land measurement chart

number theory

Number theory

Although Euclid handed down a precedent for number theory in Books VII–IX of the Elements, later writers made no further effort to extend the field of theoretical arithmetic in his demonstrative manner. Beginning with Nicomachus of Gerasa (flourished c. 100 CE), several writers produced collections expounding a much simpler form of number theory. A favourite result is the representation of arithmetic progressions in the form of “polygonal numbers.” For instance, if the numbers 1, 2, 3, 4,…are added successively, the “triangular” numbers 1, 3, 6, 10,…are obtained; similarly, the odd numbers 1, 3, 5, 7,…sum to the “square” numbers 1, 4, 9, 16,…, while the sequence 1, 4, 7, 10,…, with a constant difference of 3, sums to the “pentagonal” numbers 1, 5, 12, 22,…. In general, these results can be expressed in the form of geometric shapes formed by lining up dots in the appropriate two-dimensional configurations (see figure). In the ancient arithmetics such results are invariably presented as particular cases, without any general notational method or general proof. The writers in this tradition are called neo-Pythagoreans, since they viewed themselves as continuing the Pythagorean school of the 5th century BCE, and, in the spirit of ancient Pythagoreanism, they tied their numerical interests to a philosophical theory that was an amalgam of Platonic metaphysical and theological doctrines. With its exponent Iamblichus of Chalcis (4th century CE), neo-Pythagoreans became a prominent part of the revival of pagan religion in opposition to Christianity in late antiquity.

Polygonal numbersThe ancient Greeks generally thought of numbers in concrete terms, particularly as measurements and geometric dimensions. Thus, they often arranged pebbles in various patterns to discern arithmetical, as well as mystical, relationships between numbers. A few such patterns are indicated in the figure

An interesting concept of this school of thought, which Iamblichus attributes to Pythagoras himself, is that of “amicable numbers”: two numbers are amicable if each is equal to the sum of the proper divisors of the other (for example, 220 and 284). Attributing virtues such as friendship and justice to numbers was characteristic of the Pythagoreans at all times.

Of much greater mathematical significance is the arithmetic work of Diophantus of Alexandria (c. 3rd century CE). His writing, the Arithmetica, originally in 13 books (six survive in Greek, another four in medieval Arabic translation), sets out hundreds of arithmetic problems with their solutions. For example, Book II, problem 8, seeks to express a given square number as the sum of two square numbers (here and throughout, the “numbers” are rational). Like those of the neo-Pythagoreans, his treatments are always of particular cases rather than general solutions; thus, in this problem the given number is taken to be 16, and the solutions worked out are 256/25 and 144/25. In this example, as is often the case, the solutions are not unique; indeed, in the very next problem Diophantus shows how a number given as the sum of two squares (e.g., 13 = 4 + 9) can be expressed differently as the sum of two other squares (for example, 13 = 324/25 + 1/25).

To find his solutions, Diophantus adopted an arithmetic form of the method of analysis. He first reformulated the problem in terms of one of the unknowns, and he then manipulated it as if it were known until an explicit value for the unknown emerged. He even adopted an abbreviated notational scheme to facilitate such operations, where, for example, the unknown is symbolized by a figure somewhat resembling the Roman letter S. (This is a standard abbreviation for the word numberin ancient Greek manuscripts.) Thus, in the first problem discussed above, if S is one of the unknown solutions, then 16 − S2 is a square; supposing the other unknown to be 2S − 4 (where the 2 is arbitrary but the 4 chosen because it is the square root of the given number 16), Diophantus found from summing the two unknowns ([2S − 4]2 and S2) that 4S2 − 16S + 16 + S2 = 16, or 5S2 = 16S; that is, S = 16/5. So one solution is S2 = 256/25, while the other solution is 16 − S2, or 144/25.

greek trignometry and mensuration

Greek trigonometry and mensuration

After the 3rd century BCE, mathematical research shifted increasingly away from the pure forms of constructive geometry toward areas related to the applied disciplines, in particular to astronomy. The necessary theorems on the geometry of the sphere (called spherics) were compiled into textbooks, such as the one by Theodosius (3rd or 2nd century BCE) that consolidated the earlier work by Euclid and the work of Autolycus of Pitane (flourished c. 300 BCE) on spherical astronomy. More significant, in the 2nd century BCE the Greeks first came into contact with the fully developed Mesopotamian astronomical systems and took from them many of their observations and parameters (for example, values for the average periods of astronomical phenomena). While retaining their own commitment to geometric models rather than adopting the arithmetic schemes of the Mesopotamians, the Greeks nevertheless followed the Mesopotamians’ lead in seeking a predictive astronomy based on a combination of mathematical theory and observational parameters. They thus made it their goal not merely to describe but to calculate the angular positions of the planets on the basis of the numerical and geometric content of the theory. This major restructuring of Greek astronomy, in both its theoretical and practical respects, was primarily due to Hipparchus (2nd century BCE), whose work was consolidated and further advanced by Ptolemy.

To facilitate their astronomical researches, the Greeks developed techniques for the numerical measurement of angles, a precursor of trigonometry, and produced tables suitable for practical computation. Early efforts to measure the numerical ratios in triangles were made by Archimedes and Aristarchus. Their results were soon extended, and comprehensive treatises on the measurement of chords (in effect, a construction of a table of values equivalent to the trigonometric sine) were produced by Hipparchus and by Menelaus of Alexandria (1st century CE). These works are now lost, but the essential theorems and tables are preserved in Ptolemy’s Almagest (Book I, chapter 10). For computing with angles, the Greeks adopted the Mesopotamian sexagesimal method in arithmetic, whence it survives in the standard units for angles and time employed to this day.

Srinivasa Ramanujan findings:

Srinivasa Ramanujan was one of the world’s greatest mathematicians. His life story, with its humble and sometimes difficult beginnings, is as interesting in its own right as his astonishing work was.

1. The book that started it all
   Srinivasa Ramanujan had his interest in mathematics unlocked by a book. It wasn’t by a famous mathematician, and it wasn’t full of the most up-to-date work, either. The book was A Synopsis of Elementary Results in Pure and Applied Mathematics (1880, revised in 1886), by George Shoobridge Carr. The book consists solely of thousands of theorems, many presented without proofs, and those with proofs only have the briefest. Ramanujan encountered the book in 1903 when he was 15 years old. That the book was not an orderly procession of theorems all tied up with tidy proofs encouraged Ramanujan to jump in and make connections on his own. However, since the proofs included were often just one-liners, Ramanujan had a false impression of the rigor required in mathematics.
2. Early failures
   Despite being a prodigy in mathematics, Ramanujan did not have an auspicious start to his career. He obtained a scholarship to college in 1904, but he quickly lost it by failing in nonmathematical subjects. Another try at college in Madras (now Chennai) also ended poorly when he failed his First Arts exam. It was around this time that he began his famous notebooks. He drifted through poverty until in 1910 when he got an interview with R. Ramachandra Rao, the secretary of the Indian Mathematical Society. Rao was at first doubtful about Ramanujan but eventually recognized his ability and supported him financially.
3. Go west, young man
   Ramanujan rose in prominence among Indian mathematicians, but his colleagues felt that he needed to go to the West to come into contact with the forefront of mathematical research. Ramanujan started writing letters of introduction to professors at the University of Cambridge. His first two letters went unanswered, but his third—of January 16, 1913, to G.H. Hardy—hit its target. Ramanujan included nine pages of mathematics. Some of these results Hardy already knew; others were completely astonishing to him. A correspondence began between the two that culminated in Ramanujan coming to study under Hardy in 1914.
4. Get pi fast
   In his notebooks, Ramanujan wrote down 17 ways to represent 1/pi as an infinite series. Series representations have been known for centuries. For example, the Gregory-Leibniz series, discovered in the 17th century is pi/4 = 1 - ⅓ + ⅕ -1/7 + … However, this series converges extremely slowly; it takes more than 600 terms to settle down at 3.14, let alone the rest of the number. Ramanujan came up with something much more elaborate that got to 1/pi faster: 1/pi = (sqrt(8)/9801) * (1103 + 659832/24591257856 + …). This series gets you to 3.141592 after the first term and adds 8 correct digits per term thereafter. This series was used in 1985 to calculate pi to more than 17 million digits even though it hadn’t yet been proven.
5. Taxicab numbers
   In a famous anecdote, Hardy took a cab to visit Ramanujan. When he got there, he told Ramanujan that the cab’s number, 1729, was “rather a dull one.” Ramanujan said, “No, it is a very interesting number. It is the smallest number expressible as a sum of two cubes in two different ways. That is, 1729 = 1^3 + 12^3 = 9^3 + 10^3. This number is now called the Hardy-Ramanujan number, and the smallest numbers that can be expressed as the sum of two cubes in n different ways have been dubbed taxicab numbers. The next number in the sequence, the smallest number that can be expressed as the sum of two cubes in three different ways, is 87,539,319.
6. 100/100
   Hardy came up with a scale of mathematical ability that went from 0 to 100. He put himself at 25. David Hilbert, the great German mathematician, was at 80. Ramanujan was 100. When he died in 1920 at the age of 32, Ramanujan left behind three notebooks and a sheaf of papers (the “lost notebook”). These notebooks contained thousands of results that are still inspiring mathematical work decades later.

mathematical innovations

Top 10  mathematical innovations:

Introduction:

Of all the mathematical innovations since ancient times, only some are worthy of multicentenary celebrations. Certainly logarithms, celebrating their 400th anniversary this year, are among them. Ranking where logarithms rate among the rest is subjective, of course, but I’d put them 10th (they’d be higher if everybody still used slide rules, though). Here are the rest of my Top 10 mathematical innovations, which you might as well read here because David Letterman isn’t going to get to them before he retires:

10. Logarithms

A great aid to anybody who multiplied or messed with powers and roots, logarithms made slide rules possible and clarified all sorts of mathematical relationships in various fields. Napier and Bürgi both had the basic idea in the late 16th century, but both spent a couple of decades calculating log tables before publishing them. Napier’s came first, in 1614. Briggs made them popular, though, by recasting Napier’s version into something closer to the modern base-10 form.

9. Matrix algebra

An ancient Chinese math text included matrix-like calculations, but their modern form was established in the mid-19th century by Cayley. (Several others, including Jacques Binet, had explored aspects of matrix multiplication before then.) Besides their many other applications, matrices became extremely useful for quantum mechanics. In fact, in 1925 Werner Heisenberg reinvented a system identical to matrix multiplication to do quantum calculations without even knowing that matrix algebra already existed.

8. Complex numbers

Before Cardano, square roots of negative numbers had shown up in various equations, but nobody took them very seriously, regarding them as meaningless. Cardano played around with them, but it was Bombelli in the mid-16th century who worked out the details of calculating with complex numbers, which combine ordinary numbers with roots of negative numbers. A century later John Wallis made the first serious case that the square roots of negative numbers were actually physically meaningful.

7. Non-Euclidean geometry

Gauss, in the early 19th century, was probably the first to figure out an alternative to Euclid’s traditional geometry, but Gauss was a perfectionist, and perfection is the enemy of publication. So Lobachevsky and Bolyai get the credit for originating one non-Euclidean approach to space, while Riemann, much later, produced the non-Euclidean geometry that was most helpful for Einstein in articulating general relativity. The best thing about non-Euclidean geometry was that it demolished the dumb idea that some knowledge is known to be true a priori, without any need to check it out by real-world observations and experiments. Immanuel Kant thought Euclidean space was the exemplar of a priori knowledge. But not only is it not a priori, it’s not even right.

6. Binary logic

Boole was interested in developing a mathematical representation of the “laws of thought,” which led to using symbols (such as x) to stand for concepts (such as Irish mathematicians). He hit a snag when he realized that his system required x times x to be equal to x. That requirement pretty much rules out most of mathematics, but Boole noticed that x squared does equal x for two numbers: 0 and 1. In 1854 he wrote a whole book based on doing logic with 0s and 1s — a book that was well-known to the founders of modern computer languages.

5. Decimal fractions

Stevin introduced the idea of decimal fractions to a European audience in a pamphlet published in 1585, promising to teach “how all Computations that are met in Business may be performed by Integers alone without the aid of Fractions.” He thought his decimal fraction approach would be of value not only to merchants but also to astrologers, surveyors and measurers of tapestry. But long before Stevin, the basic idea of decimals had been applied in limited contexts. In the mid-10th century, al-Uqlidisi, in Damascus, wrote a treatise on Arabic (Hindu) numerals in which he dealt with decimal fractions, although historians differ on whether he understood them thoroughly or not.

4. Zero and 3. Negative numbers

Brahmagupta, a seventh-century Hindu astronomer, was not the first to discuss negative numbers, but he was the first to make sense of them. It’s not a coincidence that he also had to figure out the concept of zero to make negative numbers make sense. Zero was not just nothingness, but a meaningful number, the number you get by subtracting a number from itself. “Zero was not just a placeholder,” writes Joseph Mazur in his new book Enlightening Symbols. “For what may have been the first time ever, there was a number to represent nothing.”

2. Calculus

You know the story — Newton gets all the credit, even though Leibniz invented calculus at about the same time, and with more convenient notation (still used today). In any event, calculus made all sorts of science possible that couldn’t have happened without its calculational powers. Today everything from architecture and astronomy to neuroscience and thermodynamics depends on calculus.

1. Arabic numerals

Did you ever wonder why the Romans didn’t do much creative quantitative science? Try doing a complicated calculation with their numerals. Great advances in Western European science followed the introduction of Arabic numerals by the Italian mathematician Fibonacci in the early 13th century. He learned them from conducting business in Africa and the Middle East. Of course, they should really be called Hindu numerals because the Arabs got them from the Hindus. In any case, mathematics would be stuck in the dark ages without such versatile numerals. And nobody would want to click on a Top X list. (Wait — maybe they would. But you won’t see any list like that on this blog.)