History of Mathematics
Sumerian/Babylonian Mathematics
Sumerian/Babylonian Mathematics
Sumerian/Babylonian Mathematics
Sumerian/Babylonian Mathematics
Can you see how their number system works?
Sumerian/Babylonian Mathematics
What about numbers greater than 59??
Sumerian/Babylonian Mathematics
Which number is shown?
Sumerian/Babylonian Mathematics
Some more mathematics:
Egyptian Mathematics
Egyptian Mathematics
The Rhind Papyrus, dating from around 1650 BCE, is a kind of instruction manual in arithmetic and geometry, and it gives us explicit demonstrations of how multiplication and division was carried out at that time. It also contains evidence of other mathematical knowledge, including unit fractions, composite and prime numbers, arithmetic, geometric and harmonic means, and how to solve first order linear equations as well as arithmetic and geometric series.
The Berlin Papyrus, which dates from around 1300 BCE, shows that ancient Egyptians could solve second-order algebraic (quadratic) equations.
Egyptian Mathematics
Egyptian fractions
Egyptian Mathematics
Area of a Circle
The Egyptian approximated the area of a circle using shapes they knew. They were even able to approximate pi to within 1% accuracy!
Egyptian Mathematics
The Pyramids required some math to build as well
Greek Mathematics
Greek Mathematics
Greek Mathematics
Greek Mathematics
The Paradox of Achilles and the Tortoise
A theoretical race between Achilles and a tortoise. Achilles gives the much slower tortoise a head start, but by the time Achilles reaches the tortoise's starting point, the tortoise has already moved ahead. By the time Achilles reaches that point, the tortoise has moved on again, etc, etc, so that in principle the swift Achilles can never catch up with the slow tortoise.
** eventually disproved.
Pythagoras
-often called the first mathematician
-established a school in Italy around 530BC. Mostly based on math but also mystical.
-Left no mathematical writings, his thoughts have come to us through the writings of his scholars, so we don’t actually know if his theorems were solved by him or his followers.
Pythagoras
Pythagoras
“All is Number”
Pythagoras
Pythagoras
Pythagoras is also credited with the discovery that the intervals between harmonious musical notes always have whole number ratios. For instance, playing half a length of a guitar string gives the same note as the open string, but an octave higher; a third of a length gives a different but harmonious note; etc. Non-whole number ratios, on the other hand, tend to give dissonant sounds
Plato
Plato
Plato the mathematician is perhaps best known for his identification of 5 regular symmetrical 3-dimensional shapes, which he maintained were the basis for the whole universe, and which have become known as the Platonic Solids: the tetrahedron (constructed of 4 regular triangles, and which for Plato represented fire), the octahedron (composed of 8 triangles, representing air), the icosahedron (composed of 20 triangles, and representing water), the cube (composed of 6 squares, and representing earth), and the dodecahedron (made up of 12 pentagons, which Plato obscurely described as “the god used for arranging the constellations on the whole heaven”).
Hellenistic Mathematics
Euclid
Euclid
Euclid set the model for mathematical arguments. From initial assumptions to logical deductions. He called these axioms (all math) and postulates (geometry). These were unproven truths which he used to prove other theorems.
Euclid
Euclid
Among many other mathematical gems, the thirteen volumes of the “Elements” contain formulas for calculating the:
-volumes of solids such as cones, pyramids and cylinders;
-proofs about geometric series, perfect numbers and primes;
-algorithms for finding the greatest common divisor and least common multiple of two numbers;
-a proof and generalization of Pythagoras’ Theorem, and proof that there are an infinite number of Pythagorean Triples;
-and a final definitive proof that there can be only five possible regular Platonic Solids.
-Proof there are an infinite number of prime numbers
Euclid
Fundamental Theorem of Arithmetic: Any integer greater than 1 is either prime or can be represented by a unique representation of prime numbers.
Euclid
Archimedes
Archimedes
Military Inventions helped Syracuse hold off a Roman siege for 3 years
Archimedes Claw
Archimedes Death Ray
catapult
Archimedes
Other Inventions:
An Archimedes' screw, also known by the name the Archimedean screw or screw pump, is a machine used for transferring water from a low-lying body of water into irrigation ditches. Water is pumped by turning a screw-shaped surface inside a pipe.
“Give me a place to stand on and I will move the Earth”
Archimedes
Mathematics
Archimedes
Archimedes
Death
Roman Mathematics
Hypatia
Hypatia
Hypatia
Chinese Mathematics
Chinese Mathematics
Magic Squares
Lo Shu Square- each row and column adds to 15 dates to 650 BC
Eventually came up with even bigger magic squares
Chinese Mathematics
Chinese Mathematics
Liu Hui
Indian Mathematics
ex)
Indian Mathematics
Used a place value system (like Chinese and us today) as early as 3rd century BC
Beginnings of 9 numerals we use today
Indian Mathematics
The Indians were also responsible for another hugely important development in mathematics. The earliest recorded usage of a circle character for the number zero is usually attributed to a 9th Century engraving in a temple in Gwalior in central India.
The use of a number 0 in calculations would revolutionize mathematics!
Indian Mathematics
Golden age of Indian Mathematics (5th to 12th centuries)
-many discoveries predated Western discoveries by several centuries (plagiarism by Western mathematicians?)
-made advances in trigonometry. Used to calculate distances between sun, moon and Earth
-Bhaskara II credited with explaining the previously misunderstood concept of dividing by 0
Brahmagupta
-7th century Indian Mathematician and Astronomer
-from the state of Rajasthan of northwest India
-Most of his works are composed in elliptic verse, a common practice in Indian mathematics at the time, and consequently have something of a poetic ring to them.
-the first to use zeros and negatives in computations. First to use the rule that two negatives multiplied together make a positive (although no proof given)
-his works brought to the newfound centre of learning in Baghdad and helped lead the expansion of Muslim mathematics
Islamic Mathematics
-The Islamic Empire established across Persia, the Middle East, Central Asia, North Africa, Iberia and parts of India from the 8th Century onwards
-made significant contributions towards mathematics.
-They were able to draw on and fuse together the mathematical developments of both Greece and India.
-House of Wisdom set up in Baghdad in 1810
Islamic Mathematics
Mastered the art of symmetry in the form of art to decorate their buildings
Muhammad Al-Khwarizmi
-one of the directors of the House of Wisdom in the early 9th century
-oversaw translation of Greek and Indian works into Arabic
-The word “algorithm” is derived from the Latinization of his name, and the word "algebra" is derived from the Latinization of "al-jabr", part of the title of his most famous book, in which he introduced the fundamental algebraic methods and techniques for solving equations.
Muhammad Al-Khwarizmi
-strongly advocated for the Hindu number system (1-9 and 0) and the Hindu-Arabic number system was soon adopted by the entire Islamic world and later on the European world (through Fibonacci)
-published in about 830 called “Al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala” (“The Compendious Book on Calculation by Completion and Balancing”. Al-Khwarizmi wanted to go from the specific problems considered by the Indians and Chinese to a more general way of analyzing problems, and in doing so he created an abstract mathematical language which is used across the world today. (not same notation we used today but with words and pictures)
Medieval European Mathematics
-5th to 15th Centuries
-While Chinese, Indian, and Islamic mathematics flourished, Europe was stuck in the dark ages where science, math, and almost all intellectual endeavors stopped
-all trade done using clumsy Roman Numeral system and using an abacus based on Roman and Greek models
-things starting to look up:
-12th Century beginning to trade with East
-15th Century printing press invented (some math texts translated)
Fibonacci
-The 13th Century Italian Leonardo of Pisa, better known by his nickname Fibonacci
-son of a customs official, travelled around North Africa with his father where he learned Islamic Mathematics
-started to share this knowledge upon his return to Europe, setting in motion a rejuvenation of European Mathematics
-In 1202, he wrote a hugely influential book called “Liber Abaci” ("Book of Calculation"), in which he promoted the use of the Hindu-Arabic numeral system, describing its many benefits for merchants and mathematicians alike over the clumsy system of Roman numerals then in use in Europe.
-First to use horizontal bar to separate numerator and denominator
Fibonacci
1,1,2,3,5,8,13
-discovered this sequence while researching a hypothetical problem for his book.
-evidence it was known in India already
-as time went on other people began to discover even more real life examples of this sequence
-Sequence named after him in 1877
Fibonacci
in the 1750’s Robert Simson discovered the Golden ratio from this sequence
16th Century Mathematics
Albrecht Durer
Supermagic square
16th Century Mathematics
-Common math symbols begin to appear and spread
-Friar Luca Paciola
-book of puzzles
-symbols for + and - first seen in a printed book
-exploring the golden ratio: “a message from God and a source of secret
knowledge about the inner beauty of things.”
-Simon Steven
-first to use decimal arithmetic
Niccolo Tartaglia
-first translation of Euclid’s ‘Elements’ in a modern European language
-investigation of cannonball paths
-revealed formula for solving all cubic equations at a math competition
-first understanding of square roots of negatives (imaginary numbers)
-Gerolamo Cardano published his solution to solving quartics without his knowledge
-Ferrari (protege of Cardano) challenged Tartaglia to public debate. Tartaglia didn’t show up.
-Tartaglia died penniless and unknown
-Ferrari got a prestigious teaching post and became rich
-Cardano wrote first book on probability...short on money whole life due to gambling problems
17th Century Mathematics
-At end of Renaissance there was an explosion of Math and Science. Often called the ‘Age of Reason’
-Scientists like Galileo Galilei, Tycho Brahe and Johannes Kepler were making equally revolutionary discoveries in the exploration of the Solar system (Copernicus had just shown that the Sun was the centre of the Solar System)
-Math was needed to help Scientists
-Invention of logarithm by John Napier helped make difficult calculations easy for Scientists as they dealt with large numbers
The logarithm of a number is the exponent when that number is expressed as a power of 10 (or any other base). It is effectively the inverse of exponentiation.
Ex) Log10 100 = 2 since 102=100
Rules for adding and subtracting logarithms make it easier to calculate with larger numbers
Rene Descartes
-”Father of Modern Philosophy”, but also a key figure in the Scientific Revolution and considered the first of the modern school of mathematics.
-mercenary soldier for Catholic and Protestant armies
-eventually concluded real path was true wisdom and science
-believed key to ambiguities of philosophy was to build it on indisputable facts of math
-moved from Catholic France to more liberal Netherlands to explore ideas
Rene Descartes
In 1637, he published his ground-breaking philosophical and mathematical treatise "Discours de la méthode" (the “Discourse on Method”), and one of its appendices in particular, "La Géométrie", is now considered a landmark in the history of mathematics
-introduced standards for algebraic notation (using a,b,c for known quantities and x,y,z for unknown quantities)
-proposed that each point in two dimensions can be described by a horizontal and vertical location (Cartesian Plane and Cartesian/Analytic Geometry)
-soon became easier to plot equations and see their shapes. Now had a link between geometry and algebra
-popularized the use of a superscript for exponents
Pierre de Fermat
-lawyer and small town “amateur” mathematician, effectively developed modern number theory from 1601-1655
-mathematical work mainly included in letters to friends, limited proofs of theorems.
-Fermat’s Little Theorem is often used in the testing of large prime numbers, and is the basis of the codes which protect our credit cards in Internet transactions today.
-Fermat's pièce de résistance, though, was his famous Last Theorem, a conjecture left unproven at his death, and which puzzled mathematicians for over 350 years
Blaise Pascal
-1623-1662
-Child prodigy (wrote a treatise called Pascal’s Theorem when he was 16 and built a machine to help add and subtract numbers)
-Scientist, philosopher, and mathematician (international unit for pressure measurement named after him)
-In philosophy, Pascal had a pragmatic approach to believing in God on the grounds that is it is a better “bet” than not to.
Blaise Pascal
-many different patterns in Pascal’s Triangle
-used Pascal's triangle to help solve probability problems (subject of mathematical theory of probability was born)
-Pascal, along with Fermat, brought previous probability knowledge together to solve unsolved problems
Problem of Points:
Blaise Pascal
Sir Isaac Newton
1643-1727
Physicist, mathematician, astronomer, natural philosopher, alchemist and theologian, Newton is considered by many to be one of the most influential men in human history.
His 1687 publication, the "Philosophiae Naturalis Principia Mathematica" (usually called simply the "Principia"), is considered to be among the most influential books in the history of science
Over two miraculous years, during the time of the Great Plague of 1665-6, the young Newton developed a new theory of light, discovered and quantified gravitation, and pioneered a revolutionary new approach to mathematics: infinitesimal calculus
Unlike the static geometry of the Greeks, calculus allowed mathematicians and engineers to make sense of the motion and dynamic change in the changing world around us, such as the orbits of planets, the motion of fluids
Sir Isaac Newton
Derivatives (fluxion)
Integral (fluents)
Sir Isaac Newton
-chose not to publish his findings on calculus for fear of being ridiculed for unconventional ideas (finally published later on in 1693)
-Leibniz published his own version of the theory first
-credit for discovery to Newton and for first publication to Leibniz.
-wrote a number of books on a literal interpretation of the Bible
-first scientist ever knighted
-mercury poisoning from experiments led to eccentricity and death
Gottfried Wilhelm Leibniz
1646-1716
-philosopher, logician, mathematician, politician
-developed early ideas of Calculus (along with Newton) although we still use his notation in Calculus today
-arranged linear equations in a matrix in order to solve them
Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz
Developed binary number system in order to convert verbal logic statements into mathematical ones (True and False, 1 and 0)
Binary is the basis of our computer systems today!
18 Century Mathematics
Bernoulli Brothers
-prosperous family of traders and thinkers from Basel Switzerland
-Jacob and Johan resisted their fathers wishes to go into the family spice trade or enter medicine to instead focus on mathematics
-Had a very competitive rivalry (Johann jealous of Jacob’s position as a professor at Basel University)
Bernoulli Brothers
-among the first to not only study calculus, but apply it to problems. Helped make calculus a cornerstone of math today.
Also discovered an approximate value for e. While exploring the compound interest on loans.
Leonhard Euler
-1707-1783
-student of Bernoulli Brothers
-did most work in Russia and Germany (due to dominance of Bernoulli family in Switzerland)
-his collected works comprises over 900 books across all topics in math(in 1775 comprised one mathematical paper every week!)
-had a photographic memory
-considered one of the greatest mathematicians of all time
Leonhard Euler
-helped standardized common math notation. Which made it easier for collaboration on problems
-The most remarkable equation in math. Euler’s Identity (combines arithmetic, calculus, trigonometry, and complex numbers)
Leonhard Euler
19th Century Math
Galois
-French mathematician
-lacklustre performance in school (twice failed entrance exams to the University)
-studied works of other mathematicians in his spare time
-hot-head and radical republican (twice arrested for political acts)
-died in a duel at age 20 (spent the night before outlining math ideas to a friend)
Galois
-proved there was no general formula for solving a degree 5 (quintic) equation
-Galois’ breakthrough in turn led to definitive proofs (or rather disproofs) later in the century of the so-called “Three Classical Problems” problems which had been first formulated by Plato and others back in ancient Greece: the doubling of the cube and the trisection of an angle (both were proved impossible in 1837), and the squaring of the circle (also proved impossible, in 1882).
Gauss
-Carl Friedrich Gauss
- “Prince of Mathematics”
-child prodigy, made his first groundbreaking mathematical discovery while still a teenager
-At the age of 3 he found an error on his Father’s payroll calculations and he was looking after his Father’s accounts by age 5.
-At the age of 7 he amazed his teachers by summing the numbers 1 to 100 almost instantly.
How did he do it!? 1+2+3+.....+98+99+100=????
Gauss
-First to see a pattern in the occurrence of prime numbers
-Most contributions were in the area of number theory
-popularized practice and standard notation for complex numbers
-proved the fundamental theorem of algebra
Riemann
-Bernhard Riemann
-another child prodigy
-extremely shy and timid
-devoutly religious, tried to mathematically prove the correctness of the Book of Genesis
-studied under Gauss
Riemann
-developed a non-Euclidean geometry called elliptic geometry
-vision of mathematics that was in more than just 2D and 3D space into multi-dimensions
-this later enabled the development of general relativity by Einsten
George Boole
George Boole (1815-1865) British Mathematician
First to see logic as a mathematical discipline
Boolean logic-form of algebra in which all values can reduced to be true or false
Helped establish modern symbolic logic and whose algebra of logic, now called Boolean algebra, is basic to the design of digital computer circuits.
Ideas largely ignored until almost 70 years later that electromechanical relay circuits could be used to solve Boolean algebra problems. The use of electrical switches to process logic is the basic concept that underlies all modern electronic digital computer
George Boole
True AND False
True AND True
True OR False
False OR False
NOT True
1+3 =5 OR 2+6=8
NOT (2+5=7 AND 3+6=9)
20th Century Mathematics
Hardy and Ramanujan
-Hardy was a British Mathematician, who is credited with reforming British mathematics
-child prodigy, used to amuse himself at Church factorizing hymn numbers
-Ramanujan was an Indian, self taught, mathematician, wrote to University of Cambridge, claiming some of his exploits...Hardy was the only one to recognize genius and brought him to the University
Hardy and Ramanujan
-Ramanujan conjectured or proved over 3,000 theorems, identities and equations, including properties of highly composite numbers, the partition function and its asymptotics and mock theta functions
-He also carried out major investigations in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory.
-Among his other achievements, Ramanujan identified several efficient and rapidly converging infinite series for the calculation of the value of π, some of which could compute 8 additional decimal places of π with each term in the series. These series (and variations on them) have become the basis for the fastest algorithms used by modern computers to compute π to ever increasing levels of accuracy (currently to about 5 trillion decimal places).
Alan Turing
-most famously known for his work on breaking the enigma code at Bletchley Park in WW2 (Imitation Game movie about him)
-developed the Turing Machine and Turing Test before computers were invented
-worked on some of the earliest computers
-arrested in 1952 and charged with homosexuality, committed suicide in 1954
Alan Turing