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 much easier to grasp that very abstract (but fundamental) idea of 'number'. That it is not just 1,2,3...
Because otherwise, we get people thinking "Complex numbers do not exist, that is just a silly thing used by mathematicians that has nothing to do with the real world, it is useless."
The idea that real numbers are real, imaginary numbers are not real, and complex numbers that have a real number part and an imaginary number part are also not real is a natural consequence of unfortunate naming choices.
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 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.
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 love ops explanation because it separates infinity (as a number) and infinite process (counting).
Thinking back to that bug last week, where someone forgot that you can't just compare ever-incremented sequence numbers with 'x < y'... Maybe we don't want to encourage that idea :)
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...
"Do that process twice" "Perhaps after you're done putting apples " "..."
the point is, that the process is never finished ... just like you can never have a biggest number...just like the table has no end.
Defined as infinity. Kids do get that.
What they usually don't get are abstract mathematic concepts.
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 adding.