I’m not the most math-y type person in the world, so I wanted to pose a question to the geniuses on here.

Let me explain the genesis of this: Last week, as I was studying, I had to calculate a ratio on a company, and I don’t remember the exact ratio, but I remember thinking in was zero over zero. (Whatever the ratio was is irrelevant.)

That got my head spinning. I know that zero divided by anything is zero, and anything divided by zero is undefined.

What, then, is zero divided by zero? "Twas a problem that bothered me greatly for a couple of minutes.

Both x/0 and 0/0 are undefined but in a different way. x/0 -> infinite, while 0/0 is satisfied by any number, so it’s not a huge number as in x/0, but every number. Hence undefined too.

Technically it’s undefined, because the zero in the denominator is simply undefined.

However, you can often use l’Hôpital’s Rule to find the limit of something where the denominator approaches zero, so if you have a formula or function that happens to be zero at one point, you can take the limit of the numerator and the limit of the denominator to see whether f(0+delta) / g(0 + delta) is infinite, or finite or zero. f(x) is simply the numerator of the expression, expressed as a function, and g(z) is the denominator. That’s often a useful substitute.

You can look up l’Hôpital’s rule on wikipedia, it’s not that tricky if you’ve ever taken calculus, athough you may have forgotten it from high school.

As an example, you could have the expression y=(x^2 - 1)/ (x-1), which technically is undefined at x=1. However, if you rearrange the expression as y=(x+1)(x-1)/(x-1), it’s clear that the plot looks like a line y=x+1, but there’s an undefined point (a hole, if you will) at x=1. But you know if you set x to 1+delta and make delta closer and closer to zero, the function will get closer and closer to y=2, and that’s often more useful than just throwing up your hands and saying it’s undefined.

L’Hopital’s rule is a little more complex than that, but the example gives you the basic idea of how it works.

I’m not sure how helpful L’Hôpital’s rule would be, given that he is dealing with financial ratios.

Anyway, there’s some misleading stuff on here. A number divided by itself equals 1 only if there exists a multiplicative inverse for the number. In abstract algebra, you would say that the real numbers are a field, which defines the properties of addition and multiplication, including the properties of the multiplicative inverse. The condition for the multiplicative inverse in the field is that it applies to all non-zero numbers. Why? Because mathematicians care about things working more than anything else and algebra usually doesn’t make sense if you define a multiplicative inverse for zero. It is possible to have a ring (another thing in abstract alegbra) where there are no multiplicative inverses, so all division is undefined. Think of it like doing subtraction with the set of positive integers. We may know that 3-5=-2 over the set of all integers, but in the set of positive integers 3-5 is undefined. No bigger worry for dividing by zero. And x/0 is undefined, same as 0/0.

Officially (according to IEEE-754 floating point used in computers),

+ve / zero = +infinity

0 / 0 = NaN (literally, “not a number”)

Then they have rules such as NaN + number = NaN, NaN + infinity = infinity and so on.

There is also “infinitesimal” when you divide a small number by a large one and the computer runs out of digits (to be precise, the mantissa + exponent range) to record the result.

You can’t always trust what computers tell you. For the above issues, I’m guessing a symbolic math toolbox would give different answers. Alternately, take the variance of {1,2,3,4,5}, should be 2.5 using standard formula. Add 1E10 to each value, you should still get 2.5. However, different formula (taking advantage of that Var(X)=E(x^2)-E(x)^2) will give you 0 in the second case because of rounding errors that the computer makes.

X / 0 -> infinite doesn’t need further explanation. 0 / 0 is different because while it’s undefined too, it’s not positive infinite.

Best explanation is that if 0 / 0 = x (expression i), then 0 / 0 = 2, because 2 * 0 = 0, therefore satisfyting (i). But 0 / 0 = 3 too, because 3 * 0 = 0. Therefore 0 / 0 = every real number, and not only infinite. Both undefined, but quite different.

You can’t just say X/0 is infinite without further explanation. If X/0=infinity, then X=0*infinity, so you’re saying that 0 times infinity is any number. That seems silly. It makes more sense to say lim x->infinity of c/x equals zero. You can also approach the problem from the other side and say lim x->0+ of c/x equals infinity, but that does not mean that c/0=infinity.

The problem with your second proof is that you can’t say 0/0=x. c/0 is undefined for any c. Even if you could take 0/0 why would it equal 2? For any c, c/c=1. What you’re actually trying to do is say 0*c1=0 and 0*c2=0, so 0*c1=0*c2. But then if you divide by zero, then any c1=c2. The problem is you can’t divide by zero as it is undefined.

I didn’t say X/ 0 = infinite, I said X / 0 -> infinite as a lazy interchangeable expression of lim. That’s what I meant when I said it doesn’t need further explanation. Both X / 0 and 0 / 0 are undefined. All I’m saying is that the origins of both expressions’ lack of definition are different.

When I said 0 / 0 = 2, all I’m saying is, “what number multiplied by the divisor 0 is equal to zero? “ the answer to that question may be 2, or 3, or 69. I didn’t suggest any further arithmetic treatment since that is a known fallacy that ends up with things like c1 = c2, as you mentioned.

Indeterminate. But the limit as x approaches 0 is 1. I think you can get this by taking the natural log of both sides of the equation y = x^x and differentiating.

This reared its ugly head again today. I was doing a Texas Franchise Tax return, and the tax return asked for sales everywhere divided by sales in the state of Texas. Due to my input error, I didn’t put anything in these two fields, so the tax return showed zero divided by zero.

Apparently, according to the state of Texas, 0/0 is 1.0000. That is, if you put zero total sales and zero Texas sales, then Texas sales = 100% of total sales.

(When I noticed the error, I put in the total sales and the Texas sales, and that fixed the problem. So now the universe won’t end because Texas says that 0/0 = 1.0.)