What Even Is a Number?
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What Even Is a Number?
1–10 of 173 posts
Re: What Even Is a Number?
#2I was surprised by no discussion of ordinality. Cardinality was something I also specifically expected.
Counts of things are described here, but the greeks went to great lengths with geometry to discover numbers (and even killed people who came up with irrational answers)
Re: What Even Is a Number?
#3> You can see some of that history of what we call some of the different sorts of numbers: e.g. Negative, irrational, and imaginary numbers. Each of these represents a bitter argument about whether a new type of number should be allowed, all of which were eventually won by the people on the side of the new numbers...
Was there ever a proposed type of number that is today not considered a number? Like i=squareroot(-1), was there ever an attempt to do math with something like y=1/0 or another illegal construct?
Re: What Even Is a Number?
#4> You can see some of that history of what we call some of the different sorts of numbers: e.g. Negative, irrational, and imaginary numbers. Each of these represents a bitter argument about whether a new type of number should be allowed, all of which were eventually won by the people on the side of the new numbers... Was there ever a proposed type of number that is today not considered a number? Like i=squareroot(-1)…
> was there ever an attempt to do math with something like y=1/0 or another illegal construct?
Wheels are a type of algebra where division by 0 is defined.
https://en.wikipedia.org/wiki/Wheel_theory
https://math.stackexchange.com/questions/994508/wheel-theory...
To call wheels "obscure" is to make them out to be better-known than they are, however.
Re: What Even Is a Number?
#5I read this as "What is an even number" and spent the entire article stoked for when he got to the part about a number being even.
Still, lovely article.
Re: What Even Is a Number?
#6> You can see some of that history of what we call some of the different sorts of numbers: e.g. Negative, irrational, and imaginary numbers. Each of these represents a bitter argument about whether a new type of number should be allowed, all of which were eventually won by the people on the side of the new numbers... Was there ever a proposed type of number that is today not considered a number? Like i=squareroot(-1)…
Surreals and Hyperreals have lots of "extra" numbers that don't exist in the reals. They're a bit niche though.
Re: What Even Is a Number?
#7It's extremely obvious that the author is a computer scientist
Re: What Even Is a Number?
#8Article seems very verbose and only really addresses the natural numbers (it mentions negatives and rationals in passing).
In case anyone just wants actual answers without reading pages and pages of prose, here is one set of constructions (there are other competitors too).
The natural 0 is the emptyset. The natural 1 is the singleton {0}. The natural 2 is {0,1}. In general, the natural n+1 is {0,...,n}. Exercise to the reader: define appropriate arithmetical functions on the naturals.
Define a relation ~ on pairs (a,b) of naturals by saying (a,b)~(c,d) if and only if b+c=a+d. Exercise to the reader: ~ is an equivalence relation. Its equivalence classes are called "integers". The equivalence class containing (a,b) represents the integer b-a. For example, the integer -1 is the equivalence class {(1,0),(2,1),(3,2),...}. Exercise: define appropriate arithmetical functions on the integers.
Define a relation ~ on pairs (m,n) (n nonzero) of integers by saying (m,n)~(p,q) if and only if mq=pn. Exercise: Show ~ is an equivalence relation. Its equivalence classes are called "rationals". The equivalence class containing (m,n) represents the rational m/n. For example, 1/2 is the equivalence class {(1,2),(-1,-2),(2,4),(-2,-4),...}={(k,2k) for all nonzero integers k}. Exercise: define arithmetic operations on the rationals.
To get from the rationals to the reals, see https://en.wikipedia.org/wiki/Dedekind_cut
Re: What Even Is a Number?
#9> You can see some of that history of what we call some of the different sorts of numbers: e.g. Negative, irrational, and imaginary numbers. Each of these represents a bitter argument about whether a new type of number should be allowed, all of which were eventually won by the people on the side of the new numbers... Was there ever a proposed type of number that is today not considered a number? Like i=squareroot(-1)…
You can consider almost anything a number if you can do some internally consistent operations with it.
Hamilton attempted a theory of triplets before setting on 4d quaternions, so that would be one example of a failed number system off the top of my head.
Re: What Even Is a Number?
#10I had a fun mind-play with my kids; I asked them if numbers like "one" and "two" really exist. They said yes, of course.
OK, I said, show me a plain old "one". Nope, that's one fork; nope that's one ball; nope, you get the idea.
You cannot show just "one" unattached to anything else.
The concept of "one" is an idea. It only exists as a "real" concept in your mind because it exists as the same concept in the minds of others, and it's a very useful concept in describing the world around us.
This odd fact becomes ever more apparent when you find out about cultures that have very different approaches to quantitative reasoning, like the Pirahã in Brazil [1]. They don't have number words beyond "one", words roughly meaning "some" and "many" are used for anything more than one.
[1] https://www.sciencedaily.com/releases/2008/07/080714111940.h...