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Although I agree with the above statement, the purpose of this thread is to highlight a fundamental distinction between 0/0 and 1/0.

To highlight this difference between 0/0 and 1/0 it requires approaching division with what I call the reversal perspective. This perspective considers division to be the reversal of multiplication, specifically:

A x B = C then C/B = A.

If we consider that 0/0 = A with A being the unknown then A x 0 = 0. As soon as you start to insert numbers in for A, for instance 1 x 0 = 0 or 534 x 0 = 0, all of them hold. 0/0 is undefined because the answer could potentially be any value.

Now when we look at 1/0, 534/0 or any other value divided by 0, there is a solid distinction:

A x 0 = 1 and A x 0 = 534 are invalid, A x 0 always = 0

The reality is that 1/0 is undefined because there are no values to which it equals; in complete contrast to 0/0 which could be any value.

Although this distinction is clear in the above rationale, it has been overlooked when graphing functions. When graphing y= (x^2-1)/x-1 mainstream mathematics states that the graph is equivalent in appearance to the function y=x+1, with the exception of when x=1 where it is equivalent to 0/0. As 0/0 is considered to be undefined in the same way that 1/0 is, the mainstream depiction is of a hole on graph at that point.

In consideration of the highlighted distinction between 0/0 and 1/0, you would think that the hole would be more suited to 1/0 and that 0/0 would be better represented by a vertical line. Simply because 0/0 could result in any of the values on the number line, whereas 1/0 is not equal to any value on the number line.

As some may not be convinced by the rationale presented or convinced that a vertical would be more appropriate in a graph where a point equals 0/0. I have come up with the horizontal line test.

The rationale of this test is simple. When you have two equations equal each other and solve for an unknown variable, it allows you to determine the intercept point on a graph. One way to plot an equation is to have them equal to a mass of different horizontal lines, such as y = 1 or y = 2 and then mark the intercepts on a Cartesian plane. So lets to the test with y = 3, y = 2 and y = 1 and see where the intercepts are.

Where does y = 3 and y =(x^2-1)/x-1 intercept?

(x^2-1)/x-1 = 3

x^2-1 = 3x-3

0 = x^2-3x+2

0 = (x-2)(x-1)

their for x= 2 and 1

Intercepts detected at the points (1,3) and (2,3)

Where does y = 2 and y =(x^2-1)/x-1 intercept?

(x^2-1)/x-1 = 3

x^2-1 = 2x-2

0 = x^2-2x+1

0 = (x-1)(x-1)

their for x= 1

Intercept detected at the point (1,2)

Where does y = 1 and y =(x^2-1)/x-1 intercept?

(x^2-1)/x-1 = 1

x^2-1 = x-1

0 = x^2-x

0 = x(x-1)

their for x= 0 and 1

Intercepts detected at the points (0,1) and (1,1)

Even if you don't mark out each of the intercepts on a Cartesian plane, its rather obvious that there was intercepts detected every time a horizontal line passed though x = 1. this would be consistent with their being a vertical line at x = 1 and completely inconsistent with a hole being present at the point (1,2).

If you agree there is a difference between 0/0 and 1/0, then do pass this information onto your math buddies. If you don’t think there is a difference, please elaborate your reasoning in the comments.

ヽ༼ຈل͜ຈ༽ﾉ Thanks for reading ヽ༼ຈل͜ຈ༽ﾉ