I’d firstly like to refer to one of my favourite blogs, Greg Ross’ Futility Closet, that describes itself to be “an idler’s miscellany of compendious amusements.” In its Bullseye item, it cleverly proves that 0.999… (the ellipsis denotes repetition) is precisely identical to 1.
To many this is nonsense while some might find it rather intuitive. The point of this piece is not to share this oh-so-compelling idea specifically, but to recognize the progress of mathematics and how it is all related. I’ll get into that in a moment.
If you read the short proof on Futility Closet, you will notice that the desired result is achieved by simple algebraic manipulation. You might wonder whether such a manipulation was legitimate. For example, here is a “proof” that 2 x 2 = 5, from the same blog. That proof is fallacious because it incorporates division by zero.
Let us first assume the 0.999… = 1 proof wasn’t erroneous. What makes the result more assuring is that there are several other ways of proving this intriguing fact, using theorems in many other areas of mathematics. Check out the Wikipedia article on 0.999... for some examples. This simple idea seems to epitomize how connected the field of mathematics is, and how sophisticated it has become; because we can use facts from entirely different areas of mathematics to make equivalent inferences.
How about another example. The greek letter π (“Pi”) represents the ratio between the diameter of a circle and its circumference (recall that the circumference of a circle is given by C = π x d, or that π = C / d.) One way of picturing the latter formula is to reflect on what differentiates a circle from, let’s say, a square. Well, a circle has no corners, of course. How, then, can we make a square look more like a circle? We simply strip the corners of the square and connect the leftover sides; this yields a hexagon. Repeating this process once more gives us an octagon. Six iterations later produces an icosagon (a twenty-sided polygon!) If you see a picture of one, you will agree that this is already looking very much like a circle. So imagine reiterating this process an infinite amount of times. Not just a million more iterations; not a trillion; but forever more – and we precisely get a circle. We conclude that π is equal to the perimeter of an infinite-sided polygon divided by its diameter. Now recall that the area of a circle is found by A = π x r2. A unit circle is a circle with radius of 1 unit. Using calculus we can evaluate areas under curves (for those with more math background, you know that the equation y2 + x2 = 1 represents the unit semicircle). Using some calculus and trigonometry, we would find that π is actually equal to the area under that curve multiplied by 2 (since it’s only a semicircle). If you skim through the Wikipedia article on Pi, you will see that π is alternatively equal to 6arcsin(0.5), for example, along with many other representations deduced from areas such as geometry, probability, complex numbers, and infinite series.
I suppose this is one way how mathematics is different from a field like biology. We know that every mathematical principle we use is always true in general; in contrast with “studies” that are published every day that are completely dependent on experiments and surveys. One day coffee is healthy to drink everyday and the next day it is a health risk. Without math, everything in our lives would be based on uncertainty. Mathematics is really the only certain thing I know of.
