Couldn’t the result of division by zero be “defined”, just like the square root of -1?
Edit: Wow, thanks for all the great answers! This thread was really interesting and I learned a lot from you all. While there were many excellent answers, the ones that mentioned Riemann Sphere were exactly what I was looking for:
https://en.wikipedia.org/wiki/Riemann\_sphere
TIL: There are many excellent mathematicians on Reddit!
In: 1691
This is basically a slightly involved question. My mathematical answer would be that including a square root of -1 simply creates a larger ring. Not only that, but starting with something nice (like the integers, the rationals, or the real numbers), and including i, gives you something equally nice or possibly nicer.
The integers (a Euclidean domain) become the Gaussian integers (also a Euclidean domain).
The rationals (a field, namely the field of fractions of the integers) become another field, the field of fractions of the Gaussian integers.
The reals (another field, this one being “Cauchy complete”) become the complex numbers (also a Cauchy complete field, but it’s also algebraically closed – even better).
To put this in ELI5 terms, you don’t really “lose” that much by including a square root of -1. And in the case of real to complex, you actually gain something extremely important (algebraic closure).
On the other hand, including something like 1/0 (a multiplicative inverse of 0) breaks a lot of things. Your structure can no longer be a ring, unless it’s the trivial ring (in other words, if you want “normal” arithmetic, you are forced to have 0 = 1, which it turns out is super boring).
Here’s a quick proof that should be ELI5 compatible. Suppose x = 1/0, a number we want to include. By definition, we are saying that 0x = 1 (it “cancels out” the 0). The problem is that 0 = 0+0, so we can substitute that in: (0+0)x = 1. Now, an important thing we want multiplication to do is distribute (expanding the brackets, in school math terms). So we should be able to say 0x+0x = 1. Now we have a “problem”, because 0x = 1 at the start, so this equation says 1+1 = 1, in other words 2=1, and 1=0.
So, again in ELI5 terms, being able to divide by 0 causes you to “lose” a whole lot. Your nice number system can no longer have normal arithmetic, unless you want 0 to equal 1, which it turns out means this is the only number (every number is equal to this “0 & 1” number). So you have to start making weird arithmetic rules, but in doing this you lose other nice things, and so on. You can do this, but these number systems are not as nice as the things I talked about with the imaginary number i.
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