Further progress was made possible by the development of trigonometry. In the 14th century the Indian mathematician, Madhava, used trigonometry to discover the following series (which continues forever:
π/4 = 1 – 1/3 + 1/5 – 1/7 + …
This can be used to find π, but it is a very inefficient method. Using a variant of the series Madhava was able to calculate π to 11 decimal places.
Until the 20th century all the calculations were done by hand but with the invention of computers, much greater accuracy is possible. In 1949, the ENIAC calculated π to 2,037 decimal places, taking 70 hours to do so. Modern computers have calculated π to well over a million places.
The number π occurs throughout both pure and applied mathematics. Often these applications have nothing to do with the measurement of circles. For example, the equation of the normal or bell curve, which is central to statistics, is:
See: ENIAC, page 181; The Normal Distribution, pages 94–95
6th century BC Greece & Italy
The Pythagoreans
The Pythagorean slogan was: All Things Are Numbers.
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The Pythagoreans were a religious, mystical, and scientific sect mainly based in Southern Italy in the 6th century BC.
Their leader, Pythagoras himself, may or may not have existed. Many incredibly important discoveries are credited to the Pythagoreans, of which some will appear in this book.
The Pythagoreans are credited with discovering the following:
• That the Earth is a sphere.
• That the Earth is not the center of the universe.
• That musical harmony depends on the ratio of whole numbers.
No one knows what the Pythagoreans’ slogan, All Things Are Numbers, means.
Does it just mean that all things can be described in terms of numbers?
Or is it something stronger, that the solid world is an illusion and that the reality behind it consists of numbers?
More important, however, than any single discovery is the Pythagoreans’ contribution to mathematics, extending it from a practical subject concerned with areas of land or weights of corn to the study of abstract ideas.
6th century BC Greece
Pythagoras’s Theorem
For a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
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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.
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.
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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.
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.
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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