What is Zero divided by Zero?

Any number you like - it's indeterminate but personally I prefer "0"

Please note: "Indeterminate" is not the same as "Any number you like." The precise mathematical answer is that it is undefined. You can't just pick an answer that suits you and continue on with your calculations.

Steve wrote: "You can't just pick an answer that suits you and continue on with your calculations."

Exactly, Steve. But, au contraire, let's buck the System.

In keeping with the original poster's assertions that 0/0 = 1, let's prove that for any real number x, 2 equals 1:

x² − x² = x² − x²

• 2) Factoring both sides in two different ways:

(x − x)(x + x) = x(x − x)

• 3) Dividing both sides by x − x, giving (0 / 0):

(0 / 0)(x + x) = x(0 / 0) <-- Assumption that 0/0 = 1 here, leading to #4

• 4) Simplified, yields:

(1)(x + x) = x(1)

• 5) Which is:

2x = x

• 6) Since this is valid for any real number x, we can divide both sides by x (including x = 0, as we've already admitted this possibility from the outset):

2 = 1

The fallacy here is the assumption in step 4 that (x − x) / (x − x) -- which is (0 / 0) -- simplifies to 1. This proof is for the special case of dividing by zero when the numerator is zero. The fallacy results from the assumption that 0 / 0 = 1 -- an assumption that generates the absurdity that 2 = 1.

(Original proof courtesy of Wikipedia)

Here's a simpler variation which proves for any real number x, 0 = 1 and 0/0 = 1, simultaneously:

x − x = x − x

• 2) Evaluating the left-hand side only:

0 = x − x

• 3) Dividing both sides by x − x:

0 /(x - x) = 1

• 4) Simplified (noting that for any number a, 0/a = 0), yields:

0 = 1

• 5) Except that when we evaluate the denominator on the L/H side, we get:

0/0 = 1

• 6) Proving that 1 is simultaneously equal to zero and 0/0.

-e

The logic is fine but it is obvious by inspection that x-x =0.

It would be less obvious if we used a and b instead.

Lets set up an equation, say a²-b² = a(a-b)

and factorising, thus (a+b)(a-b)=a(a-b)

and dividing both sides by (a-b), we get (a+b) = a ....(A)

Then we say if a = b, and substituting for b in (A) we get (a+a) = a ...(B)

and dividing (B) by a we get 2 = 1

That's what happens when 0/0 = 1.

Zero divided by its self is zero. zero does not have a mathematicle value.

Infinity has no single value so there will never be a single equation.

You wrote: "Zero divided by its self is zero. zero does not have a mathematicle value."

What, exactly, do you actually mean by "zero does not have a mathematicle (sic) value?" More specifically, what does "mathematicle value" mean to you? Please be specific.

-e

just as i thought if you have to ask the question you will never understand the answer.

Ok What do I mean by Zero has no value, think zero is the absence of a value you can not perform a mathematical operation on it you can not put a sign to it it has no existance.

A mathematicle valueis any thing but zero ie:- 1,2,3,4,5 etc. you can have a positivevalue or minus value even a complex value. You can have an interger value a decimal value, even theoretical value. BUT NOT a zero value zero is neutral it has no asigned funtion. It can not take part in it's own righ it exists only to aid in showing the decimal place of some other value 1 vesus 1000. 3 versu 3,000,0000. or .006 versus -.005. It is a postional factor. Ok?

BrainWave wrote: "just as i thought if you have to ask the question you will never understand the answer.

Ok What do I mean by Zero has no value, think zero is the absence of a value you can not perform a mathematical operation on it you can not put a sign to it it has no existance. Try to use what little brain cell you have to think it through."

Disregarding your astonishing arrogance for a moment, please allow me to point out two things:

You wrote: "Zero divided by its self is zero. zero does not have a mathematicle value."

and,

You wrote: "you can not perform a mathematical operation on it ..."

It's your hole; you dig it. I'm outta here.

-e

BrainWave wrote: "Infinity has no single value so there will never be a single equation."

Infinity has no value at all, not even the value of any single value. Infinity is a direction, not a destination.

-e

You wrote: "What is Zero divided by Zero?"

The classical answer (insofar as Rings and Fields are concerned) is that the operation is undefined. When you are discussing division by zero in a mathematical context (alternatively, a framework) called a Wheel, division by zero is defined, but the meaning of the division operation itself is different. Infinity also takes on the characteristic of having no sign, ie, there is neither a -∞ nor a +∞.

You asked: "Similarly, what is Infinity divided by Infinity."

The problem here is not 'Infinity' per se, but the division operation itself having been sandwiched between two non-numbers. The division operator operates on numbers (however represented). But infinity is not a number - not at all! If it were a number, then you could always add one to it to make a different infinity - and both infinities would be necessarily be finite in order for the addition operation to have any meaning! The answer is not one, as you suggest, because the division operation itself is meaningless in this context. You might as well ask "what is an apple divided by an apple?"

You asked: If not One, why?

Please see the foregoing.

You asked: "And what would happen if it was One?"

A lot of mathematicians would be out of work.

-e

Europium - nice answer.

I recall a math professor once saying: Whenever you get tempted to treat "infiinity" like a number, replace it with the phrase "and so on, endlessly." Then try to perform the operation on it.

Steve wrote: "Whenever you get tempted to treat "infiinity" like a number, replace it with the phrase "and so on, endlessly." Then try to perform the operation on it.

Thinking about infinity from another angle: "infinity is a direction, not a destination." Consequently, whenever one is tempted to treat infinity as a number, consider answering the question: "How far is out there?"

-e

Nice one Europium, not sure about this 'brainwave' guy though.

Whenever someone says something such as "If you have to ask you wouldn't understand the answer".

That to me means they haven't a clue either!!

John.

The physicist Richard Feynman once said something like

If you can't explain something to a layman, then you don't really understand it yourself.

I recall Richard Feynman once saying (about the CERN atom-smasher built to test his theories) '...they could have saved 6 billion dollars if they had taken my word for it...'

In the most fundamental sense, mathematical frameworks are purely artificial constructs. Even numbers themselves are artificial constructs. Say, for example, you're walking along the beach and see two starfish sitting side-by-side in the sand. Without you there to ascibe the number 'two,' there is nothing intrinsic to the starfish themselves that ascribes a property of "two-ness." That is your doing, not theirs.

Mathematical systems, like the one we use day-to-day in which addition, subtraction, multiplication, division, and so forth, have their usual meanings are, at their essence, systems of man-made symbols and operations on those symbols. Without Man, no mathematical systems would exist and yet, somehow, the world will get along just fine without them. Mathematics is strictly a human invention. It is completely artificial.

The system we use every day defines a set of operations on numbers. This system also defines what are not numbers or are not meaningful operations on numbers in the context of the system; Division by zero and operations on infinity are defined as not meaningful.

You may wish to believe that 0/0 and ∞/∞ are equal to '1'. but your insistance that this is so is, to put it bluntly, irrelevant. If you would like to define your own framework in which these operations are meaningful, then by all means, go ahead and do so. It is fair to warn you, however, that you may find it quite difficult to communicate in a meaningful way with others who do not share your mathematical conventions, but that's a chance you'll have to take. No one will stop you.

From another angle, think of a mathematical system as an artificial construct in much the same way as a board game is an artificial construct. Say we're discussing the game of Monopoly. You have the board, the pieces, and the rules. But let's assume for a moment that I took exception to a number of the game's rules and instead played the game according to my rules. There's certainly nothing to stop me from doing so, except that must now pay the penalty: playing the game alone.

Taking this analogy one step further, let's say I take exception to all the game's rules and, out of disgust proceed to invent my own game with its own rules. If I've done a good job of it, others may take an interest in my game and maybe want to play it with me. If I've done a super job of it I may find that I have a blockbuster product on my hands that will make me million$.

Likewise with mathematical systems. Some make it, most don't.

However you want to look at it, arguing over Monopoly's rules is pointless. Monopoly has a certain set of rules by which it is played and none other. Like it or not, its rules are its rules and the game is played by following those rules. And the same goes for mathematical systems. Constructs like 0/0 and ∞/∞ make no sense in our system, nor does it matter whether you think they should make sense or not. Our system defines such operations as having no meaning. Consequently, if you don't like the game, play something else. Or invent your own. Who knows? You may come up with a blockbuster!

nice, i was waiting for someone to take the philosophical/ontological approach to the whole idea of "math" in general. what guarantee do we have that gravity will continue to happen, that particles will continue to behave in predictable ways, or even, at the risk of sounding corny, that anything is real? it all depends on you taking a leap of faith.