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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)

#351

Post by davrosthedalek »

jameshart wrote: post #203
zorgmonkey wrote: post #174

Earlier quoted context omitted.

No, any rational number with a denominator that has only the prime factors of 2 and 5 will have a finite and exact decimal representation in base 10.

That's exactly the same thing as I said, using different words. And anyway, is Sqrt(2)/2 such a number?

That's not quite what you have said. 3/2 does not have an integer power of 10 as the denominator, but can be written as a decimal. Of course, it can also be written as a fraction with a integer power, for example 15/10. (You are of course right on the 1/sqrt(2) issue)

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

#352

Post by wpietri »

mulmen wrote: post #332
wpietri wrote: post #277

Earlier quoted context omitted.

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.)

Phrases like "you won't always have a calculator at hand" only serve to erode trust in the educator. It's simply not compelling, and for all practical intents and purposes is untrue. Even on backpacking trips I have a cell phone, even if it is off. If you believe mental math is useful then say that and explain the benefits. Students can smell a lie.

That sounds like an excellent thing to somebody who actually said "you won't always have a calculator at hand". Maybe you should find someone like that.

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

#353

Post by catiopatio »

ironSkillet wrote: post #346
dbtc wrote: post #287

Earlier quoted context omitted.

> And then infinitely later There can be no 'and then' after infinitely later. I don't see why stepwise is important but that must be the key to Cantor's proof. If he gives me 1.1 1.2 1.3 and I pair with 1 2 3, then he gives me 1.11 and I pair with 4, that seems fine as far as counting is concerned. The ordering could be entirely random, I don't see how it makes a difference. There will always be enough integers to m…

Mathematicians consider two sets to be of the same size or more precisely "cardinality", if it is possible to construct a 1-1 map of elements from the first set to the second set. These maps can obviously be constructed for sets with finitely many elements, and they can be constructed for sets with an infinite number of elements as well. For instance, the set of all integers has the same cardinality of the set of all…

> It is in this sense that there are infinities of different sizes.

They aren’t actually different sizes, though.

All this proves is that under specific set theoretic assumptions, a contradiction arises if you define “size” as “cardinality” and assume that a particular bijective relation exists between your two infinite sets.

It doesn’t actually mean the sets have different sizes, it just means they differ under a set of assumptions that may (or may not) be useful for your purposes.

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

#354

Post by cabalamat »

MichaelMoser123 wrote: post #333
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

there is a difference between infinity and not a number, NaN isn't event equal to itself, in nodejs you get: > Infinity + Infinity Infinity > Infinity - Infinity NaN > Infinity == Infinity true > NaN == NaN false The fun thing: NaN to the power of zero (because technically the type of NaN is a ... number) > NaN ** 0 1 > typeof(NaN) 'number' By the same logic, NaN is also even because > NaN + NaN NaN but i think it's…

    > NaN == NaN
    false
Reminds me of SQL, where NULL isn't equal to NULL.

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

#355

Post by jjeaff »

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 only thing that is a little off here to me is that I don't think there is mathematical notation for "many" or "few". And yet infinity does have mathematical notation and is used in some equations, no?

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

#356

Post by [deleted] »

catiopatio wrote: post #353
ironSkillet wrote: post #346

Earlier quoted context omitted.

Mathematicians consider two sets to be of the same size or more precisely "cardinality", if it is possible to construct a 1-1 map of elements from the first set to the second set. These maps can obviously be constructed for sets with finitely many elements, and they can be constructed for sets with an infinite number of elements as well. For instance, the set of all integers has the same cardinality of the set of all…

> It is in this sense that there are infinities of different sizes. They aren’t actually different sizes, though. All this proves is that under specific set theoretic assumptions , a contradiction arises if you define “size” as “cardinality” and assume that a particular bijective relation exists between your two infinite sets. It doesn’t actually mean the sets have different sizes, it just means they differ under a s…

[deleted]

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

#357

Post by 1nvic7u5 »

Why can't infinity be both even and odd? The field of numbers should allow a possibility for both even & odd to happen at the point of infinity. Of course, infinity would then be a point where the definition of a number could break down. If it doesn't, then it can be both even and odd.

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

#358

Post by pclmulqdq »

MichaelMoser123 wrote: post #338
mikepurvis wrote: post #336

Earlier quoted context omitted.

Huh. Most of those make sense to me, but infinity == infinity being true definitely feels like risky business. Algebraic limits is full of even some pretty trivial scenarios where infinity divided by a lesser-infinity turns out to be a real number— those cases where the two infinities are definitely not equal to each other.

Also: in javascript you can divide by zero > 1/0 Infinity But python throws a divide by zero error: >>> 1/0 Traceback (most recent call last): File " ", line 1, in ZeroDivisionError: division by zero Still you do have infinity and nan in python - because these are part of the floating point spec. >>> float('inf') - float('inf') nan >>> float('inf') == float('inf') True >>> NAN=float('inf') - float('inf') >>> NAN == N…

Dividing by 0 or -0 is a valid floating-point operation because there's an infinity in the number system, and JS uses double precision floating point for all numbers. Python has an integer type and a double type, and division by 0 is disallowed for integers, but okay for doubles.

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

#359

Post by feoren »

lisper wrote: post #271
feoren wrote: post #268

Earlier quoted context omitted.

> 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.

Of course you cannot actually end up at ω by taking two steps at a time, but your point is still well taken.

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

#360

Post by feoren »

dbtc wrote: post #287
feoren wrote: post #264

Earlier quoted context omitted.

> Whatever method you use to generate your decimals, you can just slap an integer on each step of the way. You'll never run out of integers. Exactly correct! This holds true of everything you can generate stepwise, even infinite sets. Cantor proved that you cannot "generate" (stepwise) all Reals between 0 and 1. Any infinite set you can generate stepwise is Countably Infinite. > I'll put Cantor and his proof in a box…

> And then infinitely later There can be no 'and then' after infinitely later. I don't see why stepwise is important but that must be the key to Cantor's proof. If he gives me 1.1 1.2 1.3 and I pair with 1 2 3, then he gives me 1.11 and I pair with 4, that seems fine as far as counting is concerned. The ordering could be entirely random, I don't see how it makes a difference. There will always be enough integers to m…

> If he gives me 1.1 1.2 1.3 and I pair with 1 2 3, then he gives me 1.11 and I pair with 4, that seems fine as far as counting is concerned.

Exactly correct! Any bijection between the naturals and the reals would suffice to show that they're the same cardinality; the order does not matter. I think where you're getting confused is just in who's trying to do what; who's the "protagonist" and "antagonist" in the proof.

Cantor is not trying to overwhelm you with so many real numbers that you run out of integers. Instead, he completely accepts and agrees with everything you're saying. And then he says: okay, pick any numbering of the reals you like. 1.11 is 4, 1.111 is 76, and 1.1111 is 445662323. It doesn't matter. You pick the pairing. Write your pairing down on an infinitely long sheet of paper. If the reals and integers have the same cardinality, there must be some way to write them all down on an (infinitely long) list. Pick any one and write it down.

Cantor's only job now is to show you that any real number exists that is not on your list. To do this, he constructs a number a digit at a time. He looks at the 1st digit of the 1st number, and writes down a different digit for his 1st digit. He looks at the 2nd digit of the 2nd number, and writes a different one for his 2nd digit. He looks at the nth digit of the nth number and writes a different one down for that digit, for every digit. Real numbers never run out of digits, so this goes on forever.

If this number he has written down is on your list, you should be able to point to a number on your list and say "Aha! You see, that is just real # 65,334,649!" but you can't, because it's different from that number in its 65,334,649th digit. It is truly different from every number on your list. And so there are more reals than integers.

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