“Not everything that can be counted counts, and not everything that counts can be counted.” — Albert Einstein
Limitless in every dimension but seemingly always limited in our appreciations.
Since prehistoric times, our ancestors have sought to describe quantities. At first, it began with simple cave illustrations: We perceived three buffalo, and so, we drew three buffalo. In correspondence, the notion of natural numbers—one, two, three, and so on—grew to their modern maturity as we discovered more about this subliminal art of counting.
Eventually, this keenness to quantity spawned several symbolic representations. The Ancient Babylonians, Egyptians, Greeks, and Romans each took it upon themselves to demonstrate proper counting methodology (i.e. to illustrate numbers in universally recognized fashions). Consequently, dots, lines, fingers, swirls, crowns, and even frogs were used to represent the “counting” numbers. Given time for proper evolution, these numbers eventually became the familiar figures we use today—1, 2, 3, etc.
Of course, numbers extend far beyond the idea of “countability.” In other words, there are some numbers out there for which we simply do not say, “There are ‘x’ number of students in the classroom,” or, “That tree has ‘y’ number of leaves.” These numbers could be, for example, -3, pi, and ½. To illustrate this point more fully, I call upon the famed Greek mathematician Pythagoras, a man who struggled with the notion of what we now call irrational numbers.
To be a touch more precise than previously stated, Pythagoras went through several internal battles because of the number sqrt(2), an irrational number very important to trigonometry. Using modern technology, we can approximate the number sqrt(2) fairly easily. (It is roughly equal to 1.41421356237.) However, despite how powerful our technology may be, even we cannot fully obtain the value sqrt(2) because its decimal component never ends (i.e. an infinite amount of numbers exist after the 7 in 1.41421356237). This concept troubled Pythagoras greatly because he could clearly visualize what sqrt(2) equaled geometrically using a theorem which bears his namesake. (See the diagonal line in the diagram.)
So, having seen sqrt(2) drawn as a line segment, I now present to you the very same question Pythagoras asked himself countless numbers of times: How is it that a number which theoretically never ends fits into a perfectly finite geometric observation?
The answer lies within the beautiful, and careful, construction of our number system. To read more about how we organize numbers (and reach a more satisfying conclusion to my question), click here.