Why Can’t We Divide By Zero?
Ah, division by zero. One of the first things that math teachers start ramming into pupils’ heads. They say “You cannot divide by zero!” with as much vengeance as Gandalf shouting “You cannot pass!”.
But they usually don’t give you any reasons! It’s more like “You cannot never ever divide by zero!” rather than “Here, look at why division by zero wouldn’t work…”
In the better cases, they tell their students “division by zero is undefined”. That’s not strictly always true, but it’s close enough. But quite often, the teachers say “Anything divided by zero is infinity.” No, no, NO. That’s completely, UTTERLY WRONG.
Infinity is not a number. Seriously. It’s an abstract concept. It’s a shorthand for “gets as big as you like”. You can’t treat it as a number, and you most definitely can’t get it as a solution to an equation. In fact, if you start throwing “infinity” into equations, you can get stuff like:
infinity = infinity + 1
infinity + 0 = Infinity + 1
subtract infinity from both sides
0 = 1
Yay!
But I won’t rant about infinity in this post. Instead, I will attempt to explain exactly why division by zero goes wrong – and why the only time it does NOT go wrong, it gives you a completely useless result.
I’m going to start at the very beginning…
The origins of negative numbers
When humans first started counting, they used only natural numbers, and they only used addition. They would ask questions like “I used to have three cows. A salesman came along, and now that he left, I have 5 cows. How many cows did I get from him?”
In other words, “How much do I add to 3 to get 5?”, or “3+x=5″.
That all worked fine, until one day, a farmer asked this question: “I used to have three cows. A salesman came along, and now that he left, I have only two cows. How many cows did I get from him?”
In other words, “3+x=2″. And we run into trouble, because no matter what you try adding to 3, you will never end up with 2.
So the farmer invented the notion of “minus one”, or, as I like to call it, “anti-one”.
Now pay attention, because this is going to be really important. There is no subtraction. In fact, “subtract five” actually means “add anti-five”. We call “minus five” the additive inverse of “five”. But don’t worry about the buzzwords too much. Just try to get wrap head around the concept that what we call “subtracting x” actually means “adding anti-x”.
A quick side-note: There are two elementary operations. Addition and Multiplication. Sutraction is actually Anti-Addition, and Division is actually Anti-Multiplication. (And no, multiplication is NOT just repeated addition. But that’s a topic for another day. And “powering” as in “x squared” also comes into play, but let’s not worry about that for now.)
So subtracting actually means “anti-adding”.
“5 – x = 3″ actually means “What number do I need to add to three to get 5?”
Okay, with that background, let’s tackle division in general, and then specifically division by zero.
Why we can’t divide by zero
When you have 2 * 3 = x, it means “What do I get when I multiply 2 by 3?”
And because division is actually anti-multiplication, 6 / 3 = x means “What number do I need to multiply 3 by to get 6?”
You with me so far?
Therefore, 1 / 0 = x means “What number do I need to multiply 0 by to get 1.” And there is no possible answer to that. We call this “undefined.”
And here comes the next fun bit. What if you have 0 / 0 = x? Obviously, you can multiply 0 by ANY number and get 0. The answer is very well defined… unfortunately, you have no way of telling which answer it is. And no, you can’t just pick one that suits you. Let me explain…
Let’s play a game! I’m thinking of a number, and when I multiply that number by itself, I get 4. What number am I thinking of?
Say it.
Okay, got it? The number I was thinking of was… minus two! If you said “two”, you lose. If you said “either two or minus two”, you were right… but you still lose, because I was thinking of only one number. If you said “minus two”, you win, you tricky bastard. 
The point is, if I say “x squared is 4, what is x?”, you have no way of knowing the answer. It could be either 2 or -2, but you can’t tell which.
Analogously, if you try to find 0 / 0 … it could be any real number, and you have no idea which one. So if you get 0 / 0 = x in an equation, all you know is that x actually exists. But you have no idea what it is.
Phew. So much for division by zero. Let’s tackle something easier next time. Like octonion analysis…
February 16th, 2009 at 12:30 am
Great articles! They’re interesting and informative.
I wonder what would happen if we invented a form of number that could be divided by zero. After all, negatives and imaginaries were created to make the mathematical impossible possible, weren’t they?
February 16th, 2009 at 8:50 am
@Holly:
That’s the thing. I think that’s one of the few things that can’t be done.
Remember what I said about division being just anti-multiplication? Well, that means that “divided by three” means just “multiplied by anti-three” (that is, multiplied by one third). And there’s no anti-zero under multiplication.
Then again, people used to think that square roots of negative numbers were ridiculous, or that you can’t compare the sizes of infinite things. (Turns out there is the same “size” infinity of natural numbers as there are fractions, but there is a “bigger” infinity of real numbers. Fun stuff!)
So if you find a way to divide by zero that doesn’t make all mathematics break down afterwards (like the event horizon of a black hole), please let me know
September 26th, 2009 at 4:46 pm
I’m not great at math or anything, but this part bugged me; you say “Infinity is not a number. Seriously. It’s an abstract concept. It’s a shorthand for “gets as big as you like”. You can’t treat it as a number” and then immediately go on to plug it into an equation where you treat it just like a number, rather than the ‘abstract concept’ it is. Correct me if I’m wrong, but I’m fairly certain you can’t subtract infinity from itself.
September 26th, 2009 at 5:41 pm
@Steve:
You’re absolutely right. I was just demonstrating a simple example of what can go wrong if you DO treat infinity as a number.
December 31st, 2010 at 3:42 pm
8/10 for the explanation, which was rather nice if incomplete
& 10/10 for the humor
keep going, you rlly make math fun!
February 25th, 2011 at 8:32 am
I would like to discuss the problem of division by zero in the set of real numbers. So far the best explanation I’ve found to see why a quotient like a/0 cannot be defined in the set of real numbers is in an old textbook which shows that division by zero is undefined because division is defined by multipication which becomes an identity in the case of the denominator equaling zero.
a/b = c is defined by a = b*c.
“If a/0 = c, then a = 0*c.
But 0*c = 0.
Hence, if a is not equal to 0,
no value of c can make the statment a = 0*c true,
while if a = 0,
every value of c will make the statement true.
Thus, a/0 either has no value or is indefinite in value.”
Yours respectfully.
February 26th, 2011 at 12:36 am
@Rob:
That’s absolutely spot on. I have nothing to add.
February 26th, 2011 at 1:39 pm
Dear Sir,
When mathematicians say that division by zero is “undefined” I would like to find out in what sense are the using the word “undefined,” something not defined yet can be considered “undefined.”
My question is what are the different ways something is considered to be “undefined” and which of these is being used in this case?
Yours respectfully,
Rob
February 28th, 2011 at 10:21 pm
Dear Sir,
I have realized that:
One must note here that “undefined” means both not defined and indefinite.
Yours respectfully,
Rob
February 28th, 2011 at 11:08 pm
As far as I’ve heard it, “undefined” means there is no answer (like for 1/0). However, for the case of 0/0, I’ve heard mathematicians instead use the word “indeterminate”.
In other words, it exists, but we can’t tell which one of many possible answers it is.
March 4th, 2011 at 6:22 pm
Definition of Division
For every real number a and every nonzero real number b, the quotient a/b is defined by:
a/b = a*1/b.
The given “Definition of Division” is merely a formal rule.
For example it tells us that we may replace 12/3 with 12*1/3.
The form a/b may be replaced with the form a*1/b.
But it does not tell us how to evaluate that division and in questions about dividing by zero the questioner must realize that this is a question about evaluating that division ie. finding a value for a/0.
For that, we must return to arithmetic, and to the relationship between division and multiplication.
If a divided by b is some number n
ie. a/b = n
then n is that number such that n times b is equal to a
ie. n*b = a.
If a/0 = n, then n*0 = a.
But n*0 = 0.
Hence, if a =/= 0,
no value of n can make the statement n*0 = a true,
while if a = 0,
every value of n will make the statement true.
Thus, a/0 either has no value or is indefinite in value.
March 5th, 2011 at 9:54 pm
Division is not always possible in the system of numbers consisting of the integers (6 is divisible by 2 and 3 but not by 5), but in those cases where it is, the result is always uniquely determined. In the system of all rational numbers (that is, the integers and fractions) division is not only unique but is always realizable with one exception—division by zero. On the basis of the definition of division given above, it is apparent that it is not possible to divide a number different from zero by zero. The result of dividing zero by zero, according to the definition, can be any number since c . 0 = 0 in all cases. It is usually preferable in algebra (in order not to violate the uniqueness of division) to consider division by zero to be impossible for ALL cases.
March 5th, 2011 at 11:35 pm
In mathematics the art of asking questions is more valuable than solving problems
March 19th, 2011 at 4:37 pm
0*1=0
0*2=0
0*1 = 0*2
0/0*1 = 0/0*2
therefore:
1=2
(wikipedia
)
March 20th, 2011 at 12:47 am
The fallacy is the implicit assumption that dividing by 0 is a legitimate operation
May 26th, 2011 at 7:04 pm
I found this file on line:
http://dynamic.uoregon.edu/~teddy/OnDivisionByZero.pdf
July 29th, 2011 at 5:55 pm
I simply loved this article, as it not only gave clarification on my question, but explained things in a simplified form.
Thanks a lot.:)
January 9th, 2012 at 12:02 pm
I always new maths teachers were frauds. great work!