Why is division by $0$ not defined in division theorem?

Question

In the general division theorem, there is a condition that when considering $\frac{a}{b}$, we require $b\ne 0$. My question is, why is $0$ left out? In school we were taught that anything divided by $0$ was infinity so we carried out the calculations by taking eg. $\frac{10}{0}=\infty$. Thanks for the insights!

But that's the correct answer. You must be looking for an incorrect answer. What would you like it to say? — MJD, Aug 11 '20 at 02:54
"In school we were taught that anything divided by 0 was infinity so we carried out the calculations by taking eg. $\frac {10}0=∞$" But you can't do anything with that. And it doesn't work. $2 = \frac {10}{5} = \frac {10}0\cdot \frac 0{5} =\frac {10}0\cdot \frac {1}{\frac 50} = \infty \cdot \frac 1 {\infty} = \frac {10}0\cdot \frac 1{\frac {10}0} = \frac {10}0\cdot \frac 0{10} =\frac {10}{10} = 1$.... It's pointless. Utterly pointless. — fleablood, Aug 11 '20 at 04:05
"it's not that it's pointless, it's wrong." Well.... that depends on what is right. We can define anything anyway we want. All that matters is its consistent. So we could make up some things we call numbers and some operation we call division and some rule that any number divided by $0$ is the number we call $\infty$ and that wouldn't be "wrong" .... but it wouldn't be the arithmetic we know and love and the arithmetic it would be would be .... utterly pointless. — fleablood, Aug 11 '20 at 04:34
And yes, if he was taught that anything divided by $0$ was infinity, then he was lied to. .... Now that I think about it I was taught many inconsistent thing is school and I think I was taught that. But I was also taught it was wrong as well. And I was even taught that we couldn't divide by $0$ because it would be infinity. .... they meant well but teachers don't always know what they are talking about. It is not true (and in fact it is fairly meaningless) to say "$\frac {10}0 = \infty$". It's just not true. — fleablood, Aug 11 '20 at 04:39

score 1 · Accepted Answer · answered Aug 11 '20 at 02:56

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Let us suppose that we could define division like that. Let $a$ and $b$ be two numbers and say that $$ \frac{a}{0}=\infty\quad\textrm{and}\quad\frac{b}{0}=\infty. $$ Therefore, we have that $$ a=0\cdot\infty=b.$$ That is, $a=b$ for any two numbers $a$, $b$. There are many contradictions like this you can come up with. But lets think about it differently. What is division? The expression $a\div b$ is saying "what number times $b$ is equal to $a$?". So, what number times $0$ is equal to $10$?

answered Aug 11 '20 at 02:56

is it the fact that we do not know what number times 0 is equal to 10 that it is left undefined? – kofhearts Aug 11 '20 at 03:04
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It's not that we don't know, it's that one doesn't exist. – Aug 11 '20 at 03:05
how do you prove it doesn't exist? is there a proof? – kofhearts Aug 11 '20 at 03:06
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$x\cdot 0 =0$ for all $x$ – Aug 11 '20 at 03:06
what is the proof of x.0 = 0 for all x? – kofhearts Aug 11 '20 at 03:09
It depends what axiomatic theory you begin with. There will always either be a proof that $x\cdot 0=0$ or it will itself be an axiom. The latter is the case for Peano's axioms, the former for set theoretic models. – Aug 11 '20 at 03:15
thanks for help! – kofhearts Aug 11 '20 at 03:17
Let $0\cdot x = K$ Then $K = 0\cdot x = (0 + 0)\cdot x = 0\cdot x + 0\cdot x = K + K$. So $0 = K-K = K + K - K = K + 0 = K$. So $K=0$. So $0\cdot x = K = 0$. That's the proof. – fleablood Aug 11 '20 at 04:00
Infra-ten. It's a secret to everybody. – Trevor Aug 11 '20 at 05:51
@tomasliam The answers at Why not to extend the set of natural numbers to make it closed under division by zero? discuss in detail what kinds of things go wrong if we try to allow division by zero. It can be made to work, but not in any useful way. – MJD Aug 12 '20 at 19:27
@MJD You can define anything you like. My point is it won't match with the usual properties of division and multiplication, and if you expect it to do so, many contradictions will arise. – Aug 13 '20 at 22:11
Yes, we are in agreement. As you say, many contradictions will arise; the answers I linked to give details. – MJD Aug 14 '20 at 00:49

score 1 · Answer 2 · answered Aug 11 '20 at 02:57

Say you define $a / 0 = x$, so $a = x \cdot 0$, i.e., $a = 0$, regardless of what $x$ may be. No other $a$ can be "divided" by 0, and the "result" of that "division" is anything whatsoever. The idea of "inverse of 0" just can't be twisted into making sense without throwing out much of the rest of the properties of the operations on numbers.

Why is division by $0$ not defined in division theorem?

2 Answers2