Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Is infinity an odd or even number? (2011)

math.stackexchange.com

Post reply on HN
Print view View original

51–60 of 398 posts

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

#51

Post by singularity2001 »

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

Transfinite ordinals also known as hyperreals should really be taught in school as they make many parts of math easier: algebraic definition of derivatives (including algebraic derivative of step functions without dirac 'density') and yes: natural addition and multiplication.

https://en.wikipedia.org/wiki/Hyperreal_number

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

#53

Post by gorkish »

warent wrote: post #25
alexb_ wrote: post #12

Earlier quoted context omitted.

No it isn't. If you ask a child what comes after infinity, "Infinity + 1" is pretty much the default answer. Any kid who knows multiplication knows "Infinity + Infinity" is the same as "Infinity Times Two". The answer of "Infinity TIMES Infinity" is also popular for kids to say when they know a number bigger than their friend (who just proclaimed infinity is the largest number).

Six-year-olds know multiplication?

They dont usually know formal arithmetic multiplication but they well understand the concepts of repeated addition and subtraction. Most places in the world do start teaching multiplication at age 6/7.

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

#54

Post by nh23423fefe »

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

> I would focus on the principal idea: whether finite or infinite, a number is even when it can be divided into pairs.

why misquote someone and claim their idea is hard to understand?

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

#55

Post by dkarl »

irrational wrote: post #14

>To explain the idea to a child, I would focus on the principal idea: whether finite or infinite, a number is even when it can be divided into pairs. For finite sets, this is the same as the ability to divide the set into two sets of equal size, since one may consider the first element of each pair and the second element of each pair. The answer this quote came from is amazingly obtuse, but it does make me think that…

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 doesn't lead to any insights.

The definition based on transfinite ordinals explained in the same answer does seem interesting, and I wouldn't be surprised if it were useful. I think this is a case of simplification gone wrong, where everything interesting was lost in the translation to more accessible terminology.

A more honest thing to say to a child would be that the way even and odd are defined only make sense for finite numbers. It's true for the definition they know, and it introduces them to the important insight that logical rules that are created for one kind of thing might not work when applied to something else. I think this would be more accessible and stimulating for a six-year-old than giving them a half-baked verbal imitation of a result from transfinite mathematics.

They'll be thrilled later if they study math and discover that there are definitions of "infinity" and "even" that yield an answer to their childhood question.

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

#56

Post by taneq »

irrational wrote: post #14

>To explain the idea to a child, I would focus on the principal idea: whether finite or infinite, a number is even when it can be divided into pairs. For finite sets, this is the same as the ability to divide the set into two sets of equal size, since one may consider the first element of each pair and the second element of each pair. The answer this quote came from is amazingly obtuse, but it does make me think that…

Infinities aren’t comparable for equality… are they?

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

#58

Post by gus_massa »

The problem with transfinite is that you lose commutatively.

Flowing the standard notation, where the usual infinite in the integer or the real line is "ω = ∞ = 1,2,3,..."

ω+1 = ω+1 , i.e. "the next thing after infinity"

1+ω = ω , i.e. "the same infinity as before"

2ω = ω , i.e. "the same infinity as before", so it's even

1+2ω = ω , i.e. "the same infinity as before", so it looks odd, but don't fall in that trap

ω2 = ω2 , i.e. "two infinities chained together", that is weird

Two more weird example from https://en.wikipedia.org/wiki/Even_and_odd_ordinals

> Unlike the case of even integers, one cannot go on to characterize even ordinals as ordinal numbers of the form β2 = β + β. Ordinal multiplication is not commutative, so in general 2β ≠ β2. In fact, the even ordinal ω + 4 cannot be expressed as β + β, and the ordinal number

> (ω + 3)2 = (ω + 3) + (ω + 3) = ω + (3 + ω) + 3 = ω + ω + 3 = ω2 + 3

> is not even.

For a six year old, I'd tell that infinite is not a number so it's not even or odd. If s/he even get's a Ph.D. in math, s/he will understand.

Moreover, I remember when I was a graduate T.A. that one day before lunch I went to a class to learn about the https://en.wikipedia.org/wiki/Alexandroff_extension in the morning. (The idea is that you add one ∞ to a set of numbers to get a compact set. And in the new set ∞ is (almost) a number as good as the other numbers.) After lunch, I went to teach limits to first years students, and with a total straight face I told them that ∞ is not a number.

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

#59

Post by remram »

sukilot wrote: post #4
remram wrote: post #3

Infinity is not a number. If you want to extend even/odd to it, you can pick whatever you want.

I suggest you reread the OP and ask questions, because you seem to have overlooked some of the ideas explained therein, such as transfinite ordinals.

You can build those and then pick if you want them even or odd, which are regular-number concepts. That is exactly what they did, and what I described. You go and re-read it.

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

#60

Post by gus_massa »

warent wrote: post #25
alexb_ wrote: post #12

Earlier quoted context omitted.

No it isn't. If you ask a child what comes after infinity, "Infinity + 1" is pretty much the default answer. Any kid who knows multiplication knows "Infinity + Infinity" is the same as "Infinity Times Two". The answer of "Infinity TIMES Infinity" is also popular for kids to say when they know a number bigger than their friend (who just proclaimed infinity is the largest number).

Six-year-olds know multiplication?

My six-year-old likes Numberblocks https://en.wikipedia.org/wiki/Numberblocks https://www.google.com/search?q=Numberblocks . She knows a little more about multiplication than what I expected, probably 2x and 3x when x is small, (but as other sibling comments say not a general theory or how to calculate 287263 * 137167).
Post reply on HN