Mathematical equality

Until the year 1557, the mathematical equal sign ‘=’ did not exist. Up to that point, people simply wrote ‘is equal to’ instead. The Welshman Robert Recorde got tired of the constantly repeating words and decided something shorter should replace it. According to Recorde, the sign’s two stripes seemed most appropriate because ‘ nothing can be more equal than two straight lines’. Initially, he drew the symbol much wider to demonstrate the parallelism clearly.

Enough history. Growing up with mathematics, I always presumed a comprehensive equality. E.g. anything on the left side of the sign should be equal to anything on the right side. This presumption turns out to be false.

Suppose, you have a function:

f(x) = x

This is a straight line of 45 degrees, with a two-sided, infinite domain. For any value of x, there is a corresponding y value.

Suppose, you multiply x by one, then x remains the same value. More specifically:

x = x * 1

Another mathematical law is that when a division’s nominator and denominator are equal, the division is equal to one, e.g.:

5/5 = 1

x/x = 1

Combining the two laws (of multiplying by one and about the same nominator/denominator), we have the following:

(x*1)/(x*1) = 1

However, the characteristics of both sides suddenly change in the following example:

x = x * (x-1)/(x-1)

Now, both sides have different domains. On the left side, you can pick any value for x. Yet, that is not the case anymore on the right side. There, x cannot be equal to one. To see why suppose that x would be equal to one, then we would get the following:

x = (1) * (0/0)

Since you cannot divide by zero, no solution exists for x = 1 on the right side.

This might seem strange.

From this, you can conclude that, following a sequence of legitimate mathematical operations, a whole line is equal to a line with holes.

Hopefully the same does not hold for your tires..