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How to explain infinity to kids

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Re: How to explain infinity to kids

#21

Post by kovrik »

hutzlibu wrote: post #17
kovrik wrote: post #13

Earlier quoted context omitted.

That's an infinite process (which is an awkward way to describe anything), but not infinity. You'll get more and more natural numbers, but will never actually get infinity. I love ops explanation because it gives more natural picture: there are _other_ numbers beyond naturals. Then you tell kids there are also negative numbers (integers) and rational numbers and real and complex and so on. I believe that way it is mu…

But the sum of them is infinity, right? And I thought we are talking about kids? There are not many kids who would understand the professors way. So isn't it first about making them understand the concept of infinity? That they can imagine it, before you bombard them with other abstract math concepts?

I think that describing infinity as an infinite process is bad because that way we are basically saying that this infinite process and infinity is the same thing!

I love ops explanation because it separates infinity (as a number) and infinite process (counting).

Re: How to explain infinity to kids

#23

Post by monitorman »

kovrik wrote: post #13
hutzlibu wrote: post #8

Why so complicated? Just ask them to do i++ and never stop. That is also infinity. (or in simpler words, imagine you have a table than you put one apple to it, then one again, amd again and again and never stop. And yes, they would also soon understand, that the table needs to be infinitly big) edit: of course it is about the sum of the process. they should imagine the pile of things or the number if you never stop a…

That's an infinite process (which is an awkward way to describe anything), but not infinity. You'll get more and more natural numbers, but will never actually get infinity. I love ops explanation because it gives more natural picture: there are _other_ numbers beyond naturals. Then you tell kids there are also negative numbers (integers) and rational numbers and real and complex and so on. I believe that way it is mu…

So this article should be titled "How to Explain Infinity to Adults"?

Re: How to explain infinity to kids

#24

Post by scandox »

But before you do that it's worth just asking them what they think comes after everything else and see what they say. Because kids often have really interesting ideas on those kinds of topics and once you tell them something, then their ideas get pushed out.

Re: How to explain infinity to kids

#25

Post by romwell »

hutzlibu wrote: post #8

Why so complicated? Just ask them to do i++ and never stop. That is also infinity. (or in simpler words, imagine you have a table than you put one apple to it, then one again, amd again and again and never stop. And yes, they would also soon understand, that the table needs to be infinitly big) edit: of course it is about the sum of the process. they should imagine the pile of things or the number if you never stop a…

>Why so complicated?

Because ifinity is complicated.

>Just ask them to do i++ and never stop. That is also infinity.

What, specifically, is infinity here? You are being vague here. Is it the process of counting up and never stopping? (that's the calculus infinity) The number you get in the end? (that's the infinite ordinal, but that is, as you noted, complicated) The number of numbers? (that's the cardinality of the set of integers)

Note that you can't say that the infinity is all that. The infinities I defined here are all radically different notions!

>In simpler words, imagine you have a table than you put one apple to it, then one again .. and never stop. [..] they would also soon understand that the table needs to be infinitely big

That's another infinity here - we're talking about size of geometric object here.

For starters, what happens if each next apple is half the size of the previous one? (Your assumption is false then!).

With equally-sized apples, it's still close enough to the notion of cardinality.

The problem with this explanation is that it doesn't help understand the nature of infinity, so it's not an explanation really - but a good start.

Some questions to explore from here:

1. Do that process twice. Now you have two infinitely big tables, but each has an end. Put them together end to end. Did you just get a longer table? Are there more tables on this combined table than you had on just one table?

2. Perhaps after you're done putting apples on the table, you realize that one apple rolled under the table. The table is absolutely full, but you want to put another apple on it. What do you do?

3. You find a pile of apples, each having a label consisting of a rational number on them. You notice that no two apples have the same label. Will they fit on the table? Can you put them without removing any apples already them? (The answer here is, mind-bogglingly, yes!)

4. Same question, but now the labels are infinite strings. (The answer is no, by the way, but it's by no means ovbious![3])

Etc, etc, etc... Essentially, Hilbert's Grand Hotel[1][2] with apples. I highly recommend reading this to get a decent understanding of one of the infinities.

[1]https://medium.com/i-math/hilberts-infinite-hotel-paradox-ca...

[2]https://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert...

[3]https://en.wikipedia.org/wiki/Long_line_(topology)

Re: How to explain infinity to kids

#26

Post by antidesitter »

romwell wrote: post #11

Picking the first infinite ordinal as the infinity to explain to kids might not be the best choice for every kid, though. Oridnals are tricky. I am not too comfortable with them myself, and I say that as an adult with a degree in math! The thing is, this kind of infinity just doesn't come up that often when dealing with other objects in math. Even though, as Tom Lehrer sang, one can count up to infinity - or somewher…

I agree. Cardinal numbers (and bijections) are probably easier to start with.

Re: How to explain infinity to kids

#27

Post by peteretep »

If we're going for useful lies to tide kids over, I quite like the explanation in the Postgres docs:

> infinity (date, timestamp) later than all other time stamps

I think you could tell a kid who wasn't quite ready for Aleph numbers that:

> infinity is a useful made-up number that's bigger than all other numbers

Which is useful when kids first hear about it, I guess.

I think my first "practical" introduction to infinity came with "Space is infinitely big", and both mjd's explanation and my own fails at that point.

Re: How to explain infinity to kids

#28

Post by Izmaki »

I don't understand this explanation, and I'm an adult with an interest in mathematics and a degree in software development (although not mathematical per se, they usually go hand-in-hand). I like how "w" is the smallest number you cannot count to - if I understand it right it's almost 0 but not quite - but I don't understand the bonus questions and answers.

If this is the best "explain it like I'm 5" explanation of infinity, I believe I can think of a few examples that give a better idea of it. Heck, even the concept of "never ending" seems simpler to me. "Never ending" + 1 is still never ending.

Re: How to explain infinity to kids

#29

Post by duncancarroll »

I like the concept, but if I may editorialize, I feel the phrasing needs work. "The smallest number you can't count to" is a negative statement, which makes it confusing right off the bat.

What do you mean, a number I can't count to? If I'm 8 years old this is like throwing a null pointer exception in my brain.

Re: How to explain infinity to kids

#30

Post by throwawaymath »

kovrik wrote: post #18
dragonwriter wrote: post #15

Earlier quoted context omitted.

I think the problem with people thinking complex numbers don't exist actually stems from the imaginary component and the same problem of imaginary numbers, which is rooted not in failing to teach about other kinds of numbers but in the name. Especially given the name of the real numbers. The idea that real numbers are real, imaginary numbers are not real, and complex numbers that have a real number part and an imagin…

Agree. Scientists love confusing and weird names. For me, back in school days (or was it university?) it was a revelation when I came across quaternions. It suddenly clicked. I finally understood that there was nothing special about complex numbers (despite their special names and weird look). It was just an extension and a very intuitive one! I finally understood that real numbers are the same thing: tuples. They ju…

> I finally understood that real numbers are the same thing: tuples. They just happen to have exactly one element, hence we omit parens and everything else and just write that element (number)! Complex numbers have 2 elements (real and imaginary). Quaternions - four. And so on.

Yes, but if you go too far with thinking this way it becomes easy to confuse complexes, quaternions, octonions, etc with 2-,4-, and 8-dimensional vectors. The additive and multiplicative behavior of complexes, etc changes in increasingly pathological ways as you increment the dimension. This is not the case with vectors.

In many cases you can safely replace R^2 with C. But there are specific cases where you cannot, because treating the complexes as just a pair of real numbers doesn't work the same way as if it was just a vector. Differentiable functions come to mind because they behave differently in R^2 than C, and when you're working with rings (instead of fields and vector spaces) they are also different. In a lot of places the isomorphism between R^2 and C is actually a happy accident rather than a definition.

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