The Attraction of Mathematics (Tuklas Vol. 14, No. 1 - Nov. 17, 2012)

THE ATTRACTION OF MATHEMATICS

The physicist Eugene Wigner expressed awe at what he considered a mystery and a miracle, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences." This is in fact the title of a now famous essay that he had written. It expounds on the power of mathematics when applied to the various fundamental theories in Physics. Wigner had in mind Newton's calculus as applied to his theory of mechanics and gravitation, Maxwell's equations on electromagnetism, Dirac's mathematical formalism of quantum theory, and Einstein's General Theory of Relativity. Wigner's work itself is about the so called group theoretical model of elementary particles and requires several years of study for students of Physics and Mathematics to understand.

For those of us without much mathematical preparation, mathematics can also be a source of awe, similar to what Wigner had experienced. To see what I mean, we look at a few aspects of mathematics (albeit superficial in such a short note) which hold some of its characteristic attractions for many people.

Math is a source of understanding and illumination. For instance, what does it mean for an Olympic sprinter to run a world record of 9.87 seconds in the 100-meter dash? If we divide both numbers by ten, we see that, on the average, the athlete ran 10 meters in less than one second.  Using a meter tape, one can see that 10 meters is a significant distance. To take another example, around 6 million Jews were killed in the Second World War, from 1939 to 1945. As the mind is unused to such a large number, try dividing 6 million by 6, then by 364. This comes out to 2747 Jews killed everyday for six years! As a last example, try to gain an understanding of the following phenomena: at the height of Michael Jordan's power and popularity in the NBA, his salary was 36 million dollars in one year.

These examples, simple as they are, show the great aid mathematics adds to our understanding. In the second example, our emotional response, one would hope, was heightened because of the real horror of more than a couple of thousand of murders daily. In the first and third, in addition to illumination, we experience that element which is common to good stories, that of surprise. Lest we belittle these small demonstrations, employing as they are only the basic operations, we can imagine that schoolchildren and accountants alike in the times of the Roman Empire had a grand time multiplying, say, XXIV (24) by XIX (19). To keep track of the Empire's population must have been quite difficult. 

More than simple understanding, mathematics shed deeper light on a few things we take for granted such as counting, the computation of areas, and the twin concepts of being outside and being inside a closed curve. 

To count the number of persons in a room, we mentally tick off one, two, three, etc. while pointing at each person in turn. We know that there are sufficient chairs in the room if there are empty chairs after everyone is seated, that is, there are chairs not corresponding to any person in the room. This one-to-one correspondence between objects counted and the abstract set of natural numbers is the essence of counting. That is why it takes a very little leap of the imagination to count to "infinity": after ticking off any number, however large, we just add 1. If we are now asked if there are many more natural numbers than positive even integers, it should not be too difficult to observe that the answer is no. One set has as much elements as the other. Indeed, the correspondence \(1\leftrightarrow 2,2\leftrightarrow 4,3\leftrightarrow 6\), etc. demonstrates the equality of the number of elements between the two sets. Similarly, there are as much natural numbers as there are odd numbers, and as much natural numbers as the squares 1, 4, 9, 16, and so on. Much more surprising is the fact that there are the same number of elements in the set of natural numbers as in the set of positive fractions. This is made much more counter-intuitive by the fact that the integers are "far" away from each other while between any two fractions, however close, there is a fraction in the middle of them.

In calculus, a very powerful result (the Fundamental Theorem) and a very simple algorithm (subtract the values of the antiderivative at the endpoints) will tell us that a figure such as the first curve shown below has a finite area, although it is not "closed" or "bounded." This considerably generalizes our notion of area and is one source of the power of the calculus. 

As a final example, look at the middle figure above and determine if the dots are inside or outside the closed curve. How do we determine the answer? The surprising solution is the same as the way we determine whether a point is inside or outside a circle. Indeed, it is not at all easy to determine if one is in or out. 

Mathematics, just like art, is a source of great creativity from its practitioners. Just witness the inexorable logic and beauty of two of the earliest mathematical proofs known to the ancient Greeks. The first is the incommensurability of the number \(\sqrt{2}\). This just means that \(\sqrt{2}\) is not a fraction. To begin the proof, suppose it is equal to the fraction \(\frac{p}{q}\) written in lowest terms.  Simple manipulation will give \(2p^2=q^2\). This implies that \(q^{2}\), hence also \(q\), is an even integer. Write \(q=2k\) and substitute it into our first equation to obtain \(2p^2=4k^2\), or \(p^{2}=2k^{2}\). This, in turn, means that \(p^{2}\), hence \(p\), is an even integer. The fact that both \(p\) and \(q\) are even contradicts our initial assumption that the fraction \(\frac{p}{q}\) is in lowest terms.

Now take a look at the second proof, this time on the fundamental result claiming the infinitude of primes. The prime numbers are the basic building block of all integers and hence of all numbers. They are the integers greater than 2 which are not divisible by any integer except 1 and itself. To show that the set of prime numbers has infinitely many elements, assume the contrary. To this end, suppose we have the complete list of all primes: \(p_1, p_2, p_3, \ldots, p_n\). Consider the number \(N = p_1p_2\cdots p_n + 1\). Then this number is not divisible by any of the primes \(p_{k}\) in our list, as the division always leaves the remainder 1. This indicates that \(N\) is itself prime. We have reached a contradiction to our assumption that we already have the complete list of all primes. 

The infinitude of primes is one of those simple statements in the Theory of Numbers whose proofs are either of great complexity or as yet undiscovered. Some of these are the Prime Number Theorem on the number of primes less than a given number, the twin prime conjecture on the infinitude of twin primes such as 11 and 13, 17 and 19, 29 and 31, etc., and Goldbach's conjecture which states that all large odd numbers can be written as the sum of two primes. Fairly recent mathematical achievements are the Fermat-Wiles Theorem (formerly Fermat's Last Theorem) and the solution by G. Perelman of the Poincare Conjecture on the possible shapes of 4-dimensional objects with given special properties. Mathematics is thus full of beautiful and immensely difficult problems the solution of which will bestow more than passing fame to those lucky and persistent enough to solve them.

ABOUT THE AUTHOR:
Job A. Nable is an Assistant Professor at the Ateneo de Manila University. He obtained his Ph.D. in Mathematics at the University of the Philippines.

OLYMPIAD CORNER

from the 51st National Mathematical Olympiad of Romania, 2000

Problem: Let \(p\) and \(q\) be positive integers such that \(1<q<p\), and \(a = \left( p + \sqrt{p^2+q} \right)^2\). Prove that \(a\) is an irrational number and that its fractional part is greater than \(0.75\).

Solution: 

It is easy to see that \(p^{2}<p^{2}+q<(p+1)^{2}\) therefore \(p^{2}+q\) is not a square, so \(\sqrt{p^{2}+q}\) is an irrational number. It follows that \[a=\left(\sqrt{p^2+q}+p\right)^2=2p^2+q+2p\sqrt{p^2+q}\] is also irrational. Let us consider the number \(b=\left(\sqrt{p^2+q}-p\right)^2\). Clearly, \(b\) is also irrational, hence, we have \(b>0\). By AM-GM inequality, \[\sqrt{p^2\left(p^2+q\right)}<\frac{p^2+\left(p^2+q^2\right)}{2}\] which leads to \[\sqrt{p^2+q}<\frac{2p^2+q}{2p}=p+\frac{q}{2p}\leq p+\frac{1}{2}.\] Then \(\sqrt{p^2+q}-p<\frac{1}{2}\) so that \(b=\left(\sqrt{p^2+q}-p\right)^2<\frac{1}{4}\). This means that \(b\in (0,1/4)\) and since \(a+b\) is obviously a positive integer, it follows that the fractional part of \(a\) is greater than \(3/4=0.75\).

PROBLEMS

1. Find the solution set in \((-\frac{\pi}{2},\frac{\pi}{2})\) of the equation \[\log_2(1+\sin^2\theta+\sin^4\theta+\sin^6\theta+\ldots) = 1.\]
2. Given that \(R\) is an inner point of \(\Delta ABC\). Prove that \[\frac{1}{2}(AB+BC+CA)<RA+RB+RC<AB+BC+CA.\]
3. Find the probability of obtaining two numbers \(x\) and \(y\) in the interval \([0,1]\) such that \[x^2-3xy+2y^2>0.\]

We welcome readers to submit solutions to the problems posed below for publication consideration. Solutions may be submitted to the PEM facilitators on the deadline date or online via mactolentino@math.admu.edu.ph. Solutions must be preceded by the solver's name, school affiliation and year level. The deadline for submission is 12:00 PM November 24, 2012.