Infinity = -1. What’s wrong with this proof?

In co-ordinate geometry at high school it is taught that if two lines are perpendicular (at right angles) then the slope of these two lines will multiply to give -1.

So let’s take a look at the co-ordinate axes. Slope is calculated as rise over run (or changes in y divided by changes in x). At the x-axis, the rise is zero and thus the slope is zero. At the y-axis the changes in y are infinite (it’s a vertical line) and the changes in x are zero. Thus the y-axis has a slope of infinity/zero.

Thus under the same logic as above, we get

infinity * 0 = -1.
If we cancel the zeros, we get infinity = -1.

This is obviously not true. In fact the first equation cannot be true either as zero multiplied by any number must be zero.

So why does the pattern break down at the axes? The answer lies in the slope of the y-axis (infinity/zero).
Anything divided by zero is undefined. Thus the slope of the y axis is not really infinite, just undefined and undefinable. Thus it is not possible to use it in any equation.


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