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I voted incorrectly on this poll, which is particularly embarrassing as I have actually taught ring theory.
The answer comes down to convention, but like many conventions in mathematics, it's not entirely arbitrary but rather is chosen to minimize the number of exceptional cases one has to mention when writing common types of mathematical arguments.
While it's impossible to divide by zero (in any field or division ring) the notion of "divisible" pertains to integers (and other rings) which do not have a division operation.
On page 22 of N. Jacobson, Basic Algebra I, 2nd Edition, the notion of divisibility is defined for rings simply as follows: If for any three ring elements a, b, c (e.g. integers, in case we are talking about ordinary arithmetic) the equation ab=c holds, then a and b divide c and are called divisors of c.
That's it. There's no exception for zero.
In particular this means that every integer is a divisor of zero, including zero.
This convention is chosen to ensure that in the integers (or any principal ideal domain) "m is divisible by n" is equivalent to "the ideal generated by n contains the ideal generated by m." In the integers, the ideal generated by n is nZ, the set of all integer multiples of n, so for example, saying that the set of even numbers 2Z contains the set of multiples of six 6Z is equivalent to saying that 2 divides 6. The ideal generated by zero is 0Z={0}, and is contained in every ideal, so we say that every integer divides zero.
On page 90 of the same book, we get a definition of a "zero divisor" which says that x is a zero divisor iff there exists a non-zero y such that xy=0. This means that even though every integer is a divisor of zero, no integer is a zero divisor, except zero.
And of course no number has a quotient by zero.
(By the way, if you're curious why I answered wrong, it's because I misremembered "non-zero" exception in the zero divisor definition and thought it stated that both factors had to be non-zero.)
So, I feel bad for @tfae and the others who had more faith in Manifold users. It turns out we are all mid on the IQ bell curve meme.
Left: 2x0=0 so 0 is divisible by 0.
Middle (including me): Noooo you can't divide by zero!
Right: 2x0=0 so 0 is divisible by 0.
I thought this was clearly no, but since so many people are voting yes I’ll explain my reasoning.
If you allow 0/0 to have any defined value, division and multiplication break entirely. Abstract algebra (the branch of math that defines what addition, subtraction, multiplication, and division mean) has a thing called a field, which is where the definition of division comes from.
It explicitly states that any division by zero is not allowed. If it were allowed, the definitions of multiplication and division would clash. Here’s why:
For division to make sense you need a single multiplicative inverse for each number, such that x * 1/x = 1.
In the definition of multiplication it states that 0 * x = 0 for all numbers.
To be clear, these aren’t just things that are true about multiplication and division, they are in the definition. If these properties don’t hold for all numbers, then you’re no longer doing multiplication and division, it’s something else.
From the first statement, we get 0 * 1/0 = 1,
and the second can be shuffled around to get 0 * 1/0 = 0. So 0=1. This contradiction is so serious that the first rule about inverses explicitly says it doesn’t work for the zero element, because it will always cause issues with the rules of multiplication.
There are alternative forms of division that allow division by zero (see Wheel Theory) but these are not standard, and have to introduce a special number for the result of 0 / 0 that lies off the number line (for some reason it’s called “bottom”, it looks like this: ⊥) which avoids the contradiction. But this isn’t standard, and is not the same kind of division people use in day to day lives
@TonyPepperoni I'm not a mathematician, but I think by some definitions (at least the one used in my discrete math class) whether 0/0 is actually defined is irrelevant to whether 0 is divisible by 0. This definition is explained in these notes. It specifically says:
"The definition we gave above implies, as we noted, that “0 divides 0”, but this is not the same as saying "you can divide 0 by 0”. The wording is close, but different. The definition in this section defines divisibility in terms of multiplication; it is not the definition of dividing in term of multiplying by the multiplicative inverse."
Maybe a mathematician can comment on whether or not this definition is the one that is most commonly used?
@Arky Good points, I was mixing the concepts of disability with division, it sounds like the two are less related than I thought. With the way divisibility is defined it looks like 0 is divisible by itself, although I still wonder if that fact can be useful in any way, since it can’t be divided.
Would an alternate definition of divisibility that says 0 is not divisible by 0, change math in any meaningful way? Because if it doesn’t, then the divisibility is undecided.
Either way, I’m leaning a lot less towards no now. Thanks for the help :)
Would an alternate definition of divisibility that says 0 is not divisible by 0, change math in any meaningful way? Because if it doesn’t, then the divisibility is undecided.
Yes it would. One example is the concept of "the set of numbers divisible by N". This is called an ideal.
One useful property of ideals is that the intersection of two ideals is itself an ideal. But to accept this in general, we must also accept that {0}, the trivial ideal, is a valid ideal. This is the set of numbers divisible by 0.
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@Najawin Choose the option you like better in this context. I think giving further context would bias the poll. But of course, feel free to argue if you think this or that context is more appropriate here!