RATIONALLY IRRATIONAL Irena Papst & Nigel Pynn-Coates
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ou use numbers every day. Most intuitive are the counting (natural) numbers, and they come to us so easily that they seem to be independent of our mathematical imagination: one apple, two dinosaurs, five golden rings. As mathematician Leopold Kronecker famously remarked, “God made the natural numbers; all else is man’s creation.” Even going one step further and considering fractions is relatively intuitive; how do we share one apple among two friends? Halve it. These numbers are called the rational numbers. More esoteric are those numbers that can’t be written as fractions, the so-called “irrationals”. The most famous of these are pi, e, and the square root of 2, but we struggle to find more concrete occurrences. How
number one was especially beautiful. It can be seen as the building block of counting numbers, since you can generate any natural number by adding one to itself enough times. So which shape could be more beautiful than a square with side length one? It’s simple, symmetric, elegant, and has an area equal to its side length: the exquisite number one. Moreover, it relates to the Pythagorean theorem, the group’s most wellknown contribution to mathematics. Yet it was precisely the connection between the unit square and the theorem that turned the Pythagorean world upside down. Its inhabitants discovered the unit square had an inherent ugliness; the length of its diagonal is not a rational number, or in other words, the square root of two cannot be expressed as a
ARTWORK BY IANITZA VASSILEVA
can we divide one pie into pi pieces? What exactly is the square root of two apples? Who knows?! Despite an apparent lack of “reality” to these numbers, they pop up very naturally in geometry, one of the most concrete branches of mathematics, as we’ll see shortly. For the Pythagoreans of Ancient Greece, numbers held a mystical significance. They were intimately connected to Pythagorean views on music, nature, and the fabric of reality. In fact, Pythagoras said that “number is the ruler of forms and ideas, and the cause of gods and demons.” Though convinced of their sanctity, the Pythagoreans had no conception of anything beyond the rational; they thought all numbers could be represented as fractions. Not only did they believe that numbers were somehow pristine, but the Pythagoreans also felt some numbers were purer than others. For instance, they believed the 8 ▪ INCITE MAGAZINE ▪ DECEMBER 2012
fraction. According to legend, a student proposed this idea while at sea and Pythagoras was so upset he threw him overboard! For a mathematician, it is not enough simply to assume that irrational numbers exist; it must be proven. Axiomatically, the square root of two must bepeither rational or irrational. Let’s assume that 2 is rational. We want to show that from this assumption, we can deduce a contradiction (a nonsensical result), and that therefore we were wrong to p initially posit that is rational. Then we can 2 p conclude that 2 must actually be irrational. p Since we have assumed that 2 is rational, it can be expressed as the ratio of two numbers, a and b, written in lowest terms (we have factored out all p common factors). Mathematically speaking, 2 = a/b, and by rearranging, a2 = 2b2. A number is even if it is a multiple of 2, so a2 is even. Thus a itself must be even, since the square of an odd number is odd. Since a is
even, we can write a = 2c and substitute that into our equation. This gives us (2c)2 = 4c2 = 2b2, so b2 = 2c2. By the same argument as constructed for a, b is also even. But, we assumed that a and b had no common factors, and we have just shown that a and b are both even, so 2 divides both of them. This is our contradiction! Therefore p our original assumption that 2 pis a rational number must be wrong. Hence 2 is an irrational number. Quod erat demonstrandum. What have we done in this proof? We used a standard technique - proof by contradiction - and facts about even and odd numbers, to prove a very powerful result about the existence of a different kind of number. You may be thinking to yourself, “Sure. Cool. The Pythagoreans were nuts for numbers and the square root of two is legitimately irrational. But what does this all have to do with creation, Incite’s theme this month?” What’s so creative about this proof and, more generally, about math? Let’s switch gears for a moment and think about music, a discipline which is widely seen as very creative. Jazz musicians, for instance, often improvise during performances. Arguably, this is artistry at its most imaginative, creating on the spot, and inspired by the rest of the piece being performed. Although musicians craft a completely new musical statement when improvising, they are not creating out of nothing. Instead, they use the original piece to inform their improvisation, and draw from a pool of musical techniques and phrases that they have developed over time. Similarly, we have proved a completely new fact about numbers, but like a musical improvisation, it did not come out of nothing. We drew from our pool of mathematical knowledge and techniques, using facts that we already knew about numbers along with a standard proof method, and integrated these in the context of this problem to create new knowledge. It is precisely this process that is at the core of a creative act: using various tools to inventively utilize prior experience in a different setting. But, as we just saw, this is at the heart of mathematical proofs, and mathematics itself! Mathematics truly is a creative discipline. Quod erat demonstrandum.