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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)

#271

Post by lisper »

feoren wrote: post #268
lisper wrote: post #86

To explain to a six-year-old I would start by telling them that there are many different kinds of infinity, not just one. Some infinities are odd, others are even, and others are neither. It matters whether you are asking "how many" (cardinals) or "in what position" (ordinals). For regular finite numbers, cardinals and ordinals are (more or less) the same, but for infinities they behave differently. Then, if they wan…

> It matters whether you are asking "how many" (cardinals) or "in what position" (ordinals) But "even" and "odd" are all about whether you can partition something into an equal number of pairs or not. If you're asking "in what position" (ordinals), you've explicitly said you're not in the realm of counting sets of things. I would argue division makes no sense in the realm of ordinals! Everyone is saying the transfini…

> But "even" and "odd" are all about whether you can partition something into an equal number of pairs or not.

Sez you. I can just as easily define even and odd in terms of whether or not I can arrive at a given position in a (potentially infinite) sequence taking by taking two steps at a time.

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

#272

Post by crazygringo »

smallnamespace wrote: post #166
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…

> arbitrarily decided Modern mathematics is all about coming up with definitions and rules that give rise to interesting (to a mathematician!) properties when further investigated. The definition given naturally lets the ordinal numbers continue the odd/even/odd... pattern. Choosing the alternative definition would not. In one sense that's 'arbitrary' because we decided on one definition over another. But another sen…

Thanks, but you may have misunderstood the definition I have for defining odd numbers, because that corresponds equally to the natural numbers as well.

So there is no better parity rule as you say, it is entirely arbitrary. It's not extending the same pattern, it's seeing that there are two ways of extending it and picking one arbitrarily that happens to prioritize even. When you could have just as easily prioritized odd.

So that's not an argument for why infinity is even, or should be. It's just a decree, an arbitrary labeling, the flip of a coin.

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

#273

Post by sawyna »

paulddraper wrote: post #219
Bilal_io wrote: post #218

Earlier quoted context omitted.

I read the title and I sarcastically thought "ask JavaScript!" Your comment didn't disappoint. Importing the is-even package is the cherry on top.

The amazing thing is that the is-even package depends on the is-odd package. I would have thought the reverse, but ¯\_(ツ)_/¯

The is-odd package depends on is-number!

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

#274

Post by mulmen »

retrac wrote: post #193
13of40 wrote: post #188

Earlier quoted context omitted.

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.

Ontario's 1990s curriculum was pretty awesome. The idea of dimension and sets were both introduced simultaneously and joined, using multiplication. Started in the 2nd grade and they just kept elaborating. Number lines and groups of items. (Tied it into geometry, too. Square numbers came up by at least 4th grade.) What is 3 x 3 but moving 3 units, 3 times in one dimension? Now, memorize these tables up to 12 x 12, you…

> you won't always have a calculator at hand.

I do though. I still blow minds when I put my iPhone calculator in scientific mode. Math education is important for many reasons. But teaching it as a practical survival skill using no tools does a disservice to the student. Either it is useful as a problem solving exercise or it is a practical skill that should take advantage of tools. "Just memorize this stuff" isn't useful because it backfires into hating learning. Nothing about math makes it ideal for memorization and none of my math teachers spent any time on study skills.

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

#277

Post by wpietri »

mulmen wrote: post #274
retrac wrote: post #193

Earlier quoted context omitted.

Ontario's 1990s curriculum was pretty awesome. The idea of dimension and sets were both introduced simultaneously and joined, using multiplication. Started in the 2nd grade and they just kept elaborating. Number lines and groups of items. (Tied it into geometry, too. Square numbers came up by at least 4th grade.) What is 3 x 3 but moving 3 units, 3 times in one dimension? Now, memorize these tables up to 12 x 12, you…

> you won't always have a calculator at hand. I do though. I still blow minds when I put my iPhone calculator in scientific mode. Math education is important for many reasons. But teaching it as a practical survival skill using no tools does a disservice to the student. Either it is useful as a problem solving exercise or it is a practical skill that should take advantage of tools. "Just memorize this stuff" isn't us…

Nah. As somebody who ends up doing a ton of mental math, I think it's valuable. Yes, they should also learn how to use tools. But developing a feel for numbers is valuable, and I think that is much harder to do if one always relies on a calculator. (And yes, of course, this should be learned in a way that doesn't involve the kids hating it. But that's possible.)

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

#278

Post by chowells »

amelius wrote: post #71
dkarl wrote: post #55

Earlier quoted context omitted.

In mathematics, you can define things in different ways to get different answers. Ways of defining things tend to be highlighted as true (in at least some context) if they are interesting and useful, and ignored if not. I don't think the definition based on "dividing into pairs" is particularly interesting or useful in the context of the child's understanding of numbers, because it's too vague to be useful, and it do…

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 the ordinal numbers.

These are things with very precise models and interpretations. No hacks at all.

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

#279

Post by [deleted] »

crazygringo wrote: post #272
smallnamespace wrote: post #166

Earlier quoted context omitted.

> arbitrarily decided Modern mathematics is all about coming up with definitions and rules that give rise to interesting (to a mathematician!) properties when further investigated. The definition given naturally lets the ordinal numbers continue the odd/even/odd... pattern. Choosing the alternative definition would not. In one sense that's 'arbitrary' because we decided on one definition over another. But another sen…

Thanks, but you may have misunderstood the definition I have for defining odd numbers, because that corresponds equally to the natural numbers as well. So there is no better parity rule as you say, it is entirely arbitrary. It's not extending the same pattern, it's seeing that there are two ways of extending it and picking one arbitrarily that happens to prioritize even. When you could have just as easily prioritized…

[deleted]

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

#280

Post by chowells »

srcreigh wrote: post #168
FabHK wrote: post #102

Earlier quoted context omitted.

Sure they are. You can define a one-to-one mapping, they're equal.

You’re thinking of isomorphic, not equal.

They meant "their cardinalities are equal". It's honestly an easy mistake to make, especially if typing on a small screen. Or especially if having a discussion where sizes of infinity are already being discussed.
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