1 fish, 2 fish
Let’s count the meters that a capybara will walk to find some shade after having a drink. He may walk a meter, another meter after that, many more meters, or he may just decide to chill in the lake instead and walk no meters at all. We call this counting of 1, 2, more, or 0: numbers.
Perhaps instead he’ll leave the lake but we’ll count him walk less than a meter before deciding to nap, or we’ll count him walk more than 5 but fewer than 6 meters. So 1⟋2 or 42⟋8 are numbers.
Let’s say we now want to measure how far he’ll walk on either side of the lake bank. Rather than measuring his movement in an eastward value and a westward value, we can measure it more conveniently as a single longitudinal value, by adding a sign to the number. So -1, -2, and on are numbers.
Similarly, we could make even more algebra convenient if we added another dimension to that number, making it a complex number. We could even describe how our capybara laid himself down with an more complex number.
One may start to wonder just how many discrete numbers you can glue together while still calling it a
number, but even these numbers all maintain a very important property: You can count to them, calculate new numbers with them, that you can then back from. This is the raison d’etre of numbers.
To infinity and beyond
Infinity is generally well accepted as not being a number. And for good reason: It cannot be counted to. Attempting to use it in algebra will break algebra. Nevertheless, it has the property of numeric order, which makes it sometimes valuable to use in place of a number. For instance, 0 < x < ∞ simply means x may be any positive number. Yet, ∞ is not a number.
Infinity has a reciprocal, too. The infinitesimal, which also has numeric order, but is less useful than infinity, so lesser known. 1⟋∞ is greater than 0, but less than any positive number. Even so, 1⟋∞ is not a number... even though it is sometimes referred to as one.
Did you know that there are many more mathematical objects that have numeric order, but are not numbers?
Pi in the sky
The ratio of a perfect circle’s diameter to circumference is often useful to know when dealing in geometery. Man has been using it for a long time. The ancient Babylonians measured it to 25⟋8. Today it has been calculated to over 202 trillion digits. Impressive, but this isn’t exactly π.
A complete formula for π is:
4 × ( 1 - 1⟋3 + 1⟋5 - 1⟋7 + 1⟋9 ... )
This is a series. A sequence of values that converge toward π as they are successively summed. Every application of π that has ever existed, or will ever exist in the future, have all used numbers that order near to π, but are not actually π. This is because you can’t count to it, so π is not a number.
Indeed, this is true of all irrational numbers
. They have numeric order, but can never be be counted to, only approximated. Irrational numbers
are not numbers.
Unreal reals
It might sound like this is just a semantic matter, but many mathematicions and logicians have treated infinity and series as real numbers, which has cemented many absurdities in to modern maths. The most affected probably being set theory, where sets of infinite count are reasoned about and not treated as the oxymoron that it is.