quote:

**stigant wrote**:

um, why? Wouldn't 0/0 = any imply that 0 * any = 0? (That's true, by the way)

I will assume that we're using real numbers with the usual meanings for *, /, and =. First off, the "

="

predicate tests two real numbers for equality and evaluates to true or false.

By "

any"

you could mean, "

There exists a real number 'any' such that (0 / 0 = any)"

, or you could mean, "

For all real numbers 'any' it is true that (0 / 0 = any)"

. I'll explain why both of these propositions are false under the interpretation given above.

It's easiest to examine the second one first. The binary predicate "

="

is a theorem of logic and without qualification refers to true equality. This means it must be reflexive (for all a, a = a), symmetric (a = b implies b = a), and transitive (if a = b and b = c, then a = c). Additionally, for it to be

*true* equality rather than a mere equivalence relation, it must be the case that (a = b) implies for any equation f: f(x,a) = f(x,b).

So, by logic, and using a couple of axioms of natural numbers, I can show 0 =/= 1 on the natural numbers:

Axiom 1: for all x, (x + 1 =/= x)

Axiom 2: for all x, (x + 0 = x)

for all x, x + 1 =/= x + 0 (by transitivity of =)

0 + 1 =/= 0 + 0 (choose 0)

1 =/= 0 (apply axioms again)

0 =/= 1 (by symmetry of =)

Let's apply this to the latter of the two possibilities first.

"

For all real numbers 'any' it is true that (0 / 0 = any)."

If there exists more than one distinct real number in our universe, this

**must** be false. Otherwise, you would have, for some x and some y =/= x:

0 / 0 = x

0 / 0 = y

x = y (by transitivity of =)

But this contradicts our assertion that there are two distinct real numbers x and y, so it must be false that 0 / 0 yields multiple solutions.

This shouldn't be surprising, however. Note that "

/"

is a

*binary ***function** on real numbers. That means it must return a

*single real number* anywhere it is defined. If it

*didn't* return a single real number, you wouldn't be able to use "

/"

on two numbers to get a result that you could compare with "

="

.

Now let's consider the former statement:

"

There exists a (particular) real number 'any' such that (0 / 0 = any)."

This one

*could* be true by logic, and in fact it is possible to extend the reals to make it true (but rarely useful so it is not often done). However, it is not true under the typical interpretation of "

/"

as "

the inverse function of *"

. Why is that? Simply put: there is no single number, that when multiplied by zero, returns zero. Multiplication on the reals is not an fully invertible function. If you want to treat "

0 / 0"

as a real number, it has to be a particular number. It can't be all of them, or you wouldn't be able to use it in calculations.

Essentially, +, -, *, and / are functions that take two specific terms and evaluate them to produce a third term. You can't treat (0/0) (or x/0 for any real number x) like a real number because there is no real number that maintains its algebraic properties and still satisfies what division by zero implies. You can't even define (0/0) to be 0 or -1 or 42 or some arbitrary value, because this will violate other algebraic properties and lead to immediate contradictions:

Assume: we can define (0/0) =

**a**
For all x, 0 * x = 0 (axiom)

0 * 1 = 0 (substitute 1)

0 * 2 = 0 (substitute 2)

0 * 1 = 0 * 2 (substitute 0*2)

1 = 2 (divide by zero, since /0 is inverse of *0)

You might be tempted to say this contradiction only proves that (0/0) =/= 1, but if that's true then / is not the inverse of multiplication, so we're no longer talking about division at all.

Here's the bottom line.

**Math is the philosophical study of what happens logically when we decide to adopt certain rules.** When those rules are inconsistent, they are worthless to us because once you produce an inconsistency you can prove anything from it, so all statements become true (0=1, 0=/=0, minimike is the best forum member ever). This works as follows, and is a basic tenet of logic:

1) For all x and y, "

x implies y"

is vacuously true whenever x is false. (If dragons exist, "

blah"

. This statement is always true if dragons don't exist, because there can't be a contradiction unless "

blah"

is false

*and* dragons exist. Basic logic, though a little different from how humans use "

if"

in English.)

2) Assume a contradiction (something that can be shown logically to be false) is proven true.

3) Therefore, every statement is true. (Just stick the contradiction in place of x, and what you want to say is true in place of y.)

Ironically, your proof that "

math is absurd when we use 0/0"

is a very good proof of why we choose to leave 0/0 undefined.

Congrats!

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*[Last edited by Trickster at 03-01-2012 10:38 PM]*