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history of math t. 1
up until the end of the greeks
| Question | Answer |
|---|---|
| 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
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