Chapter 1 of Calculus by Michael Spivak begins with twelve basic axioms of the real numbers, including the existence of multiplicative inverses (a·a¹=1). This axiom justifies that division is only valid when the divisor is a non-zero real number. The importance of restricting division to non-zero numbers really stood out to me when I was introduced to an incorrect “proof”.

Here is the “proof”:

Suppose x = y.

    x = y

Multiply both sides by x:

    x2 = x·y

Subtract y²:

    x2−y2= x·y−y2

Factor:

    (x−y)(x+y)=y(x−y)

Divide by x−y:

    x+y=y

Since x=y:

    2y=y

    so, 2 = 1

the error occurs in the step where both sides are divided by x-y. Since x = y was assumed, this means:

    x-y = 0 

so this step divides by zero, which is undefined.

2=1 seems like a one-off contradiction, but this proof made me think that if dividing by 0 was possible then it wouldn’t be limited to just 2=1.

Applying the usual rules of arithmetic while allowing division by non-zero numbers (0)

For example:

    since 1⋅0= 2⋅0 = 0

Divide both sides by zero:

    1=2

Then continuously add one to both sides:

    2=3, 3=4, 4 =5….

Another example:

    since 1⋅0= 3⋅0 = 0

Divide both sides by zero:

    1=3

Apply 1⋅0= x⋅0 = 0 for all positive integers, divide both sides by zero:

    1=3, 1=4, 1=5,…

The impact of these two examples is that without clarifying that division only by non-zero values is possible the whole number system is obliterated. This struck me because allowing a single invalid operation causes every number to become equal, destroying the idea of quantity.