My answer to my father about dividing by 0.

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My father when I was a teenage when I divide by 0 why couldn't I take 0 times undefined and obtain a value back. Here is why you can't do it through proof by contradiction.

I tried to define an answer to dividing by 0. The problem I saw was how can this be undefined even if you supplied an undefined value as an answer. I tried to make 1/0=1/0. It did not work. I obtained the indeterminant back. I did not obtain 1 back. It could not be defined, D nor could it be undefined ~D. This means whatever the answer had to be had to be here ~(D or ~D). When I use DeMorgan's law to convert this I obtained (D and ~D) which was a contradiction. There was no member that existed that could satisfy both (D and ~D). I decided to probe further.

Maybe the answer was in ~(D and ~D). This statement is a tautology because all values satisfy this. It is the complete opposite of a contradiction. When I used De'Morgan's law I ended up with (D or ~D). I ended up going in a complete circle. There was no value in D, ~D or (D and ~D) that could satisfy division by 0. What did this mean? It meant that because it leads to a contradiction I can't use ~D as a member of subset of D. This is what my father asked of me a long time ago. This was the assumption that he had. This means in math something has to be defined or undefined. It can't be both or neither. It is either D or ~D. This means in mathematical absolute terms there is no definable answer that would satisfy x/0 where x<>0. This is the final answer to my father's question of can you multiply 0 * undefined and obtain a value back? The answer is no by proof of contradiction

I stink at maths so I might be off here.

I always replied "nothing" when someone ask me
what is x divided by 0 because if I use 0 to divide
it is the same to me as not dividing at all, as if I
never performed any mathematic operation
at all.

I just sat there and did nothing, sometimes
as a joke, if someone ask "So what are you doing today?"

I reply "Dividing by zero" meaning "I am doing nothing."

Thats just how I understand it, which means
I might not get it at all, o well.

Pick a boogar.

It's "undefined" by is represented as a vertical line on a cartesian coordinate based graph.

The reason is...first understand that...

0/X = 0

Assuming X is non-zero

Using algebra we can rearrange this equation to be

0*X = 0 In English, 0 multipled by any number is always 0

But if we say X/0 = D where X is non-zero and we don't know what D is at all.

Using algebra to rearrange....

0*D = X But this statement is never true. We have already determined that 0 times anything must equal 0.

Now look at Y = a/x where a is a non-zero number. Let's say it's also positive for simplicity.


Choose any number you want for a, and any number you want for x. Perform a/x and find Y
Remember that number.

Then choose a smaller number for x and repeat.

Keep repeating. When x gets to 1 you will see that Y = a

But making x smaller, you will see that Y starts getting bigger than a.

In fact, as x approaches 0, Y approaches infinity.

So in this case, saying a/0 is undefined is the same as saying we go off to infinity.

Simple answer: Let X be non-zero if X/0 were defined then X/0 = Y for some number Y. This implies
X = Y*0. But Y*0 = 0 and X is not 0. Contradiction. And that is why we cannot divide a nonzero number by 0. It is meaningless.

ruveyn

My father was asking why couldn't undefined or no value on the cartesian plane be a value in itself? He was asking why couldn't undefined be a defined value in itself? He was asking why couldn't nothing be a form of something in itself? I just showed him the contradiction of what he was saying. That was my point. He was trying to say something was defined and undefined at the same time which is a contradiction. Otherwise, you guys are right about everything you all said.

I've always defined n/0 as infinity.

Well, you can make up any definitions you want. But if you made division by zero a defined value, then as a consequence of that, you'd mess up how multiplication works.

So you can define division by zero, but only if you don't mind giving up defining multiplication, and also don't mind that what you have isn't really division anymore.

There are a few reason why it's not given a value.

In respect to the cartesian coordinate system, which is a simple x y coordinate system...example: go x steps over and y steps up

y expresses a distance from the origin along the y axis.

If we had some curve y = 20/x and we said that undefined is going to mean 10, we'd end up with 20/0=10

Meaning if we were at 0 on the x axis our curve would be 10 units up on the y axis. But that's not true because there is already a value, that if you divide 20 by, you will get 10. That value is 2. 20/2 = 10

So in that sense we cannot arbitrarily assign an actual value to undefined. They're all taken.

In practical applications, undefined takes the form of a concept.

What if, for example, I had some strange guitar amplifier which had a gain that could be modeled as

A = 10/R where R was some resistor (this actually is not how guitar amplifiers are really modeled but it explains my point)

What if I wanted to know what happens when I make the resistance of that resistor smaller and smaller and smaller.

On my graphing calculator, I can see my gain, A, will get bigger and bigger and bigger.
I can't actually really get to R=0 because I'm using wires and all wires have some resistance, but I can see that as my resistance R approaches 0, my gain theoretically approaches infinity.

In the real world, even if an amplifier could be described by that model, other things would eventually come into play to prevent the gain from actually becoming infinite, like the fact that we can't have 0 resistance with wires, or other physical characteristics would cause the model to break down, but in other areas, such as astrophysics, a similar model might be applicable for very large values before the model breaks down.

So in understanding the nature of things, we really use the equations in a qualitative way to get a feel for how things act.














.

Infinity in the context of the real or complex number system is NOT a number.

It does not have the algebraic properties of a number.

ruveyn

exactly, which is why the question asked by my father was a contradiction. He expanded the usage of the definition when he should not have.

Chronos, do you mind if I add what you said to my writings at some point. I will give you credit.

As you indicate, there is no division without multiplication and no subtraction without addition.

ruveyn

Bottom line: division by zero leads to a contradiction.

It cannot be made any simpler than that.

ruveyn

Infinity makes sense in the context of limits. Additionally, while infinity is not normally included in the real or complex number sets, you can construct a set that includes infinity by topologically adding a single point to either of them. Look up the Riemann number sphere:

http://en.wikipedia.org/wiki/Riemann_sphere

Last edited by Jono on 24 Oct 2010, 3:22 pm, edited 1 time in total.

Divide by zero and the answer is both plus infinity and minus infinity as a graph of 1 / x when plotted closer and closer to zero gets towards plus and minus infinity.

By the way I like cubedemon6073's avatar, it looks like a unit cell of CaF2 or UO2 (uranium dioxide's unit cell has the same shape as calcium flouride)

When you take the limit, yes depending on when whether zero is approached from the left hand side or the right hand side. What didn't mention though is that when you divide zero by itself, that can be made to approach any number you wish, depending on whatever the function is that you're taking the limit of. So 0/0 is completely indeterminate.