0/0=?

x/x=1
0/x=0
x/0=infinity
0/0=who knows?

I read in some higher math book that this mathematical term
(amongst others…infinity/infinity, x/infinity, etc) which cannot be expressed definitely.
Why not?

It is not possible to define division by uniquely and still be consistent with other mathematical definitions and axioms.

For instance, division is the reverse of multiplication, so that a/b = c means a = b*c, for all numbers except when b = 0.

Say that 1/0 = c. Then 1 = 0*c = 0 is a contradiction, an inconsistency, that arises from violating the principle of division by 0.

Thus, you has slightly misstated; 1/0 is not “infinity”, it is undefined. 1/x approaches infinite as a limit as x approaches 0, but that is not the same thing at all. That says that 1/.00000000001 is a very large number.

Now, for your question, suppose 0/0 = X. That would mean 0 = 0*X which is true for EVERY X. Thus, there is no unique number that can set as the answer to 0/0.

In the world of calculus and limits, we can set up a case where 3x/x approaches 3 as x approaches 0, for example, so we can find different sequences that look like 0/0 but that have very different limits.

Hope that helps. There are several other threads on this, you might try searching around.

I wouldn’t agree that 0/0 is undefinable. I’d say it’s zero. Meaningless, perhaps, but equal to zero.

Expressions involving infinity are different, since infinity has no fixed value.

My mathmatical experiences state that x/0= (null set), an impossible function. The logic being, if you have 10 jellybeans, it is impossible to divide them into zero groups. You already have them, you can’t make them go away, even by eating them.

Now, infinity/infinity should equal 1. But x/infinity should be considered to be zero.

0/0 is not ‘undefinable’, it is ‘undefined’. That is to say that within the formal system that is mathematics, any occurrence of 0/0 has no meaning, it is undefined. If it were to be ‘defined’, that is to have some value, then it would lead to those mock ‘proofs’ that 1 = 2, etc. that almost always rely on 0/0 being effectively hidden in the proof

**DI-VI’SION,**n. … The quotient a/b of two numbers a and b is that number c such that b * c = a, provided c exists and has only one possible value (if b=0, then c does not exist if a <> 0, and c is not unique if a = 0; i.e., a/0 is meaningless for all a, and division by zero is meaningless);

This thread goes into more detail of 0/0.

and…

C K, is you a mathematician?

I’m no expert but:

0/0 surely means the same as ‘how many times does zero divide by zero?’…

…Um…zero.

Anything divided by zero is zero. Even zero itself, so your answer is… TWO HUNDRED EIGHTY FIVE THOUSAND, SIX HUNDRED AND NINETY TWO POINT SIX FIVE NINE!!!

Actually, it’s zero.

Unless there’s more to this than meets the eye, but this looks like nothing more than zero divded by zero.

Anything divided by 0 is undefined. I’m pretty sure the same goes for 0, but I have not taken calculus, so I could be incorrect.

Your probably right. I failed calculus.

As my high school physics teacher brilliantly put it,

“Trying to divide zero by zero is like asking, ‘I got no money, and no one to give it to. What do I do?’”

Wow, there are a lot of misconceptions in this thread. To express it in layman’s terms, think about what division means. “What is 20/5?” is the same thing as asking “How many fives, when added together, total twenty?”

So how many times do you have to add zero to itself in order to get zero? Zero? One? A million? All of the above? It’s a meaningless question, which the mathematicians describe as “undefined,” which just means “that makes no sense.”

No. How many times does nothing go into nothing? Two nothings can go into nothing, zero nothings can go into nothing, etc. It’s undefined. However, in the strictest arithmetical sense, 0/0 = 0 seems to work, because it leads to 0 = 0. But this leads to other bogus solutions such as how many times does (not can) an infinite or indeterminite quantity go into zero ( (1/0)/0=0*1/0=0/0=0 )? zero times? Not necessarily…
infinity/infinity does not equal 1 either, (1/0)/(1/0) = 0/0 = undef - or does infinity go into infinity zero times? Maybe a bit too simplified, but none the less, it isn’t 1.

I also don’t know about L’Hospital’s application in this case, but for instance, lets say f(x) = x - 1 and g(x) = 1 - x

lim f’(x)/g’(x) = -1
x->1

So you have two possible cases: 1 or -1, either way, it is not clearly defined.
1/0 = 0
1 = 0
? No… probably not.
Now, it might seem to work in a “real-life” sense… If you have a 10kg pile of dirt, and you wanted to split it up so that each group has a mass of 1kg, how many groups must there be? 10. If you wanted to split it up so that each group had a mass of 0, how many groups must there be? What? You CAN’T do it? ok… then the answer seems to be zero times. But you haven’t even split it up, so how can there even be an answer? Because it isn’t defined, both on paper and in real life.

In short, its easier to just say it isn’t defined, because in “reality”, it isn’t.

Division is defined in terms of multiplication.
a/b is defined to be that unique number x that satisfies:
a = b*x

If b is zero and a is not, then no x satisfies.
If a and b are both zero, then any x satisfies, so x is not unique.

This is why division by zero is not defined.

Do you know how frightening it is for me to read this stuff and actually understand it?
Hmmm, I guess this means I’m actually learning something in Calculus. Mr. Johnson will be so proud when I tell him…