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# How to Divide by Zero

Recently on digg.com, I saw an article on how to divide a number by zero. Almost every single comment for that post proved the person's math and/or logic incorrect in several different ways. Well, that got me thinking of a simple, yet elegant solution. I dare you to prove me wrong. Really! The final results are truly shocking. We may have to unlearn some wrong math if I am right.

First, let me make an a statement that everyone should agree is true.

Axiom #1:

0 = 1

Granted this is true for approaching infinity for the same reason that 1=0.999... Here is the proof for this. If you can't agree with this statement, then have fun disproving my proof.

Now, let's prove 1/0 equals something other than undefined, nullify, or error (E).

y = 1

0

Step 1: Replace 0 with the axiom:

y = 1
———
(1 / ∞)

Step 2: Solve this fraction. Since 1/(1/2)=2 and 1/(1/10)=10 etc..., then:

y = ∞

Therefore, this is true:

1

0
= ∞

Each time that I look at this equation, I get a mild headache. I think some of my brain cells are exploding.

Someone pointed out this is just my Axiom rotated by 90 degrees, and that I did not prove anything so far. I agree, however I think that my math is correct. Just not a proof. This will be useful for what I try to prove below.

Now, let's try a more general equation. Let's assume n is any constant, so a new axiom would be:

Axiom #2:

0 = n

Now, let's prove what 1/0 equals again using the second axiom.

y = 1

0

Step 1: Replace 0 with the second axiom:

y = 1
———
(n / ∞)

Step 2: Solve this fraction.

y =
——
n

Therefore, this is true:

1

0
=
——
n

Therefore, if n is a positive number, then y = +∞. If n is a negative number, the y = -∞. Finally, if n is equal to zero, then you can recursively start back to the top of this proof. This could be an infinite loop.

As a result, I do not think this is the definition of undefined that states something is undefined if it has no meaningful results. You decide.

Again, someone has pointed out that my conclusion is just my Axiom #2 rotated 90 degrees, therefore is not a proof at all. Yet, the logic of my math equations appear correct.

Let's try looking at this another way. Let's graphically draw this equation:

y = 1

x

First, let's get some data points.

For x = 4, then y = 1 / 4 = 0.25
For x = 3, then y = 1 / 3 = 0.333...
For x = 2, then y = 1/2 = 0.5
For x = 1, then y = 1/1 = 1
For x = 1/2, then y = 1/(1/2) = 2
For x = 1/3, then y = 1/(1/3) = 3
For x = 1/4, then y = 1/(1/4) = 4

For x = -4, then y = -0.25
For x = -3, then y = -0.333...
For x = -2, then y = -0.5
For x = -1, then y = -1
For x = -1/2, then y = -2
For x = -1/3, then y = -3
For x = -1/4, then y = -4

Now, let's draw this in a graph. Well, it looks like when x is 0, y approaches both positive and negative infinity.

So far, my proof holds up to my investigation. One divided by zero approaches positive and negative infinity.

Now, let try a more general proof to divide any number (x) by zero.

y = x

0

Then, we can substitute the second axiom to get:

y = x
———
(n / ∞)

Then:

y = x * ∞
———
n

Therefore, when x is not equal to 0 and n is not equal to 0 then:

y = ±∞     or      x

0
= ±∞

and when x = 0 and n is not equal to 0, then:

y = x

0
= 0 * ∞
————
n
= Undefined      since      0 * ∞ = Undefined

Finally, when n = 0 for any value of x, then:

y = x * ∞
————
0
= ±∞
——
0
= x

0

I do not think this really proves anything either. Therefore, infinity divided by zero is also undefined.

Let me summarize what I think I have proven.

Any number divided by zero is ±∞ and undefined. However in practicality, any number divided by zero is just undefined.

I apologize if I just gave you a migraine.

Thank you!

by Phil for Humanity
on 12/07/2006