🧵 what is a number?
Anonymous at Mon, 1 Apr 2024 18:51:15 UTC 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 at Mon, 1 Apr 2024 18:53:31 UTC No. 16108055
All of the above are numbers except for the so called "complex numbers". Those are an abomination and definitely not numbers.
Anonymous at Mon, 1 Apr 2024 19:02:54 UTC No. 16108072
>>16108052
A number is a quantification of a state of some system
whydoiexist at Mon, 1 Apr 2024 19:09:58 UTC 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 at Mon, 1 Apr 2024 19:12:47 UTC 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 at Mon, 1 Apr 2024 19:13:47 UTC No. 16108086
>>16108052
An abstract valuation. Look up the Pythagoreans.
Anonymous at Mon, 1 Apr 2024 20:16:18 UTC No. 16108199
>>16108052
A number is a measure in a space possibly zero.
Specifically recursively measured from the unit measure
/thread
Anonymous at Mon, 1 Apr 2024 22:05:14 UTC No. 16108376
>what is counting
we need a new rule for this board against posts that are overly low effort reply bait
Anonymous at Mon, 1 Apr 2024 23:23:15 UTC No. 16108490
>>16108052
This is a really helpful diagram, perfectly explains the concept
Anonymous at Tue, 2 Apr 2024 00:01:27 UTC 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 at Tue, 2 Apr 2024 00:07:24 UTC No. 16108538
>>16108376
>yeah I’d like to have [code]4.5-3i+\sqrt{2}j-\pi k[/code] apples, please
Anonymous at Tue, 2 Apr 2024 00:11:10 UTC 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 at Tue, 2 Apr 2024 00:14:43 UTC 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 at Tue, 2 Apr 2024 00:18:44 UTC 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 at Tue, 2 Apr 2024 00:25:51 UTC No. 16108571
>>16108558
The ELEMENTS of an extension can be anything.
Anonymous at Tue, 2 Apr 2024 00:27:29 UTC No. 16108580
>>16108052
anything that can be counted
Anonymous at Tue, 2 Apr 2024 00:30:42 UTC 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 at Tue, 2 Apr 2024 00:35:33 UTC No. 16108595
>>16108052
Abstract states of a series of added ones or partial states between said states
Anonymous at Tue, 2 Apr 2024 00:36:57 UTC 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 at Tue, 2 Apr 2024 00:38:37 UTC 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 at Tue, 2 Apr 2024 00:39:41 UTC No. 16108601
>>16108600
It has to be in order to define partial states in the negative territory
Anonymous at Tue, 2 Apr 2024 00:43:00 UTC 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 at Tue, 2 Apr 2024 00:43:22 UTC 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 at Tue, 2 Apr 2024 00:55:02 UTC 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 at Tue, 2 Apr 2024 00:56:22 UTC 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 at Tue, 2 Apr 2024 00:59:54 UTC 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
>>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 at Tue, 2 Apr 2024 01:02:28 UTC 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 at Tue, 2 Apr 2024 01:10:02 UTC 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 at Tue, 2 Apr 2024 01:15:46 UTC 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 at Tue, 2 Apr 2024 01:24:38 UTC 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 at Tue, 2 Apr 2024 01:33:05 UTC No. 16108658
>>16108625
Do basic arithmetic operators respect different symmetries or can they be bound uniformly In set theory?
Anonymous at Tue, 2 Apr 2024 01:38:06 UTC 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 at Tue, 2 Apr 2024 01:41:29 UTC No. 16108664
>>16108662
So the set interactions based on their composure derive rational numbers. Interesting stuff thanks!
Anonymous at Tue, 2 Apr 2024 08:52:15 UTC No. 16109023
>>16108055
If matrices are numbers, then so are complex numbers since they are basically just matrices.
Anonymous at Tue, 2 Apr 2024 08:53:16 UTC No. 16109025
>>16108072
So a barrel is a number since beer can be quantified by the barrel?
Anonymous at Tue, 2 Apr 2024 09:01:48 UTC No. 16109039
>>16109023
Nobody said matrices were numbers.
Anonymous at Tue, 2 Apr 2024 09:05:19 UTC No. 16109042
>>16109039
*Except of course, the anon who said all of the above when the above structures included matrices.
Cult of Passion at Tue, 2 Apr 2024 09:48:45 UTC 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 at Tue, 2 Apr 2024 09:51:53 UTC 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 at Tue, 2 Apr 2024 09:56:41 UTC 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 at Tue, 2 Apr 2024 09:58:59 UTC No. 16109074
>>16109072
>math is just too objective
Where can I find the object formerly known as 1?
Anonymous at Tue, 2 Apr 2024 09:59:48 UTC 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 at Tue, 2 Apr 2024 10:02:01 UTC No. 16109080
>>16109074
>formerly Chuck's
Anonymous at Tue, 2 Apr 2024 10:04:55 UTC No. 16109082
>>16108055
>Quaternions are numbers
>Complex numbers aren't
what is it about complex numbers that causes brainlet seethe?
Cult of Passion at Tue, 2 Apr 2024 10:05:42 UTC 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 at Tue, 2 Apr 2024 10:08:27 UTC 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 at Tue, 2 Apr 2024 20:20:30 UTC No. 16109651
>>16109025
If you define "barrel" as a unit of measurement equal to X quantity of beer, yes.
Anonymous at Tue, 2 Apr 2024 20:24:12 UTC No. 16109654
An imperfection to the left or right of a time zone
Anonymous at Tue, 2 Apr 2024 20:27:22 UTC No. 16109657
>>16109654
Would 1 ever mean anything without '='
It's an imperfection on the equatorial of a time zone.
Anonymous at Wed, 3 Apr 2024 05:25:49 UTC 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 at Wed, 3 Apr 2024 05:52:48 UTC 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 at Wed, 3 Apr 2024 05:57:02 UTC 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 at Wed, 3 Apr 2024 06:02:33 UTC 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 at Wed, 3 Apr 2024 15:29:45 UTC 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 at Thu, 4 Apr 2024 00:02:22 UTC No. 16111358
>>16108538
>>16108616
This, numbers should belong to totally ordered sets.
>I would like [-1, 4, 3, -5] apples please
Utterly deranged.
Anonymous at Thu, 4 Apr 2024 00:23:24 UTC 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.