The divide-by-zero paradox

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Consider these statements to be true:

1. Any number divided by zero gives an undefined output.
2. Zero divided by any number equals zero.
3. Any number divided by itself equals 1.

Now, let's take a look at the following expression:

0/0
That's zero divided by zero.

According to rule number 1, the result would be undefined.
According to rule number 2, the result would be 0.
According to rule number 3, the result would be 1.

Discuss.

0 is not a number. It was invented by the Indians to express the absence of something. Not sure it is a paradox really. Those statements are true but:

1. Something can't be divided by nothing but nothing divided by nothing is nothing
2. Nothing divided by any number (or nothing) equals nothing.
3. Any number divided by itself equals 1. Nothing is not a number.

It is, however, interesting to note how many people do treat zero as a number. The symbol "0" is more of a placeholder than anything else. When you have nothing, 0 is there to hold the place. Same with when you have a number such as 100 or 1000. There is nothing in the ones or tens digit, so the zeros are there to help distinguish the difference between 1 and 100.

We could always go back to counting with abacuses

Ah but in an abstract you can treat it as number or more a number type. The character "0" is something not nothing it is a merely an assumption that people want it to equate to nothing.

Therefore, zero in itself becomes a paradox, if it stands for both something and nothing at the same time.

LOL, the most awesome image ever!

Not that is not a paradox it is an abstract. For example is saying the that \x30 in ascii represents the character 0 \x0 represents NUL. But the character 0 is something, how you choose to interpret it is up to you. But they are place holder as is decimal dot/comma. How you write numerals is up to you. But 0 is not a number.

It seems that the set of 3 statements is true, and the 2nd and 3rd statements are incomplete truths in this case (true only as part of the set of 3). In other words, 0/0 equals both 0 and 1, and therefore it is undefined.

That's because statements two and three are not quite right. A better formulation would be:

Let R be the set of all real numbers. Let R* = R\{0}. Then:
1. For all x in R, x / 0 is undefined.
2. For all x in R*, 0 / x = 0
3. For all x in R*, x / x = 1

So, consider 0 / 0.

According to rule 1, the result is undefined.
Rule 2 does not make a prediction about the result because 0 is not in R*.
Rule 3 does not make a prediction about the result because 0 is not in R*.

You can divide by zero... in calculus...

LOL
I was going to post something derogatory about these two statements.
But to be fair, i can see what you mean.

I would say zero is a number and was not invented by anyone.

LOL all you like. You're just used to the type of arithmetic you do now (positional numerology) there were other systems.

Zero was invented by the Indians in 'arabic' numerals which are actually Indian. There was use of something similar to zero as By the Babylonians.

http://en.wikipedia.org/wiki/Indian_numerals
http://www.andrews.edu/~calkins/math/bi ... iozero.htm

How do you know what type of arithmetic i am used to ?
I stopped doing mathematics as a profession many years ago.

But there are some who would construct the whole number system using just zero. Which is what i found amusing.
As for whether mathematics is invented; that's a long standing philosophical question. We will probably never hear the last of that. But it is really not much of an issue.

it is not about whether mathamatics was invented it is about the use of the zero symbol as is currently used. But yes you can do what you like in china they have 3 different sets of numerals

Yes, i should read the whole thread really before posting. But i did find that little excerpt funny ( i have a strange kind of humour).

I would certainly treat zero as a number, then construct the rest. Of course you only have to do it once, and then forget about it. Rather counter-intuitive and not following the history of mathematics ( which i think is what you guys are doing).
But if you are going to construct 'numbers', then you must start somewhere. May as well start off with nothing (since that is what you have) and go from there.

As far as going back to using abacuses. I don't know if this was meant in jest.
But a colleague of mine used to collect them. It's amazing how fast he could compute things with them.
Multiplication, division, roots and logarithms of large numbers and decimals. Much faster than computers of the time ( granted over ten years ago), but quite impressive.