Following the current trend of math topics

So there's the common explanation as to why one cannot divide by zero:

6/3=2
and so it follows that the number 2 would be suitable in this equation:
3x?=6

And then they say
6/0=?
And propose that for
?x0=6
and then that "any number multiplied by 0 is 0 and so there is no number that solves the equation."

But my problem with this logic is that it assumes that 0/0=1
Think about it, how did you learn to carry over a number to the other side? By multiplying both sides by the reciprocal of the number!

If it isn't clear, follow the logic as such:
6/0=?
Multiply both sides by zero
(6x0)/0=?x0
Simplify
6=?x0
Fallacious, no?

It's even on wikipedia ;o

I am not sure what your problem is.. this particular argument about dividing by zero is simplistic but still essentially correct.

If you think about it, to solve 0*x = 6, you would need a really big number, so big, in fact, that it would be infinite. Thus we sometimes call this result of division by zero undefined, and note the limiting case give infinity.

If you have 0*x = 0, then note that this equation is satisfied for any x, by definition. Thus we sometimes refer to this as indeterminate, and more information is required (such as using L'Hospital's Rule) to determine the value in the limiting case.

The problem is that when going from 6/0=? to 6=0x? in the proof, you have to divide zero by zero, which is indeterminate, and then they go as far to say that 0/0=1 by cancelling them out in it. I'm just saying it's not ever ok to do that in a proof.

The same problem could be said with square roots (or roots to any even power for that matter) if you don't know imaginary numbers exist, so maybe some day they'd invent another imaginary value of (n/0) thus making that logic up there true ;o

Now think about a clock.

The numbers are ordered in a cyclic fashion, with 12 = 0.

Now if you think about this:
3 * 4 = 12 = 0, so 3 / 0 = 4!

This really wasn't on topic, but I couldn't help myself.

You know that makes no sense. If you're equating 12 to 0, then it would be 12 / 3 = 4, or 0 / 3 = 4. I'm also assuming you didn't mean four factorial.

3 / 12 != 4
3 / 0 != 4

More accurately, your scenario is describing modulus. A clock's hour is x % 12.

In any case, your post was slightly amusing and primarily nonsensical (which was most likely a mistake swapping around the 3, 12 and 0 like you did).

You're right, you can't just cancel out 0 / 0 and say it equals 1.

But take an even simpler approach: if you're multiplying both sides by zero, your equation is going to turn into 0 = 0.

I noticed that too, Roy. Who multiplies a number by 0 and still keeps them separate?

I'll elaborate on what DT is saying.

Division by zero isn't impossible, despite what many may have you believe. It's "undefined". This word just means a particular expression doesn't mean anything. In any case, when you learn about division by zero in algebra, they do not add a necessary "in a set of real numbers". This is because if we expand our set to include, for example, hyperreal numbers, then we can add meaning to the expression.

Inside a set of real numbers, we are stuck contemplating with a fixed representation of zero. But when we push those boundaries to calculus (which suddenly introduces thinking about the infinite), we begin to realize that not only are all infinities not equal, but neither are all zeroes equal. Calculus darts around this issue by playing with limits and summations, but only to accomplish its task, and does not explore them for its own sake.

Post has been edited 1 time(s), last time on Nov 7 2011, 7:12 am by Sacrieur.

Just use the real part of the complex plane, throw a sphere in there, add a little conformal mapping spice, mix it all up and define the point at infinity. GG no re

Nah but seriously, DTBK is GG. Course, you could just let x be defined such that 0*x = 6, its just the case that x would no longer be an element of the real numbers.

As for your actual proof, it really has nothing to do with 0/0. For every given element, a, in our field of numbers there exists an inverse element such that a * (a^-1) = 1.

So for 6/0, we're really multiplying 6 * (0^-1), the inverse element of 0.
Then 6(0^-1) = ?
Luckily this field has ab = ba, so when we multiply by the inverse element of (0^-1) we can move stuff around within the term
6*(0^-1)*((0^-1)^-1) = ? * ((0^-1)^-1)
The inverse element of an inverse element is the element itself, so
6*(0^-1)*(0) = ? * (0)
then for a = 0, a * (a^-1) = 1
6*1 = ? * (0)
We just use the convention "division" because its easier to write really. Everything can be done without dividing by 0, its really multiplying by the inverse element of 0 thats the problem, not the assumption that 0/0 = 1, that is inherent in the definition of the field.

Post has been edited 2 time(s), last time on Nov 7 2011, 6:53 am by Vrael.

The real problem is that division is not really a well defined operation from R x R to R. The field axioms do not require the existence of a multiplicative inverse of 0, and considering z_∞ doesn't really completely resolve the problem. Various branches of mathematics will do different things to better define a case where you want to "divide by zero" but the best answer is that it is just not well-defined.

Subtraction and division are not really operations

Well, true, but they almost do, especially if you forget to write "OH and this is true except for (0) lolz". If you meddling kids and your pesky mutt hadn't interfered I would've gotten away with this too!


Sure it does, for 1 element anyway. I was just trying to be creative though in order to show that the world isn't limited to one little batch of numbers all in a single pot, you can add sugar and spice and lots of things nice.


GG no re.

You could argue that multiplication and exponents aren't operations either...
Multiplication is just condensed addition. 3x3 is equal to 3+3+3. Exponents are just condensed multiplication. 3+3+3=3x3=3²

I hate to admit it but I have no idea what you guys said. Just not in the math that I've done so far.
That's not to say everything was above my head, just some of it.

Basically, in math all the concepts which you're familiar with pertaining to numbers, addition and multiplication and commutivity and whatnot, are all the results of a set of assumptions and the application of these assumptions, or axioms, to a set. Instead of thinking of numbers as natural objects that interact with each other, mathmaticians wanted a more rigorous and strict set of rules to govern the numbers, so they made up these things called groups and fields which apply to any set, not just numbers. If a given set satisfies the rules, then it can be called a field, or a ring, or a group, whichever rules it happens to satisfy. The real numbers happen to be a field. One of the rules of the field is that every element in the field has an inverse such that when you multiply the two elements together you get the identity element, which we know as 1. Unfortunately DTBK is technically correct, that 0 is the exception to this rule, so in a sense my earlier post is not correct, because the inverse element of 0 doesn't have to exist because it isn't required to exist by the rules. The rule is literally written "For every element EXCEPT 0, there exists an inverse element." My point though, was that division isn't really an operation, so the idea of "dividing by 0" isn't really a problem. You can always change division into multiplying by an inverse.

In the field of real numbers, multiplication and addition happen to be related as you say, but in other fields that might not necessarily be the case, because the elements being multiplied have more structure inherent to them than the ordinary numbers.

The real problem with z_∞ is that even though now z/0 is well defined for z != 0, you still have the problem that z_∞ (and 0) do not have multiplicative inverses, and worse, z_∞ doesn't have an additive inverse either since z_∞ - z_∞ is also undefined. It is nice, though, the freedom the Riemann sphere gives you in complex analysis.

I think the interesting thing though is the difficulty of defining a multiplicative inverse of 0. It actually strikes me as very interesting that the additive identity need have this property, and this leads to finer points of mathematics theory I really shouldn't touch since I'm not a math major.

This falsely implies that math majors know what they're talking about.

Fine, technically it is 0 = 12 mod(12)
But the whole reason I brought it up is because modular arithmetic is different than normal arithmetic. In modular arithmetic, the only way to avoid being able to divide by zero is to only have prime sets of integers. Because that way when you multiply things together you won't get the modulus. This makes prime modular groups very important tools.

I hate being those people that picks out their own thing in a topic, but...