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Is infinity an odd or even number? (2011)

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Re: Is infinity an odd or even number? (2011)

#291

Post by gowld »

bmacho wrote: post #255
zw123456 wrote: post #202

The way I would explain it to a 6 year old would be like this: Infinity isn't a number really, it's a concept, like the word many or the word few. If someone says they have many of something, you don't think is that odd or even you just know they have a lot of it. Infinity is kind of like that, it explains the idea of things going on forever, not an exact quantity of things like the number 10 or 11.

The way I would explain it to a 6 year old would be like this: There are natural numbers, like 0,1,2 and so on. Natural numbers can be odd or even. There is no such natural number as infinity. Therefore the question if 'infinity' is odd or even is meaningless. It does not even type-check. In math people like well-formed questions, and generally don't like ill-formed questions.

The OP clearly explains why the question is meaningful.

Re: Is infinity an odd or even number? (2011)

#292

Post by amelius »

chowells wrote: post #278
amelius wrote: post #71

Earlier quoted context omitted.

An even more honest thing to say is that infinity when used as a number is a hack introduced by mathematicians to make notation and reasoning simple in some cases, but that it can be dangerous in other cases, like any other hack. If you want to use infinity in a safe way, then use limits around your expressions. (And this quickly resolves the case of this article, since lim x->inf x-2*floor(x/2) does not exist).

It's not a hack to create a new set and work out rules for how to use it which are both internally consistent and support easy morphisms with more familiar sets. It may not be easy , but it's hardly a hack. It's one of the big ways math works, really. Are negative numbers a hack? Rational numbers? Algebraic numbers? Well then neither is the two-point compactification of the reals or extending the natural numbers into…

But nobody said that hacks cannot have precise interpretations. It's the unreasonable cognitive load that is the problem.

Re: Is infinity an odd or even number? (2011)

#293

Post by technothrasher »

aaronharnly wrote: post #99
hammyhavoc wrote: post #67

Earlier quoted context omitted.

I had it explained at a very early age as "three lots of three", and to imagine it like three boxes of three ice-creams. Treating the multiplication symbol as one would to indicate quantity in a list, thus calculating how many ice-creams there are.

Found the Brit! As an American I’d never heard the “lots of __” phrasing until I watched Numberblocks (a British show) with my kid…

I've never watched Numberblocks, but I do like to watch a good game of Numberwang!

Re: Is infinity an odd or even number? (2011)

#294

Post by freehorse »

warent wrote: post #10

In my experience with children, one of the easiest-to-grasp concepts of infinity is provided by the transfinite ordinals, since it can be viewed as a continuation of the usual counting manner of children, but proceeding into the transfinite: 1,2,3,⋯,ω,ω+1,ω+2,⋯,ω+ω=ω⋅2,ω⋅2+1,⋯,ω⋅3,⋯,ω2,ω2+1,⋯,ω2+ω,⋯⋯ Presumably this person has no experience with 6 year olds? This explanation is horrendous haha

He has children (not sure about age right now) and discusses mathematics often with them. His tweets have had many interesting examples.

I do not think he means he would use symbols to explain to children, but that the notion of counting natural numbers that children have easily generalises to counting transfinite numbers.

Re: Is infinity an odd or even number? (2011)

#295

Post by gowld »

RHSeeger wrote: post #215
sritchie wrote: post #199

Earlier quoted context omitted.

It follows from the way addition is defined on top of set theory. "a + b" is implemented as "increment a (the set that represents a) b times". A number is represented in set theory as a set that contains all of the numbers before it. 0, 1, 2 is {}, {{}}, {{} {{}}}... SO! If you start with a finite "a" and increment it infinite times, you still have infinity; you haven't broken out. But if you start with Infinity, the…

Is addition defined _by_ set theory, or is set theory one way of defining addition? If it's the later, then there could be other ways of defining addition that don't have the same results for infinity (because our math system doesn't really "work" for infinity, or 0, depending on the circumstances). I am in no way a mathematician. My question about the definition of addition as it relates to set theory is just that;…

There are many ways of definiting everything. Most of them are equivalent in the ways that matter, which is why math "works" so well as the language of science. Some of them are different in critical ways, which opens up vistas of new objects and concepts.

Re: Is infinity an odd or even number? (2011)

#296

Post by gowld »

13of40 wrote: post #188
jasonjayr wrote: post #40

Earlier quoted context omitted.

Some do, yes. If they have an aptitude for basic sums then pointing out that 3 x 3 is the same as 3 + 3 + 3 sets them down the right path ...

The technique used by the Oregon public school system in the 80s went something like "Hand the child a 10x10 grid of numbers, then tell them, absent of any other context, that they must be memorized." I like your way better.

Mathematics should be taught be mesmerization, not memorization.

Re: Is infinity an odd or even number? (2011)

#297

Post by crazygringo »

jetunsaure wrote: post #167
crazygringo wrote: post #159

Earlier quoted context omitted.

OK, so I guess I'm just understanding that mathematicians arbitrarily decided to prioritize "even" over "odd"? Because as I stated in another comment, you could just as easily say odd cardinality exists if you can find two subsets with the same cardinality and there's one element left over, and otherwise we call it even. So at the end of the day, what you're saying is that ultimately infinity would be even just becau…

Evenness is a more natural condition, so to speak, in that it has a simple definition and is easy to generalize. Having defined an even number, if an integer isn't even, it's odd. To get a feel for why this is convenient, consider that you can generalize by replacing "multiples of 2" with "multiples of n". Then, instead of splitting everything into two sets (even/odd), we can naturally split the integers into n sets…

I understand what you're saying, so thank you, but I still find myself disagreeing.

There are just as many odd numbers as even, so there's nothing more natural about either. They alternate. Yes you can extend to higher multiples, but there's still nothing more natural about multiples of 7 vs. multiples of 7 with remainder 3.

And it's just as easy to say that infinity is divisible by 7, as it is to say that infinity is divisible by 7 with remainder 3:

  [1, 2, 3, 4, 5, 6, 7], [8, 9, 10, 11, 12, 13, 14], ...
  1, 2, 3, [4, 5, 6, 7, 8, 9, 10], [11, 12, 13, 14, 15 16, 17], ...
So the entire idea I'm arguing against is that there's anything more natural, more default, more basic about the concept of "evenness" next to "oddness". The very first natural number, 1, is odd -- not even -- so it's just as easy to say that oddness comes first. But really they're fundamentally complementary -- they require each other, neither is more primitive.

Re: Is infinity an odd or even number? (2011)

#298

Post by quantified »

tobiasSoftware wrote: post #36
Eisenstein wrote: post #9

I want to know if there are more decimal numbers between 0 and 1 than there are integers between 0 and infinity.

There are, and it turns out that this is a significant mathematical concept. The integers between 0 and infinity are defined as "countably infinite". Other infinities are considered countably infinite, or the "same" infinity, if and only if you can arrange it in a list such that each item in the list pairs to an integer in our 0 to infinity list. So the set of even numbers is countably infinite because for every i th…

Cantor's proof is a single attempt. Suppose you could construct a space-filling curve that did indeed map all numbers between 0 and 1 to all integers? Has there been a proof that such a curve does not exist? The fact that his proof leans on a specific set of decimal places at every juncture has always seemed a weakness of his proof, because you can always map any set of numbers from 0 to 1 with any set of decimal places to a set of integers.

Re: Is infinity an odd or even number? (2011)

#299

Post by short_sells_poo »

thriftwy wrote: post #267
tsoukase wrote: post #252

Earlier quoted context omitted.

My 6 and 7 yo's call infinity the "endless number". Well, at least it is a NaN number :) PS: they seem to _know_ that endless*endless > endless but do not dare to admit it

Infinite is just a fancy word for endless, anyway.

Not necessarily:)

A circle is endless, and yet certainly isn't infinite.

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