The Little Book of Mathematical Principles, Theories & Things. Robert Solomon

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The Little Book of Mathematical Principles, Theories & Things - Robert Solomon

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Theorem

      For a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.

      _______________

      Pythagoras is credited with the proof of this most famous theorem in mathematics.

      There are several hundred proofs of the theorem. The visual one below is just one example: Take a right-angled triangle with sides a, b, and c, where c is the hypotenuse, the longest side. Make four copies of this triangle. Draw a square of side a + b. The four triangles are arranged inside the square in two ways. In both cases, look at the region left uncovered by the triangles.

illustration

      In the upper diagram, the triangles are put in the four corners. The region left uncovered is a square of side c, which has area c2.

      In the lower diagram, the triangles form two rectangles, at the top left and bottom right. The uncovered region consists of two squares, one of side a, the other of side b. The area is a2 + b2.

      The region left uncovered must be the same in both diagrams. Hence c2 = a2+ b2.

      The theorem (though probably not its proof) may have been known long before Pythagoras. There are Babylonian clay tablets dating from about 2000 BC, which seem to provide numerical instances of the theorem.

      6th century BC Greece

      Irrational Numbers

      An irrational number cannot be expressed as the ratio of two whole numbers. Many numbers, such as √2, the square root of √ are irrational.

      _______________

      The Pythagoreans thought that everything could be explained in terms of whole numbers and their ratios – fractions, in other words. It was a great shock when it was shown that this is not true.

illustration

      Take a right-angled triangle in which the two shorter sides each have a length of one unit. According to Pythagoras’s theorem, the length of the hypotenuse h is given by h2 = 12 + 12. So h2 is 2, and hence h itself is √2, the square root of 2. This number is not the ratio of two whole numbers, and hence is an irrational number.

      The proof is the earliest example of a “proof by contradiction.” It assumes that √2 is a rational number, i.e. that √2 = a/b, where a and b are whole numbers, and derives a contradiction.

      This proof is one of the most important in the history of ideas. It destroyed the notion that everything could be described in terms of whole numbers. The actual person who made the discovery remains unnamed but his fellow Pythagoreans were so appalled by his impudence that they drowned him in the Aegean Sea.

      See: The Pythagoreans, page 17

      6th century BC to Present Global

      Perfect Numbers

      A number is perfect if it equals the sum of its proper divisors.

      _______________

      The search continues for perfect numbers, especially an odd perfect number.

      Numerology is the magical side of mathematics and some traces – such as perfect numbers – remain in modern mathematics. Perfect numbers were thought to be mystically superior to all others and this can be seen by the following quotation from St Augustine’s City of God (420 AD):

       Six is a perfect number, not because God created the world in six days, rather the other way round. God created the world in six days because six is perfect…

      A perfect number is equal to the sum of its proper divisors. The first two perfect numbers are 6 and 28.

      The divisors of 6 are 1, 2, and 3.

      6 = 1 + 2 + 3

      The divisors of 28 are 1, 2, 4, 7, and 14.

      28 = 1 + 2 + 4 + 7 + 14

      The next perfect numbers are 496 and 8128, the only ones known before the 13th century. The next three were found (along with three incorrect numbers) by Arab mathematician Ibn Fallus.

      Finding even perfect numbers is comparatively easy. There is a formula for them, which essentially appears in Euclid’s Elements. The formula is 2n–1(2n – 1), provided that the term inside the brackets is a prime number.

      All the perfect numbers that have so far been discovered are even; an odd perfect number, if it exists, remains to be found. This is the oldest unsolved problem in mathematics.

      Certainly there are no odd perfect numbers up to 10300 (1 followed by 300 zeros). They may not exist, but if one is ever found, mathematicians will already know a lot about it: that it has at least nine prime factors, for example.

      6th century BC Greece

      Regular Polygons

      A regular polygon has equal angles and equal sides.

      _______________

      Examples of regular polygons are the equilateral triangle (all sides equal, all angles equal to 60°) and the square (all sides equal, all angles equal to 90°). Then comes a pentagon, then a hexagon, and so on.

      How do you draw these shapes? Greek mathematicians were very particular about exactness in geometry, and required exact constructions. They would not allow a protractor to measure and draw angles, because one cannot do so exactly. They did not allow a ruler to measure and set out lengths, as one cannot be sure one has the exact length. These constructions had to be made with two instruments only – a straight edge and compasses.

      Constructions of an equilateral triangle and a square are part of school mathematics. The construction of a triangle is shown.

      The line AB is drawn, then arcs of the same length as AB are drawn to intersect at C. Notice that a straight edge has been used to draw the lines, and compasses to draw the arcs. We do not need to use a protractor to measure an angle of 60°.

      With a lot more work, it is possible to construct a regular pentagon. Hexagons and octagons are straightforward. Heptagons (7 sides) and nonagons (9 sides) had to wait!

illustration

       The diagram shows the construction of an equilateral triangle.

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