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Beautiful theorems

A proof of Pythagoras’ Theorem. The two large squares have the same area and contain the same four identical right-angled triangles. Thus, the area c2 of the white space in the first large square is equal to the total area a2 + b2 of that in the second. Now a, b, and c are the side lengths of each triangle, with c being the length of the hypotenuse.Photo: https://en.wikipedia.org/wiki/Pythagorean_theorem

A proof of Pythagoras’ Theorem. The two large squares have the same area and contain the same four identical right-angled triangles. Thus, the area c2 of the white space in the first large square is equal to the total area a2 + b2 of that in the second. Now a, b, and c are the side lengths of each triangle, with c being the length of the hypotenuse.Photo: https://en.wikipedia.org/wiki/Pythagorean_theorem

Bertrand Russell said, “Mathematics, rightly viewed, possesses not only truth, but supreme beauty.” The remarkable mathematician Paul Erdos said, “Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.”

Beauty is in the eye of the beholder; however, some things are widely regarded as beautiful or consistently rank highly in a particular category. In a paper published in The Mathematical Intelligencer in 1988, David Wells asked readers to evaluate 24 famous theorems for beauty. He published the results in the same journal in 1990. The following theorems are among those ranked and widely regarded as striking and appealing.

Ranked ninth is the Four Colour Theorem, stating that for any map of regions, it is possible to use at most four colours for giving each region a colour that is not given to a neighbouring region (so that regions can be distinguished). This was conjectured by Francis Guthrie in 1852 and proved by Kenneth Appel and Wolfgang Haken in 1976. It was the first major theorem proved using a computer.

The Pythagoreans (c. 6th century BC) discovered that there is no rational number (a number of the form a/b, where a and b are whole numbers) that gives 2 when multiplied by itself. This result ranked seventh.

The Basel problem, posed by Pietro Mengoli in 1644, asked for the precise value of the infinite sum 1/(1 x 1) + 1/(2 x 2) + 1/(3 x 3) + .... In 1734, Leonhard Euler proved that it is (p x p)/6, where p is pi (approximately 3.14159), the ratio of a circle’s circumference to its diameter. This result ranked fifth.

In third place, we find Euclid’s Theorem, stating that there is an infinite number of prime numbers (whole numbers that are divisible only by 1 and themselves). This was published in Euclid’s Elements (c. 300BC), a monumental, classical mathematical treatise.

Ranked second is Euler’s polyhedral formula, established in 1752 again by Euler. It states that for any polyhedron (a solid with plane faces), n + f = m + 2, where n, m, and f are the number of corners, edges, and faces, respectively.

And the top spot? Unsurprisingly, this is occupied by Euler’s identity eip + 1 = 0, where p is pi as above. Again, this was proved by Euler (1748). Rearranging the equation, we get eip = -1, featuring the additive inverse -1 of 1. The equation links five fundamental numbers: 0 (representing nothingness), 1 (the basis of all numbers), p, e (a number that is approximately equal to 2.71828 and arises naturally in mathematics, physics, and finance), and the imaginary number i (a non-real number) given by the square root of -1. Indeed, an outstanding theorem!

Did you know!

• Zero baffled the ancient Greeks. They asked, “How can nothing be something?”. This generated much philosophical debate about zero and the vacuum.

• The set of whole numbers, the set of rational numbers (fractions), and the set of real numbers are infinite; the first two have the same degree of infinity, but the third has a higher one. Unlike the whole numbers and the rationals, the reals cannot be ‘listed’. This was proved by Georg Cantor in 1874. Cantor also established an infinity of degrees of infinity.

• Suppose that a number n of lines are drawn on a plane. If no two are parallel and no three intersect at the same point, then the number of regions created is 1 + n x (n+1)/2 (for example, 10 lines create 1 + 10 x 11/2 = 56 regions); otherwise, less regions are created.

• The number pi (mentioned in the article above) is equal to 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + .... This is called Leibniz’s formula. It was proved by Gottfried Leibniz in 1673.

For more trivia see: www.um.edu.mt/think

Sound bites

• From the international scene: The partition function. A partition of a whole number n is a sum of positive whole numbers that add up to n. Two sums that differ only in the order of their numbers are considered to be the same. For example, the partitions of 5 are 1+1+1+1+1, 1+1+1+2, 1+1+3, 1+2+2, 1+4, 2+3, and 5. The partition function p(n) is the number of partitions of n. For example, p(5) = 7. The partition function was first studied by L. Euler (mentioned in the article above). Building on the work of G. H. Hardy and S. Ramanujan (1918), H. Rademacher (1937) surprisingly expressed p(n) as an infinite sum of real numbers. In 2011, K. Ono and J. Bruinier established the first formula that expresses p(n) as a finite sum of real numbers. Their work is found in the following publication: Ken Ono and Jan Bruinier, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms, Advances in Mathematics 246 (2013), 198-219.

• From the local scene: Cross-intersecting families. Extremal set theory is the study of how small or how large a parameter of a system of sets can be. A family is a collection (set) of sets. We say that two sets intersect if they have at least one common member. Let n and r be whole numbers. Let X  be the set {1, 2, ..., n}. A <r-subset of X is a subset of X that has at most r  members. Suppose that we want to construct two families A and B of <r-subsets of X such that the number of sets in A multiplied by the number of sets in B is maximum under the condition that each set in A intersects each set in B. The author (of this page) recently proved that this is achieved by taking each of A and B to be the family of all <r-subsets of X that have 1 as a member. This was done in the following publication: Peter Borg, A cross-intersection theorem for subsets of a set, Bulletin of the London Mathematical Society 47 (2015), 248-256.

To find out some more interesting science news listen in on Radio Mocha  every Monday and Friday at 13.00 on Radju Malta 2.

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