts. In addition to your standard [math]\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}[/math], you also have constructs like the quaternions, the cardinal numbers, the ordinal numbers, hyperreal numbers, surreal numbers, p-adic numbers and others. At the same time, there are concepts which are generally not considered to be numbers, such as vectors, matrices, arbitrary groups and arbitrary fields.
Is there a simple rule (in English) for what is and what is not a number?

[Anonymous No.16108055]

All of the above are numbers except for the so called "complex numbers". Those are an abomination and definitely not numbers.

[Anonymous No.16108072]

>>16108052
A number is a quantification of a state of some system

[whydoiexist No.16108078]

anything that can be used to measure or quantify something can be considered a number, but in general those who have more structure than the complex numbers are generally not considered out of pure tradition lol

in general u can check if something can be considered a number or not depending on how did it originated

reals and complexes originated from solutions to algebra, while matrixes come from transformations and linear equations

its kinda hard to define a number desu

[Anonymous No.16108084]

>>16108078
>its kinda hard to define a number desu
The thing is that a number can be defined as literally anything you want as long as that thing can be quantified, that's the whole point of numbers/math/physics

[Anonymous No.16108086]

>>16108052
An abstract valuation. Look up the Pythagoreans.

[Anonymous No.16108199]

>>16108052
A number is a measure in a space possibly zero.
Specifically recursively measured from the unit measure
/thread

[Anonymous No.16108376]

>what is counting

we need a new rule for this board against posts that are overly low effort reply bait

[Anonymous No.16108490]

>>16108052
This is a really helpful diagram, perfectly explains the concept

[Anonymous No.16108529]

The only numbers are cardinal numbers.
Everything else entirely misses the most fundamental concept of a number.
We need to reclaim the definition of "number."

[Anonymous No.16108538]

>>16108376
>yeah I’d like to have 4.5-3i+\sqrt{2}j-\pi k apples, please

[Anonymous No.16108546]

>>16108052
There's no canonical one. Frankly, my preferred interpretation of "number" is "element of a field extending the rationals". That excludes stuff like [eqn]\mathbb{H}[/eqn] or grassmann numbers.
However, I would all them number-like for being isomorphic to a vector space equipped with an associative algebra.

[Anonymous No.16108552]

>>16108546
>Frankly, my preferred interpretation of "number" is "element of a field extending the rationals".

Your definition is retarded, as that would make everything a number, since you can use anything in a field extension.

[Anonymous No.16108558]

>>16108552
The reals are a unique ordered extension of Q, and the maximal order preserving extension of Q is the surreals which are proper class sized. The only finite-dimensional field extension of the reals is the complex numbers. The only extension the rationals permit that do not involve the reals are the p-adics, to my knowledge. That is pretty damn restrictive.

[Anonymous No.16108571]

>>16108558
The ELEMENTS of an extension can be anything.

[Anonymous No.16108580]

>>16108052
anything that can be counted

[Anonymous No.16108588]

>>16108571
What do you even mean by that? What elements "are" exactly is completely immaterial if they have the correct properties. "1" can be represented as sets of sets, as a function, as a matrix, as a graph, as a knot, as a game, it literally does not matter. The complex numbers can be thought of as equivalence classes over the ring of real polynomials. And they are still numbers. What makes a number is a set of algebraic properties.

[Anonymous No.16108595]

>>16108052
Abstract states of a series of added ones or partial states between said states

[Anonymous No.16108596]

>>16108588
I mean, your definition of number allows my pet parrot to be considered a number. All I have to do is put him in a field of characteristic zero, and then voila he's a number.

[Anonymous No.16108600]

>>16108596
For that you will need to also define addition and multiplication involving your parrot. Here is a simple question: is [math]\emptyset[/math] a number?

[Anonymous No.16108601]

>>16108600
It has to be in order to define partial states in the negative territory

[Anonymous No.16108603]

>>16108601
Are you the same guy I just replied to, this anon >>16108595, or both? Bear in mind I am not speaking of 0, I am speaking of the empty set.

[Anonymous No.16108604]

>>16108600
>For that you will need to also define addition and multiplication involving your parrot.
Easy: I could just take the set of real numbers, and replace pi with my parrot.

[Anonymous Mogul No.16108616]

If you can have apples of it, it's a number.

But really, I think it's any set of things that can be ordered, possibly in several dimensions. If you can order it, then it's a number, or at least it can be mapped to a set o fnumbers.

[Anonymous No.16108618]

>>16108603
Not the same. Wouldn't a null set ultimately be the same as zero since it collapses state in multiplication and can surround a defined state in a matrix?

[Anonymous No.16108625]

>>16108604
That is, for any mathematical purpose, just the field of real numbers. If we did not silently imply "up to isomorphism" for everything we would have to distinguish between [math]\sqrt{2}\in\mathbb{N}^\mathbb{Q}[/math] and [math]\sqrt{2}\in\mathrm{No}[/math], which are the same mathematical object defined in fundamentally disagreeing ways. If you like set theory then everything is just a set of sets. So in short: your parrot literally *is* pi (up to isomorphism) because what "is" means is defined by the equality operator, which you need to modify for your thought experiment to work out.
>>16108618
You can define addition and multiplication such that the empty set is a representation of 0. That was the point essentially, the empty set is "on its own" not a number, but neither are numeral symbols or anything else. A number is a thing equipped with an arithmetic. So yes, within a carefully chosen context, the empty set is a number. But then it is a number in that context because it is isomorphic to 0.

[Anonymous No.16108628]

>>16108625
Your original definition gave no way to choose which extensions from the isomorphism classes are allowed, which matters because you were talking about the elements themselves as numbers.

[Anonymous No.16108634]

>>16108628
>gave no way to choose which extensions from the isomorphism classes are allowed,
I feel like there is a misunderstanding in something that should be extremely obvious.
>you were talking about the elements themselves as numbers.
yes, in the exact same sense people talk of elements of a group, or elements of a vector space. Technically a "group" has exactly two elements, a set and a binary operator. That's not exactly what the literature means by "element of a group" now, is it? No, they mean the element of a set in a context in which the set is equipped with additional operations. An element of a field is unaware of how the field was extended. It really is not difficult to grasp. Have not even given a single definition of what you think elements of a number set are "made of". And I know why, it is a moronic thing to try to conceive. Is square root of two a rational sequence or an ordered pair of sets of surreal numbers? Yes. Up to isomorphism.

[Anonymous No.16108644]

>>16108634
>I feel like there is a misunderstanding in something that should be extremely obvious.
No, you just weren't careful in your definition. It is a simple fact that anything can be an element of a field extension of the rationals.

[Anonymous No.16108654]

>>16108644
and that is not only correct but objectively necessary by design. Because there is no such thing as "the" representation of even rational numbers. Let alone natural numbers. It really is that easy, I am a bit concerned why you seem to struggle to understand that. So I will ask one last time: what is a rational constructed from, besides pairs of integers? What are integers constructed from? The answer is: anything, as long as you can define the arithmetic properly. If you give any one construction and claim it to be "the one", you are ust objectively wrong.

[Anonymous No.16108658]

>>16108625
Do basic arithmetic operators respect different symmetries or can they be bound uniformly In set theory?

[Anonymous No.16108662]

>>16108658
You can, in principle, define all operators and numbers as sets and sets of sets. The lit is full of these constructions. Start with the von neumann construction and work your way up the hierarchy and after a couple abstractions you are back at the rationals.

[Anonymous No.16108664]

>>16108662
So the set interactions based on their composure derive rational numbers. Interesting stuff thanks!

[Anonymous No.16109023]

>>16108055
If matrices are numbers, then so are complex numbers since they are basically just matrices.

[Anonymous No.16109025]

>>16108072
So a barrel is a number since beer can be quantified by the barrel?

[Anonymous No.16109039]

>>16109023
Nobody said matrices were numbers.

[Anonymous No.16109042]

>>16109039
*Except of course, the anon who said all of the above when the above structures included matrices.

[Cult of Passion No.16109067]

>>16108052
>At the same time, there are concepts which are generally not considered to be numbers, such as vectors, matrices, arbitrary groups and arbitrary fields.
Das my shit mang.

[Cult of Passion No.16109069]

>>16109039
>matrices were numbers.
Its 1, or 1/-1 (depending how you make Matrixes, one way is obviously Chad).

Side length or otherwise, it shall be done.

[Anonymous No.16109072]

>>16109067
>>16109069
Don't ever talk about math again. You can shit up evopsych or linguistics threads with your schizo babble but math is just too objective and rigorous to be amenable to your dysfunctional low IQ nonsense.

[Anonymous No.16109074]

>>16109072
>math is just too objective
Where can I find the object formerly known as 1?

[Anonymous No.16109075]

>>16108052
Positive real numbers are numbers. Other things are not numbers.
A negative so-called number is properly a vector (arrow on the number line) with a magnitude (number i.e. a positive real) and a sign (orientation).

[Anonymous No.16109080]

>>16109074
>formerly Chuck's

[Anonymous No.16109082]

>>16108055
>Quaternions are numbers
>Complex numbers aren't
what is it about complex numbers that causes brainlet seethe?

[Cult of Passion No.16109085]

>>16109072
>math is just too objective and rigorous
Yes, let us change that.

Obama said "Yes we can.", youre not a part of that "we".

:^)

[Cult of Passion No.16109088]

>>16109067
>arbitrary fields
And for the fields I count it as one and assume the rest filled in, like forming a circle from a radius.

And Im only interested in alignment, so varience from north/south pole.

[Anonymous No.16109651]

>>16109025
If you define "barrel" as a unit of measurement equal to X quantity of beer, yes.

[Anonymous No.16109654]

An imperfection to the left or right of a time zone

[Anonymous No.16109657]

>>16109654
Would 1 ever mean anything without '='

It's an imperfection on the equatorial of a time zone.

[Anonymous No.16110167]

>>16109651
>equal to X quantity of beer
Barrel is the quantity, the amount of beer equals the beer that can fill the barrel.

[Anonymous No.16110186]

>>16109082
Quaternions are unironically more legitimate than complex numbers. Quaternions are just a simple computational tool for rotations and a good explanation for spin. And luckily they are algebraically not a field. They don't claim to be bullshit like new solutions to previously unsolvable equations like square root of -1.

[Anonymous No.16110193]

>>16110186
Complex numbers are just a simple computational tool for rotations in the plane, dude. Also a complex number is a quaternion with the jimaginary and kimaginary parts set to zero.

[Anonymous No.16110199]

>>16110186
>Quaternions are just a simple computational tool for rotations
So are imaginary numbers? Hell, so are negatives.

Negatives are positive numbers rotated 180 degrees. Imaginary numbers are positive numbers rotated by 90 degrees.

>They don't claim to be bullshit like new solutions to previously unsolvable equations like square root of -1.
Why should it matter that the problem was previously unsolvable? Negatives were ALSO new solutions to square roots. -1 as a solution to the sqrt(1) was fucking new. Literally all imaginary numbers are is an iteration of negatives.

Also, negatives were new solutions to previously unsolvable equations, too. eg 1-2=

[Anonymous No.16110743]

>>16110186
>They don't claim to be bullshit like new solutions to previously unsolvable equations like square root of -1.
Meanwhile in the quaternions: i^2 = j^2 = k^2 = -1

[Anonymous No.16111358]

>>16108538
>>16108616
This, numbers should belong to totally ordered sets.
>I would like [-1, 4, 3, -5] apples please
Utterly deranged.

[Anonymous No.16111380]

>>16108052
[math] (x/(x/x))/((x/x)/x) = x^2 [/math]
>Is there a simple rule (in English) for what is and what is not a number?
You hallucinate numbers onto operations.
The operation of an apple releasing molecules into the air such that you smell the apple. The apple is losing countless molecules per second on its way to being shrivled and dried up. Is the apple only its dry weight? Is apple smell apple? Does apple smell count as apple?
Do you want to eat an apple in its true, static form, which is old, dry, and stale, but in relative equilibrium?
You hallucinate numbers onto operations. The universe is an operation, and you hallucinate whatever you need in order to cope. Sometimes you can make a lot of money out of hallucinatory speculation.
Can't eat a number, though.