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# Proof: 0 * Undefined = -1

Here is a wild proof that will surely leave you scratching your head.

Axiom #1: Two lines are perpendicular (intersect at 90º angles) if the slopes (or gradients) of both lines are negative reciprocals of each other. This is often written as:

Slope #1 = - 1
——————
Slope #2

For example, if Line #1 has a slope of 2 and Line #2 has a slope of -1/2, then the above equation is true. Therefore, both lines intersect and are perpendicular to each other.

Axiom #2: Alternatively, this equation can be written as:

Slope #1 * Slope #2 = - 1

Axiom #3: Slope equals rise over run between two different points on the line:

slope = rise
———
run

So, let's calculate the slope of a horizontal line. The rise between any two different points on a horizontal line is equal to 0 and the run will be any non-zero number (X), so the slope of a horizontal line is 0 divided by X for X not equal to zero. Therefore, the slope of a horizontal line is equal to zero.

Axiom #4: Conversely, the slope of a vertical line is X divided by 0 (where X is non-zero) which is equal to undefined. If you need proof of this, then read this proof.

Axiom #5: The final axiom is that horizontal lines are perpendicular to vertical lines. In other words, horizontal lines intersect at 90º angles to vertical lines.

Therefore, if we apply the equation in Axiom #2 to a horizontal line and a vertical line, then we would have this.

(Horizontal Slope) * (Vertical Slope) = - 1

Since we know what these slopes are equal to, we can do these substitutions.

0 * Undefined = - 1

I don't think this equation can be simplified any further, nor can I think of any useful application for it either. However, it is interesting.

If you used this equation instead:

(Vertical Slope) = - 1
————————
(Horizontal Slope)

Then you will obviously get:

Undefined = - 1

0

However, if you used this equation instead:

(Horizontal Slope) = - 1
————————
(Vertical Slope)

Then you will get:

0 = - 1
——————
Undefined

These 3 equations are basically identical to each other, however presented differently.

Furthermore, if you multiply both sides of the equation by any number (N), then the above equation can be generalized like this:

N * (0 * Undefined) = -1 * N

Therefore:

0 * Undefined = -N

If we divide both sides by zero. Yes.. I know how strange that sounds, and I know you were told to never do this. But please have an open mind for a moment.

0 * Undefined
——————————
0
= -N

0

Thus,

Undefined = -N

0

Substituting -N for N, since N can be any number or negative number:

Undefined = N

0

This is just what we would expect, because any number divided by zero is undefined.

by Phil for Humanity
on 04/10/2008