User:Freenaulij/Division by Zero

Okay, when we first learned addition and subtraction we learned that we couldn't subtract a large number from a smaller number right. Well, later we learned a new concept, negative numbers, and we could subtract larger numbers from smaller numbers. Later, in middle school, we learned that it wasn't possible to take the square root of a negative number. Then, about a year or so later we learned we could with the imaginary number i.

In both of these cases, we never learned why we couldn't do these things we just accepted what the math book told us because trying to do these things took a lot of thinking and so we accepted that it couldn't be done.

Now, all along, ever since we learned division, we have learned that it is not possible to divide by zero, but I have yet to have a math book give me a reason as to why division by zero is not possible. When we first learned elementary division, division by zero was not plausible because we learned division as making groups out of something. It therefore makes sense that we cannot have zero groups of something and we took division by zero as an impossibility.

So, I decided to contemplate the division of zero.

The result I came up with and the result I am currently standing by, because no one can give me a reason to why its wrong, is that 1/0 = infinity.

I am now going to try and convince you that the definition of division can be extended so that 1/0 is infinity through a mathematical proof demonstration, two examples of functions, I'm going to address the problem of elementary division, and I will give some properties that can be extended from this if it is taken to be true.

=Mathematical Proof Demonstration=

ParItalic text'Bold text'Bold text'Bold text t 1''' $$1 = 0.\bar{9}$$
'Italic text'

$$\mathbf{1: X  = 0.\bar{9}}$$

$$*(10) = *(10)\,$$

$$\mathbf{2: 10X = 9.\bar{9}}$$

$$  -(X  = 0.\bar{9})$$

$$\mathbf{3: 9X = 9\,}$$

$$  /(9) = /(9)\,$$

$$\mathbf{4: X = 1\,}$$

$$\mathbf{\therefore 1 = X = 0.\bar{9}}$$

Part 2 $$1/\infty = 0$$
$$\mathbf{1: 1 = 0.\bar{9}}$$

$$\mathbf{2: 1 - 1/(10^\infty) = 0.\bar{9}}$$

$$\qquad (1/(10^\infty)\ is\ the\ same\ as\ 0.000...0001)$$

$$\mathbf{3: 1-1/(10^\infty) = 1}$$

$$-(1) = -(1)\,$$

$$\mathbf{4: -1/(10^\infty) = 0}$$

$$/(-1) = /(-1)\,$$

$$\mathbf{5: 1/(10^\infty)= 0}$$

$$\mathbf{6: 10^\infty = \infty}$$

$$\mathbf{7: 1/(10^\infty) = 1/\infty}$$

$$\mathbf{\therefore 1/\infty = 1/(10^\infty) = 0}$$

Part 3 $$1/0 = \infty$$
$$\mathbf{1: a/b = c}$$

$$\mathbf{2: a/(b*c) = 1}$$

$$\mathbf{3: a/c = b}$$

$$\mathbf{4: 1/\infty = 0}$$

$$\mathbf{\therefore 1/0 = \infty}$$

This is the mathematicl proof demonstration for why one divided by zero is equal to inifinity. Some of the people who have completely blown this theory off have said the error lies in the last part where we divided by zero. Well, that's what I'm trying to disprove, so that idea has to be ignored.

= The Functions =

$$y = ax$$


Okay, envision in your mind a line in the form of y = ax, where a is the slope.

If you cannot I have it here for you.

If a = 0, then the line should be horizontal right?

As a increases, so does the slope of the line. The line becomes more like a vertical line, just as when the slope of the line decreases to zero, it looks more like a horizontal line.

Keeping the above statement in mind, lets pretend a = $$\infty$$ and its possible to have a slope of infinity.

What would you expect the graph to look like?

A vertical line.

When we try to find the slope of this line through the formula $$(y_2-y_1<)/(x_2 -x_1)$$, we get something like 1/0. We learned to call this "undefined". Well, why not call it infinity. Because that's what a graph with an infinite slope would look like.

$$y = 1/x$$
Here's the obvious line that must be addressed, because well its what I'm trying to compute.

When the function of y = 1/0 is graphed, you find yourself with a graph that has no point for either x = 0 or y = 0. If it cannot be graphed its not possible, right? So you say well there goes that theory.

Well, I say that this actually proves my theory. A coordinate plane is only suitable for plotting real numbers. Is infinity a real number? No. Therefore it cannot be graphed, but the graph does show that the smaller x or y gets the larger the other gets. So if x was to equal the lowest it could possibly be, zero (exluding negative numbers, those are on the opposite side of the graph along with negative infinity), then y should be the highest number it could possibly be, infinity. And vice versa.

This goes hand in hand with my above proof.

Through this the points ( 0, $$\infty$$ ) and ( $$\infty$$ , 0 ) cannot be graphed but are nonetheless points in the function.

= Problems with Elementary Divison =

We learned that division consists of taking a group of something and dividing it into groups.

For example, if we have 8 apples and 4 people, each person will receve 2 apples. 4 groups of 2 have been created out of 8 apples.

Divison does not correspond with my theory. At first glance.

There is another way to look at this problem of 8 apples and 4 people. I have a pile of 8 apples. I can take 2 apples away from that pile 4 times.

Similarly, if I have a pile of 1 apple, I can take 0 zero apples away from that pile an ifinite number of times.

If you think about it, you can also apply it to the first way of thinking about division. If I have one apple for 0 people. That's apples person. I have an infinite number of apples per person.

=Extended Properties=

From this idea that 1/0 = infinity and 1/infinity = 0, there are a number of properties that we extend from this.

The first thing that needs to be addressed is 0/0. 0/0 could be one, if we remember the rule that anything divided by itself is one.

To demonstrate this problem I will show you this equation:

$$\mathbf{1: 2 * 0 = 1 * 0}$$

$$/(0)=/(0)\,$$

$$\mathbf{2: 2 * 1 = 1 * 1}$$

$$\mathbf{3: ??2 = 1??\,}$$

0/0 cannot be one.

Well, then what could it be?

The other possibility is zero. This is pretty much accepted as the closest idea to the "truth".

The problem with that can be seen through the manipulation of the equation.

$$\mathbf{1: 1/0 = \infty}$$

$$mathbf{\therefore \mathbb{R} \div 0 =\infty}$$

$$*(0)=*(0)\,$$

$$\mathbf{0/0 = \infty * 0}$$

So 0/0 is equal to 0 times infinity, if we take 1/0 to be infinity.

Well that makes sense, doesn't it? anything times zero is equal to zero.

I answer that anything times infinity is equal to infinity.

So what do we get when we multiply these two extremes.

The answer lies in the equation 1/0 = infinity.

This equation is not the full equation. You see any real number (R) divided by infinity is zero, so any R divided by zero would be infinity.

So $$\infty$$ * 0 = R

So infinty times zero is equal to a real number.

Therefore 0/0 is equal to a real number.

Any real number? No.

A specific number.

Let's take another look at the equation 0 * 2 = 0 * 1.

If we divide both sides by zero, the 0/0 on the left is equal to 1, and the 0/0 on the right is equal to 2.

We get 2 = 2.

This is kind of odd, but one of the properties about infinity and zero is that you cannot use them to determine a variable.

For example if we had 0 * X = 0 * 5.

We cannot determine what X is equal to, even though we know what 0/0 is.

Both infinity and zero are special in that they cannot be divided out of an equation like all non-zero real numbers.

From this we get that 0/0 = 0 * infinity = infinity/infinity. (we get the last part by a similar process from 0/0 to 0 * $$\infty$$)

The unfortunate thing is that there is absolutely no use that I can think of that comes from proving and real number divided by zero is equal to infinity.

-- freenaulij 03:49, 4 December 2007 (UTC)