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Comment by kovrik | Hacker News Reader

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 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."

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

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 imaginary number part are also not real is a natural consequence of unfortunate naming choices.

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 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.

> 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.

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).

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