Math Myth: You cannot divide by 0

[Greg Bernhardt]

From @fresh_42's Insight
https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/

Please discuss!

This always sounded to me as if there was obscure mathematics police that forbids us to do so. Why? Well, there is actually a simple reason: $0$ hasn't the least to do with multiplication and even less with division. The question to divide by $0$ doesn't even arise! The neutral element for multiplication is $1,$ not $0$. That belongs to addition. And there is only one way to combine the two, namely by the distributive law $a\cdot(b+c)=a\cdot b+b\cdot c.$ Let's pretend there was a meaningful way to define $m=a:0.$ Then $$a=m\cdot 0 = m\cdot (1-1)= m\cdot 1 - m\cdot 1= m-m =0$$ So can we at least divide $a=0$ by $0$ and get $m$? I'm afraid not. Have a look at $$m\cdot a= \dfrac{0}{0}\cdot a = \dfrac{0\cdot a}{0\cdot 1}=\dfrac{0}{0}=m\Longrightarrow a=1$$ or $$2m=(1+1)\cdot m=m+m=\dfrac{0}{0}+\dfrac{0}{0}=\dfrac{0+0}{0}=\dfrac{0}{0}=m$$ and we could only multiply $m$ by $1$. So even if we define $m=0/0$ we wouldn't get anything useful in the sense that it would be connected to any number we normally use for calculations.

 

[.Scott]

The normal statement is "division by zero is not defined".
It's not so much that you "cannot" do it. You can logically deduce that the value you seek is 0/0. But when you do, you probably want to try again, since 0/0 is most commonly of no practical use.

 

[emmawarner96]

What is the maximum number of people you can feed? Since you're not really feeding them something, there's no cap. If all seven billion people on the planet turn up at your door and demand their share of pizza, you should say "No problem!" because "their share of pizza" is meaningless. You might say the same thing if you added another seven billion people. What is the maximum number of people you can feed? There’s no answer.

There is no single answer when dividing a number by 0. To divide anything means to separate it into piles of a specific size. It doesn't make sense to break anything down into size zero piles.

 

[Mark44]

What is the maximum number of people you can feed? Since you're not really feeding them something, there's no cap. If all seven billion people on the planet turn up at your door and demand their share of pizza, you should say "No problem!" because "their share of pizza" is meaningless. You might say the same thing if you added another seven billion people. What is the maximum number of people you can feed? There’s no answer.

What you are describing is not division by zero. Here, you are dividing some finite number (1 pizza) by a relatively large number, not zero. In the first case, each person would get $\frac 1{7,000,000,000}$th of the pizza. In the second case, each person would get a slice that was half as large.

Actually, there is at least a theoretical limit to how small the shares could be, since the pizza contains some large number of pizza "molecules." Once we get down to shares that are only a single "molecule" (I realize that there is no such thing as a pizza molecule, but you probably get the idea), then we can't divide the pizza any further with each person getting a piece of the pizza.

If "pizza molecule" bothers you, just take it a bit further down to the atomic level.

 

[pinball1970]

What you are describing is not division by zero. Here, you are dividing some finite number (1 pizza) by a relatively large number, not zero. In the first case, each person would get $\frac 1{7,000,000,000}$th of the pizza. In the second case, each person would get a slice that was half as large.

Actually, there is at least a theoretical limit to how small the shares could be, since the pizza contains some large number of pizza "molecules." Once we get down to shares that are only a single "molecule" (I realize that there is no such thing as a pizza molecule, but you probably get the idea), then we can't divide the pizza any further with each person getting a piece of the pizza.

If "pizza molecule" bothers you, just take it a bit further down to the atomic level.

If we cannot divide by zero then is it not considered a number?

A number on the number line -2, -1, 0, 1, 2?

The number between -1 and 1?

Dividing a number by -1 has a value also 1 and also 0.1, 0.01, 0.001 etc ie approaching infinity so what is happening between the area where everything is undefined before it appears on the negative side?
With something mysterious in between?

Ie is this an infinity is a concept not a number issue? Are Zero and infinity somehow inextricably linked?

None mathematician view of the question

 

[Mark44]

If we cannot divide by zero then is it not considered a number?

A number on the number line -2, -1, 0, 1, 2?

The number between -1 and 1?

Zero is a number. The fact that division by zero is not defined has no bearing on whether it is a number.

Dividing a number by -1 has a value also 1 and also 0.1, 0.01, 0.001 etc ie approaching infinity so what is happening between the area where everything is undefined before it appears on the negative side?
With something mysterious in between?

Not sure what you're asking here. You can divide any real number by any other real number except zero. The divisor can be as small as you like, just as long as it is not zero.

Ie is this an infinity is a concept not a number issue? Are Zero and infinity somehow inextricably linked?

Yes, infinity is a concept unless you're talking about the extended real numbers.