A thorough explanation on why division by zero is undefined?

That division by zero is undefined cannot be proven without math, because it is a mathematical statement. It's like asking "How can you prove that pass interference is a foul without reference to sports?" If you have a definition of "division", then you can ask whether that definition can be applied to zero. For instance, if you define division such that $x\div y$ means "Give the number $z$ such that $y \cdot z =x$", there is no such number in the standard real number system for $y=0$. If we're required to have that $(x\div y) \cdot y=x$, then that doesn't work when $y$ is equal to zero. In computer languages where x/0 returns an object for which multiplication is defined, you do not have that (x\0)*0 == x. So we can can a class of objects in which we call one of the objects "zero", and have a class method such that "division" by "zero" is defined, but that class will not act exactly like the real numbers do.

Another definition of division is in terms of repeated subtraction. If you take 50 apples and give one apple each to 10 people, then keep doing that until you run out of apples, each person will end up with 5 apples. You're repeatedly subtracting 10 from 50, and you can do that 5 times. If you try to subtract 0 from 50 until you run out of apples, you'll be doing it an infinite number of times.


Consider a problem where you have to divide a finite number like $5$ by zero. $5\div0$ is essentially a request for some number which when multiplied by zero gives you $5$:

$$5\div0=N\implies 0\cdot N=5.$$

Is there a number that when multiplied by zero gives you $5$? The answer is clearly no because any number times zero always gives you zero. Therefore, $5\div0$ is left undefined. "Undefined" here basically means that we can't explain what $5\div0$ really means.

What about the case $0\div0$? $$0\div0=N\implies 0\cdot N=0.$$

We know that any number times zero is zero. This means that $N$ can be any number at all. This kind of division problem gives you an infinite number of answers instead of just one as it should be. Because of this indeterminateness, $0\div0$ is also left undefined.

Here's another very simple example for good measure. You have $7$ pizzas and you want to divide them among zero people. How much pizza will each person get? Well, you have no people to give the pizzas to. You can pose that question and even write it mathematically as $7\div0$, but what could possibly be the answer to this question? Practically speaking, this is unanswerable. In other words, it's not clear what the statement $7\div0$ in this context means. In math-speak, we would say that this is undefined.


My understanding of division by zero goes back to the definition of rings. Let $R$ be a commutative ring and $a,b\in R$ with $b$ a unit in $R$. Then define the fraction $a/b$ as follows: $$\frac{a}{b} = a\cdot b^{-1}$$ i.e., division by $b$ is defined by multiplication with the inverse of $b$.

Since the zero element $0$ in a ring is absorbing (i.e., $a\cdot 0 = 0 = 0\cdot a$) and thus not a unit, division by $0$ is not defined.

Tags:

Arithmetic

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