# Dividing by zero

## Main Question or Discussion Point

I'm not sure where this would go specifically, mods move it to where you feel it needs to be if necessary.

I'm in a discussion with 2 guys about dividing by zero. I say its undefined and give the standard reasons why. They say it is. I'm pasting parts of the convo here and would like your input.

Me:
"All equations must be subject to reversiblity in some way. I know that isn't the official term.

8/4 = 2
can be re-expressed
2*4 = 8

8/0 = 0
cannot be re-expressed as
0 * 0 = 8"

Dude #1
"Actually, it does work:

8/0 = infinity -> 0 * (8/0) = infinity * 0 -> ( 0*8 )/0 = infinity * 0 -> 0/0 = 0 * infinity."

Me
"I disagree. And I'm not the only one.

http://www.math.utah.edu/~alfeld/math/0by0.html
http://mathforum.org/dr.math/faq/faq.divideby0.html

Here's a specific quote from the math forum website.

"But maybe you're thinking of saying that 1/0 = infinity. Well then, what's "infinity"? How does it work in all the other equations?

Does infinity - infinity = 0?
Does 1 + infinity = infinity?

If so, the associative rule doesn't work, since (a+b)+c = a+(b+c) will not always work:

1 + (infinity - infinity) = 1 + 0 = 1, but
(1 + infinity) - infinity = infinity - infinity = 0.

You can try to make up a good set of rules, but it always leads to nonsense, so to avoid all the trouble we just say that it doesn't make sense to divide by zero."

Perhaps elaborate on you reasons why?

Dude #2

"Mathmaticians are lazy," that is a direct quote from my Calculus professer.

Humans are imperfect, with imperfect minds, therefore we do not understand all levels of the true complexity of pure mathmatics, therefore the lazy mathmaticians who are dealing with these equations, unable to grasp infinity declare it "off limits" . (Math joke)

However in calculus if you do not know what a function is equal to at some number, you use the values you do know on either side and try to work close to the value you do not know, this is called taking the limit, if the equation seems to approach the same value when coming from either direction this value is the limit at this number. When taking the limt of 1/0 you find the function going to infinity from both directions, therefore the limit of 1/0 is infinty, the same as the limit of 4/2 is 2.

Now do you see why your calculator might display ERROR when you try this home,

you have the wrong calculator.

Dude#1
Discrete Mathematics and Calculus are two seperate branches of mathematics.

Discrete Mathematics, on which all computing machines, from pocket calculators to supercomputers are based, indeed does not allow for infinity at all.

Calculus, on the other hand, which has been essential to physics since Newton, deals infinities all the time and in that field 1/0 = infinity, and reciprocally, 1/infinity = 0.

Calculus does not provide for adding and subtracting infinities, only multiplying and dividing them, but in Quantum Physics, infinity - infinity does equal 0. This does violate the asociative property, but Quantum Physics is so strange anyway that it doesn't really matter.

in discreet mathematics such as calculus, factorials in a way support dividing by zero, although its just a part of the "factorial" law.

For instance 5!/0!

many would say indeterminate, as 5x4x3x2x1 over 0x0 is indeterminate.

However, zero factorial is defined as 1.

ahhhhh :-)

Last edited:

1/0 = positive infinity? negative infinity? zero? one ?

besides, its pointless. meaningless! and there is no room for meaningless mathematics!!

Galileo
Homework Helper
scribe said:
Me:
"All equations must be subject to reversiblity in some way. I know that isn't the official term.

8/4 = 2
can be re-expressed
2*4 = 8

8/0 = 0
cannot be re-expressed as
0 * 0 = 8"
You managed to put your finger on the sore spot.
It becomes evident when we ask the question: "What does division mean?"
It's the inverse operation of multiplication.

The expression $\frac{a}{b}$ denotes a number. Which number?
It's the number which, when multiplied by b, gives a.

Now consider $\frac{a}{0}$. What number would this be? It should be the number which, when multiplied by 0, gives a. However, any number multiplied by zero yields zero.
So you cannot assign a number to $\frac{a}{0}$ in any rational (excuse the pun) way.

Hurkyl
Staff Emeritus
Gold Member
It's undefined simply because that's part of the definition of division for real numbers. You should probably state this clearly once or twice, but usually all you can really do with these people is give clear reasons why it's not useful to make a definition for it.

Dude#2 is almost semi-correct, but I don't know if he knows why. There are two topological spaces that are often useful to use called the projective real line and the extended real line. The former admits a single point "at infinity", and the latter two.

In the projective line, it is correct to say that 1/0 = infinity... but the important thing to know is that this instance of / is a different function than / for real division. (though, both functions agree when the arguments are both real) Of course, 0/0 is still undefined.

In the extended real line, it is even incorrect to say that 1/0 = infinity. It is even incorrect to say that $\lim_{x \rightarrow 0} 1/x = \infin$ -- don't forget that when x goes to 0, x takes on both positive and negative numbers.

But, in any case, neither of these are the real numbers. These systems weren't built for doing arithmetic, they were built for doing calculus and other geometric things. They are not the relevant systems when speaking about arithmetic questions!

t!m
Galileo said:
So you cannot assign a number to $\frac{a}{0}$ in any rational (excuse the pun) way.
Except when a=0. Which is why the "inverse operation of multiplication" explanation doesn't exactly work for me.

Zero times zero equals zero, but zero divided by zero does not equal zero.

Galileo