As promised I will still write a couple of more posts about the interesting things I have discovered during the event: History of Mathematics in December. With nothing more to add, here are some more history facts:

16th December:

The Egyptians used a written numeration that was changed into hieroglyphic writing, which enabled them to note whole numbers to 1,000,000 . It had a decimal base and allowed for the additive principle. In this notation there was a special sign for every power of ten. For I, a vertical line; for 10, a sign with the shape of an upside down U; for 100, a spiral rope; for 1000, a lotus blossom; for 10,000 , a raised finger, slightly bent; for 100,000 , a tadpole; and for 1,000,000, a kneeling genie with upraised arms.

Other civilizations did not have a sign for it: for example, Roman numerals stop at 1000 ( M ). But they could write a million by using different operations. For example, the Greeks used the symbol M for 10 000 and wrote multiples of 10 000 by putting symbols above M, so they could write numbers bigger than 1 000 000.

Another interesting system is the Babylonian one. The Babylonian number system began with tally marks just as most of the ancient math systems did. The Babylonians developed a form of writing based on cuneiform. They wrote these symbols on wet clay tablets which were baked in the hot sun. Many thousands of these tablets are still around today. The Babylonians used a stylist to imprint the symbols on the clay since curved lines could not be drawn. The Babylonians had a very advanced number system even for today’s standards. It was a base 60 system (sexigesimal) rather than a base ten (decimal). Moreover, they used operations to represent bigger numbers. For example, 5 220 062 would be used as a string of signs that represent (24*603) + (10*602) + (1*60) + 2.

17th December:

Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry.

The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss’s work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Évariste Galois (one of my favorite mathematicians) coined the term “group” and established a connection, now known as Galois theory, between the nascent theory of groups and field theory. Also, he was the first to employ groups to determine the solvability of polynomial equations. Arthur Cayley and Augustin Louis Cauchy pushed these investigations further by creating the theory of permutation groups. The Rubik’s Cube is used as an illustration of permutation groups.

In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. In an attempt to come to grips with possible geometries (such as euclidean, hyperbolic or projective geometry) using group theory, Felix Klein initiated the Erlangen programme. Sophus Lie, in 1884, started using groups (now called Lie groups) attached to analytic problems.

The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of abstract algebra in the early 20th century, representation theory, and many more influential spin-off domains. The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups.

18th December: History of Quadratic Equation part 1

The Rhind Papyrus – dating from around 1650 BC, but probably based on a document 200 years older – contains a problem: “A quantity and its 1/7 part become 19. What is the quantity?” The problem is solved in the Egyptian manner of regula falsi; that is, one assumes a – probably wrong – solution. This is known as the method of contradiction now. Today we would solve this using algebra.

The Ahmes papyrus and other ancient Egyptians scrolls are mainly concerned with problems leading to first-degree equations, but also describe some second-degree equations relating to land surveying. Babylonian clay tablets from the time of the Hammurabi dynasty (about 1800 – 1600 BC) deal with quadratic equations and their solutions by the method of “completing the square”, and also describe numerical methods of solving all quadratic equations and some simpler forms of cubic equations.

19th December: History of Quadratic Equation part 2

Much of the knowledge built up by the old civilizations of Egypt and Babylonia was passed down to the Greeks, who, in turn, gave mathematics scientific form.

Between about 540 and 250 BC, the ancient Greeks, represented by Pythagoras, his followers the Pythagorean, and Euclid, gave strict geometric proofs to algebraic problems, using lines and ares for numbers and products. ( in the image you can see how they understood quadratic equations)

The Greeks had considerable difficulty in solving cubic equations since their practice of treating algebraic problems as problems of geometry led to complicated three-dimensional constructions.

20th December: Quadratic equation part 3

In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit (although still not completely general) solution of the quadratic equation as follows: “To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value.” This is equivalent to the formula we have today to solve the equation.

The Bakhshali Manuscript written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as quadratic indeterminate equations (originally of type ax/c = y). Muhammad ibn Musa al-Khwarizmi (Persia, 9th century), inspired by Brahmagupta, developed a set of formulas that worked for positive solutions. Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while providing geometric proofs in the process. He also described the method of completing the square and recognized that the discriminant must be positive, which was proven by his contemporary ‘Abd al-Hamīd ibn Turk (Central Asia, 9th century) who gave geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution. While al-Khwarizmi himself did not accept negative solutions, later Islamic mathematicians that succeeded him accepted negative solutions as well as irrational numbers as solutions. Abū Kāmil Shujā ibn Aslam (Egypt, 10th century) in particular was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation. The 9th century Indian mathematician Sridhara wrote down rules for solving quadratic equations.

21st December: Quadratic Equation part 4 (final part)

The Jewish mathematician Abraham bar Hiyya Ha-Nasi (12th century, Spain) authored the first European book to include the full solution to the general quadratic equation. His solution was largely based on Al-Khwarizmi’s work. The writing of the Chinese mathematician Yang Hui (1238–1298 AD) is the first known one in which quadratic equations with negative coefficients of ‘x’ appear, although he attributes this to the earlier Liu Yi. By 1545 Gerolamo Cardano compiled the works related to the quadratic equations. The quadratic formula covering all cases was first obtained by Simon Stevin in 1594. In 1637 René Descartes published La Géométrie containing the quadratic formula in the form we know today. The first appearance of the general solution in the modern mathematical literature appeared in an 1896 paper by Henry Heaton.

Source: Mathematics: From the Birth of Numbers by Jan Gullberg

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Don’t forget that maths is everywhere! Enjoy! ~LThMath~