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# history of math t. 1

### up until the end of the greeks

When was Babylonian mathematics? 1800-1600 BC
Babylonian number system base 60 or sexagesimal
How the Babylonians treated division as multiplication by a reciprocal a/b=a(1/b)
Where does our knowledge of Egyptian mathematics come from? Rhind Papyrus and Moscow Papyrus
When Rhind Papyrus was written? 1650 BC by scribe Ahmes copying from earlier version from 1800 BC
Origin of Geometry in Egypt rooted in measuring land gained or lost in the yearly flooding of the Nile (geo=earth, metre=measure)
Rhind papyrus shows Egyptian techniques of multiplying and dividing, using addition and doubling, all problems are numerical and many are practical
Unit fractions how the Egyptians thought of fractions where the numerator was greater than 1, but the denominators could not be the same
Rhind papyrus #28 earliest “think of a number” problem
Rhind papyrus #79 old riddle if I have 7 horses that each have 7 cats, etc.
Correct Egyptian geometry area of isosceles triangle and trapezoid formulas
Incorrect Egyptian geometry area formulas for a quadrilateral and circle
Rhind papyrus #14 the correct formula for the volume of a truncated square pyramid, Herodotus claimed that the Great Pyramid led to the golden ratio
Plimpton 322 clay tablet dating to 1900-1600 BC from Babylonians, proof they knew the Pythagorean theorem before Greeks
Cairo Mathematical Papyrus 300 BC in Egypt also has problems with the Pythagorean Theorem
Beginning of Greek mathematics about 700-600 BC
Thales (622-547 BC) first mathematician, first of 7 Sages of Greece, Father of Geometry, coined “Know thyself”, first to use logical proofs, 6 geometric propositions
Thales stories claims he measured height of the Great Pyramid by measuring shadows and using similar triangles, measured distance from a ship to show, some say he was Pythagoras’s teacher
Pythagoras (585-501 BC) founded a school for political, philosophical, and religious teaching in southern Italy, married student Theano
Pythagoras’s 4 mathemata arithmetica(number theory), harmonia (music), geometria(geometry), astrologia (astronomy)
Pythagoraean secret society communal lifestyle, symbols- 5 pointed star and 10 dots in a triangle, mathematics/philosophy way of life/religion, “everything is number”,
Pythagoras’s mathematics and philosophy that which we learned and love of wisdom
Nichomachus about 100 AD, wrote “Introductio Arithmeticae” that tells what we know of Pythagorean number theory
Pythagorean numbers had no digits, but arranged dots into shapes to represent numbers ex. Triangular numbers, square numbers
Gauss (1777-1855) discovered formula for summing numbers like n(n+1)/2= 1+2+3+…+n
Zeno (450 BC) member of Eleatic school in southern Italy, believed time and space are continua, undivided wholes, had 4 paradoxes related to this
Pythagorean theorem proofs Chinese(600 BC), same as given by Bhaskara (1114 AD) in India, Euclid’s is simpler (323-285 BC)
Formula for Pythagorean triples Fibonacci (1175-1250) x=2mn, y=m^2-n^2, z= m^2+n^2, used by both Euclid and Diophantus
Irrational numbers alien to Pythagoreans and were forbidden to be discussed, beginning of proof by contradiction, not proved until 1872 by Dedekind
Plato’s mathematics could only use a straight edge and compass,
3 problems that couldn’t be solved with straight edge and compass quadrature of a circle, duplicating the cube, trisecting a general angle
Quadratrix curve invented by Hippias(460 BC) to trisect an angle, used by Dinostratus to square a circle, first curve drawn by plotting points
Sophist tutored for a fee, not just in academics but in powers of persuasion ex. Hippias and Hippocrates
The Academy opened by Socrates’s student Plato(429-348 BC), center of learning in Greece for 900 yrs, valued logical training of mathematics
Museum 300 BC founded by Ptolemy I in Alexandria, Egypt, primarily a research institute
Hellenistic age Greek-like age, 200 years after Museum founded, Euclid was here
Euclid (328-285 BC) author of the “Elements of Geometry”, not all original, but gave a logical arrangement of theorems and proofs, based on definitions and assumptions (postulates and axioms)
Number of axioms and propositions 10 axioms and 465 propositions
Proposition 4 proves the side-angle-side theorem of congruence
Proposition 5 an isosceles triangle has congruent base angles
Proposition 16 the exterior angle theorem, assumes that a line has infinite proof
Proposition 29 first proof using parallel postulate, a line crossing 2 parallel lines, the alternate interior angles are equal, the corresponding angles equal, & interior angles on the same side sum to 2 right angles
Proposition 31 a perpendicular from the vertex of a right angle divides a triangle into 2 similar triangles
Elements studies the arithmetic of natural numbers
Primes Euclid proved theorem that there are infinitely many primes
Euclidean algorithm gcd(a,b)=ax+by denotes the greatest common divisor
Fundamental theorem of arithmetic every positive integer can be written uniquely as a product of primes
Eratosthenes (276-194 BC) Greek scholar studying in Alexandria, chief librarian at Museum, liked Geography and math, calculated Earth’s circumference
Mesolabium made by Eratosthenes, used to duplicate the cube, used similar triangles to find the mean proportional of duplicating the cube
Sieve of Eratosthenes means of finding all primes less than integer n, list all numbers less than n and strike out muliples of primes <= sqrt(n)
Claudius Ptolemy (100-170 AD) wrote Almagest where he described an earth-centered system of planetary motion, divided a spherical globe into 360 degrees with meridians and parallels
Apollonius described motion in circles, not ellipses, and epicycles explained deviations
Archimedes (287-212 BC) lived in Syracuse, Sicily, engineer, invented Archimedian screw to pump water, used levers and pulleys to protect country, killed by Roman soldier
Archimedes’s discoveries volume of a sphere is 2/3 volume of the surrounding cylinder
Archimedes’s pi 22/7 or 3.1429, found using method of exhaustion
Method of exhaustion areas of inscribed and circumscribed polygons of an increasing number of sides are calculated
-pi symbol was introduced in 1706
Lin Hui found the value for pi to be 3.14159 in 3rd century AD
Arybhata found estimate of 3.1416 in 5th century AD in India
Ludolph Van Ceulen (1540-1610) found pi to 35 places
Spiral of Archimedes in polar form it is r=aӨ and successfully calculated its area from 0 to 2π using exhaustion, used spiral to trisect an angle and square the circle
Created by: lfalkens