[gjulianm]

The usual definition of division is the multiplicative inverse, yes. But that does not mean that, as an intellectual curiosity (which is how I take this post) you can't define "division" as having other value. Sometimes that's useful, sometimes it's not. Most of the time it is fun to see how things break and don't break. For example, the infamous 1+2+3+... = -1/12 sum[1]: for most purposes it's a divergent sum but you can find ways to assign a finite value and make that useful.

1: https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B...

[godelski]

The OP says

> The field definition does not include division, nor do our definitions of addition or multiplication.

This is clearly a misunderstanding of not realizing that the definition of division is a symbolic shorthand for inverse multiplication.

Sure, you're right that we can define it another way. But I don't know what we would call an object with three operators. (Someone more knowledgeable in field theory please let me know, it's hard to Google) in the normal field we work with (standard addition and multiplication) we only have two unique operators. Everything else is a shorthand. In fact, even multiplication is a shorthand.

>For example, the infamous 1+2+3... = -1/12

Ramanujan Summation isn't really standard addition though. It is a trick. Because if you just did the addition like normal you would never end up at -1/12, you end up at infinite (aleph 0, a countable infinity). But the Ramanujan Summation is still useful. It just isn't "breaking math" in the way I think you think it is.

But I encourage you to try to break math. It's a lot of fun. But there's a lot to learn and unfortunately it's confusing. But that's the challenge :)

[gjulianm]

The fact that it's a shorthand means that it's not in the definition, just a convention. In fact, nowhere in my studies I saw anyone using 'division' when working explicitly with fields in algebra, it's always multiplicative inverse.

By the way, you cannot do the addition like normal on that series, you don't end up at anything. You can say it diverges and its limit is +infinity (not aleph0, aleph0 is used for cardinality of sets so I think no one would use it as the result of a divergent series). When I said it 'broke math' what I meant was that, in the same way that OP did, it is a way to assign values to things that usually don't have one. I know it does not actually break math.

[jle]

That's not necessarily the definition of division, there are many commonly used definitions in different contracts that are different.