You can put on the same page the definition of a smooth function.

]]>Here's another explanation why division by zero is not possible. Division is a secondary operation,

not a primary operation; that is, division is not given in the field axioms that establish the real

number system. Multiplication is given in the axioms (as well as addition, but not subtraction).

Division is then defined as multiplication by reciprocals.

For example suppose x is a real number whose reciprocal is y. Then given a number z we define

z/x as z*y. So to divide by x, x must have a reciprocal. BUT zero has no reciprocal as stated in

the multiplicative inverse axiom: For each real non-zero number x, there is a real number y such

that xy=1. Of course the usual notation for the reciprocal of x is written 1/x. But this looks like

division, so using 1/x for the reciprocal in the axiom and then later using "/" for division is a bit

confusing and makes the definition via z/x = z*(1/x) look circular.

When all is said and done, the axioms do not allow for a reciprocal of zero, hence division by zero

is a non-issue --- it could never happen since zero has no reciprocal to define division by zero in

terms of.

Having zero in the denominator of a fraction is a similar issue. The set of fractions based on the

set of whole numbers W={0,1,2,3,...} is defined as

F = { p/q | p and q are in W and q is not zero }

If someone asks why we can't have something like 2/0 the answer is simply the definition does

NOT allow zero in the denominator. It has nothing to do with "division by zero."

Analogously the field axiom for the reals do not allow zero to have a reciprocal. Hence to write

an expression one would read as "division by zero" simply violates the field axioms. Trying to

write "a number divided by zero" would be defined as "a number times the reciprocal of zero."

But a "reciprocal of zero" does not exist.

The other explanations of why one can't divide by zero illustrate the problems that would occur

if we did allow zero to have a reciprocal. As such they provide good reasons from disallowing

zero to have a reciprocal in the first place.

A nice physical example of trying to divide by zero can be seen in the operation of the old

mechanical calculators of yesteryear. They operated on division as a "repeated subtraction."

If one tried to divide 2 by zero, the calculator would subtract zero from two, add one to the

quotient, and then check to see if there was enough left to subtract zero from it again. Of course

there was enough left to subtract zero again since 2>0. So it subtracted zero from two again,

added one to the quotient, and checked to see if there was enough left to subtract zero again.

The old mechanical calculators got stuck in an "infinite loop" subtracting zero over and over

again. The quotient register looked like an old gas pump register with the dials spinning as

the quotient grew.

A friend of mine in high school rented a mechanical calculator to do lots of arithmetical homework.

On the front in bold letters it said "DO NOT DIVIDE BY ZERO!" Of course, that was an invitation.

So about half way through his assignment he starts a division by zero. The calculator was run

on electricity, so it just kept spinning the digits in the quotient. Quite fun to watch, but tiresome

after a while. So he punched "clear" and all the other buttons and levers he could find. Nothing

stopped the process. Finally he got the idea to pull the plug, and presto! it stopped.

So he thought that he'd better get back to his homework. He plugged it back in and presto! it

resumed the division calculation. He did the rest of his assignment by hand. It appears that

something had to be reset internally to get it to stop the division. I suspect when he returned

the machine he just set it on the counter and quickly headed for the door!

The old electric mechanical calculators (which were made of metal and weighed about 60 lbs)

demonstrated quite well the "repeated subtraction" algorithm for division. You could watch it

as it proceeded through a calculation as the dials spun and the carriage shifted. Slow by today's

standards, but it got the job done (if not dividing by zero!).

And that 0/0 thing ... it is like asking "how many 0s in 0?". Are there no zeros in zero, or perhaps just the one? See: Dividing by Zero

]]>Calligar wrote:

If you are talking about the 3'rd line where it says, (a-b)(a+b) = b(a-b), you are multiplying, not dividing, unless I'm missing something...?

Let a=b

a² = ab

a²-b² = ab-b²

(a-b)(a+b) = b(a-b)

a+b=b

1+1=1

2=1

In line three you have a factor of (a-b) on both sides. On line 4 you do not. Both sides were divided by (a-b) which is equal to 0. That is the misstep. Generally, you never divide by a variable unless you are sure that it cannot be 0. Sometimes such divisions are fatal, sometimes they just destroy answers. For instance

Divide both sides by x

Here the decision to divide by x, has cost us the other root which is 0.

]]>0 x 63 = 0 x 105

If dividing by zero was ok you'd get

63 = 105 ??

So in basic number theory there is a rule that you shouldn't ever divide by zero

In the fake proof

(a-b)(a+b) = b(a-b) is ok.

The problem arises at the next step because when you cancel a common factor what you are actually doing is dividing both sides of the equation by it.

eg 7 times 3 = 7 times N

divide both sides by 7

3 = N

In the fake proof to get to the next line you have to divide by (a-b), that is the false step as (a-b) = 0

0/0 has an indeterminate value. Why?

Let's go back to what division is.

48 = 6 x 8 so 48/6 = 8

You reverse the multiplication process.

So to work out 0/0 you have to think what would be the number here 0 x ? = 0

As any number will make that work, you cannot say what 0/0 is. (In calculus you have to work with limits to sort out 0/0)

Bob

]]>Also, I'm not sure I quite understand the reason for simplifying it. Everything seems fine about it too me...

]]>In line 4 to 5 of the 2=1 proof there is a division by a-b. But if a=b as stated in the first line then a - b =0 and you can not divide by zero.

]]>As for one of the posts, I don't really understand the 2=1 proof. Everything seems right until you get to a+b=b, nor do I understand how they got to that. I fail to see how this is proving 2=1, or rather, making it arguable I should say.

]]>You're welcome.

Hi MIF

Awesome page!

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