MATH 4311 Final Review
Archimedes
-used a series of triangles to prove that the area of a parabola was 4/3 the area of a triangle. -realized the series of sum of the areas of the triangles converged to 4/3
Chinese Counting Rods
2-3 symbols base 10 positional not additive
Susa Tablet
200 BCE written in cuneiform symbols discovered in the town of Shush, Iran Example of Babylonian knowledge of the Pythagorean Thm Find the radius of a circle in which an isosceles triangle is inscribed with sides 50, 50, 60
Euclidean Postulate 3 and Hyperbolic Postulate 3
A circle can be described with any center and distance
Euclidean Postulate 2 and Hyperbolic Postulate 2
A finite straight line can be produced continuously in a line.
Euclidean Postulate 1 and Hyperbolic Postulate 1
A straight line can be drawn from any point to any other point.
Elliptical Postulate 4
A triangle can have more than 1 right angle
Euclid's Proof of the Pythagorean Thm
Alexandria 325-265 BC The Elements 300 BCE, collection of all Greek knowledge of Math
Euclidean Postulate 4 and Hyperbolic Postulate 4
All right angles are equal to each other
European Influence on Hindu/Arabic Numbers (976)
Codex Vigilanus written in Spain - oldest writing in Europe containing the Hindu - Arabic Numerals
Golden Spiral
Constructed out of squares 1, 1,2, 3, 5, 8, 13,... The ratios (a+b)/a and a/b get closer to the golden ratio It was thought that the chambered Nautilus Shell followed the Golden Spiral
Euclidean Geometry
Created by Euclid (~300 BC) Model: Plane
Calculating Distance using Euclidean Geometry
Euclidean geometry says that the distance between points A and B is found using the formula: d = √[(x2 - x1)2 + (y2 - y1)2] That is, the distance between points A and B is a straight line from A to B, denoted AB.
European Influence on Hindu/Arabic Numbers (~1450)
Gutenberg invented the printing press which standardized the symbol forms.
Elliptical Postulate 2
If one continues to move on a line, you end up back at the starting place.
The fundamental counting principle
If there are a ways for one activity to occur, and b ways for a second activity to occur then there are a • b ways for both to occur.
Euclid's Parallel Line Statement (Postulate 5)
If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
Pascal's triangel
In the triangle the first and last terms are always 1. The terms in between are the sum of two adjacent terms
Square Numbers
Number of dots: 1, 4, 9, 16, 25, 36,... Rule: n^2 where n is a natural number Notation: s3n+1 represents the 3n+1 square number
Yale Tablet
Square with side length 30 1; 24, 51, 10 = square root of 2 Diagonal length 42; 25, 35 = 30(sq root of 2)
Elliptical Postulate 3
The only circles are the great circles.
Euclidean Algorithm for GCD
a= br +q
Pythagorean School
founded by Pythagoras about 585 B.C Philosophy. The study of proportion. The study of plane and solid geometry. Number theory. The theory of proof. The discovery of incommensurables (Irrational Numbers) The Pythagorean Theorem
Babylonian Problem Texts
stones with problems intended to be solved Babylonian Math based on algorithms rise of computers has lead to a more sensitive view of Mesopotamian mathematics
Hindu Number History 8th Century AD
the 'sacred' numbers or Devanagari. Symbols begin to look like what we use today.
Rosetta Stone
uncovered by Napoleon's army in 1799 3 panels (greek, demotic Egyptian, ancient heiroglyphics) could translate bc of Greek section key to understanding Egyptian civilization
Hindu Number History 5th Century AD
used place value and a zero Zero evolves from a dot to a circle
Eudoxus
"The Method" anticipated the concept of limits in Calculus. "The Method" employed the concept of exhaustion. To find the area of a circle, polygons of increasing numbers of sides would be inscribed in a circle to "exhaust" the space of the circle. The area of the polygons would increase as the number of sides increased gradually approaching the area of the circle.
Construction Problems of Antiquity
1. Squaring the Circle (construct a square equal in area to the given circle (Lindman 1882) 2. Doubling the cube given the length of an edge of a cube and the second cube could be constructed having double the volume of the first (Pierre Wantzel in 1837) 3. Trisecting an angle-divide an angle into 3 congruent angles (Pierre Wantzel 1837) *All were proven impossibel
Babylonian Cunieform
2 characters no symbol for zero base 60 positional not additive
Ancient Egyptian Hieroglyphics
7 characters base 10 not positional additive
Bhaskara's Proof of the Pythagorean Thm
Born 1114, Died 1185 India
European Influence on Hindu/Arabic Numbers (1202)
Fibonacci (Leonardo Of Pisa) wrote "Liber Abaci" (Book of Counting) explaining and promoting the use of the numerals. Changing from Roman Numerals.
Arabic Influence on Hindu/Arabic Numbers 11th Century
Gobar numerals were used
Fibonacci Sequence
Leonardo of Pisa (c. 1170 - c. 1250), or, most commonly, Fibonacci, was an Italian mathematician, considered by some as the most talented mathematician of the Middle Ages. Fibonacci is best known to the modern world for the spreading of the Arabic numeral system in Europe, as well as for a modern number sequence named after him known as the Fibonacci numbers. In his book, the Liber Abaci, he posed and solved a problem involving the growth of a hypothetical population of rabbits based on idealized assumptions. The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The next number is found by adding the two numbers before it. Example: the next number in the sequence above would be 21+34 = 55
Mayan Priest Numerals
Mayan Empire (Classic Period, 250 AD-900 AD) 3 characters symbol for zero base 20 positional not additive
Rectangular/Oblong Numbers
Number of dots: 2, 6, 12, 20, 30,... Rule: on= n(n+1) o for oblong, n is a natural number o2n+3 represents the 2n+3 oblong or rectangular number
Calculating Distance Using TaxiCab Geometry
Taxicab geometry says that the distance from one point to another is the shortest route from A to B, denoted AB*. The distances that make up AB* are either vertical or horizontal. To calculate vertical distances, subtract one y value from the other, because the x values of the points do not change: d = |y2 - y1| To calculate horizontal distances, subtract one x value from the other, because the y values of the points do not change: d = |x2 - x1| The distance between points A and B can be calculated in many ways. Below are four ways to get from B to A.
Pascal's triangle
The numbers on the triangle are coefficients to terms of the expressions equal to the binomial (x+y)^n The numbers at the ends of the rows are ones The numbers in between the ones are the sums of adjacent numbers
Eleatic School
led at one time by Zeno: 5th Century BC
Eratosthenes
(230 B.C.) of Cyrene, a famous Greek scholar, lived and worked in Cyrene and Alexandria. He directed the library in Alexandria found the measure of Earth's circumference was 250,000 Stadia or 39,350 km (24,500 miles). This is only 158 miles less than the currently accepted value
Eratosthenes
(230 B.C.) of Cyrene, a famous Greek scholar, lived and worked in Cyrene and Alexandria. He directed the library in Alexandria, and is known for many contributions to mathematics, geography, and astronomy. -Eratosthenes knew the earth was a sphere, and if he could determine the angle of the noon sun at some other location on the first day of summer, ANDif he knew the distance between these two locations, he could compute the circumference of the earth as a simple ratio. -The rays of the sun travel parallel to each other, therefore creating a pair of congruent alternate interior angles. -The measure of the angle created by the base of the Obelisk (C), the center of the earth (D) and the well at Syene (E), ∠CDE, = 7.2°. -The distance from the well in Syene and the Obelisk at Alexandria (CE) is 787 km -Made ratio with angle at well over 360 set equal to the distance from the well at Syne to the Obelisk over the circumference -His estimate was only 158 miles less than the true value of Earth's circumference
Hindu-Arabic
10 symbols number for 0 base 10 positional not additive
Chou-pei Proof
3rd Century BC The Arithmetic Classic of the Gnomon and the Circular Paths of Heaven oldest known proof of the Pythagorean Thm
Hindu Numeral History 3rd Century BC
Brahmi Numerals: 1, 2, 4, 6, 7, and 9 found on the walls of caves near Mumbai, India
Geocentric view of the universe
Earth=center of universe Aristotle-proved earth was a sphere and believed it to be the center of the universe Ptomely held that universe was centered around the earth bc it made sense the Church accepted this idea
Hindu Number History 2nd Century AD
Nasik, India The Brahmi symbols include the numbers 1-9.
Fractal Properties
Self Similar- copy of the previous Iterative-produced by doing the same thing repeatedly fractional dimensional-dimension is not an integer
Babylonian Mathematical Tablet
University of Pennsylvania Musem of Archaeology and Anthropology multiplication by 10 table a-ra (times)
School of Aristotle, called the Lyceum,
founded by Aristotle (384-322 B.C.) Aristotle is a philosopher more than a mathematician Regards a definition as a significant part of a argument Distinguished between axioms and postulates Axioms: laws of logic Postulates: must remain true when by the results derived from them. Euclid uses this. Invented logic The law of contradiction (not T and F) The law of the excluded middle (T or F)
Platonic School
founded by Plato: 387 B.C. in Athens.(not a mathematician but an advocate of mathematics) The Platonists are credited with discovery of two methods of proof, the method of analysis and the reductio ad absurdum. (contradiction) lato believed all things were made up of five different atoms. Plato identified: fire atoms with the tetrahedron, earth atoms with the cube air atoms with the octahedron cosmos atoms with the dodecahedron water atoms with the icosahedron *Euclid was a student of Plato
Ionian School
founded by Thales (c. 643- c. 546 B.C.). Thales is sometimes credited with having given the first deductive proofs. Deductive reasoning, or deduction, starts out with a general statement, or hypothesis, and examines the possibilities to reach a specific, logical conclusion. Example: Thales Measures the Great Pyramid - LM 5.1.4
Zeno's Paradoxes
1. Dichotomy- there are infinitely many 1/2 distances between the start and finish of a race so you could never finish the race ( a finite distance) 2. Achilles vs the Tortise- Achilles could never have overtaken the tortoise because of the time it took Achilles to move to where the tortoise was and the tortoise would continue to move ahead. Similar to Dichotomy-there are infinite measures for finite distances. Solution was addressed by George Cantor by his concept of transfinite sets which says that infinite sets are different sizes 3. The arrow- the belief that time is made of instants, discrete units, and nothing moves in an instant. Thus the arrow must be at rest at each instant and cannot move. The arrow can be at its start point or end point but cannot move 4. The stadium- time is finitely divisible`
Chinese Version for Estimating a Square Root (200 BC)
1. Draw a large square to represent the area of the number you want to find the square of. 2. Draw a square in the upper left corner inside the big square so the square inside has area of the largest perfect square not larger than the number for which you want to square root. The side lengths of the square are the square roots of the perfect square. 3. Subtract the area of the square from the number you want the square root of. This is the area you have left to fill in the big square. 4. Draw rectangles and a square off two sides of the smaller square to make a bigger square inside the larger square. The length of this square is the side of the square area. The width you need to choose so the area of the two rectangles and the smaller square is equal to or less than the area left to be filled. 5. Subtract the area of the two rectangles and square from the area that was left to be fill. The answer is the new area left to be filled. 6. Now your square root is the side length of the new square. 7. Repeat part 4
Golden Ratio was used for the construction of:
1. The CN (Canadian National Tower) in Toronto, the tallest tower and freestanding structure in the world, contains the golden ratio in its design. The ratio of the total height (553.33m) to the observation deck at 342 meters is the golden ratio!So is the ratio of the observation deck (342m) to the tower above the observation deck (211.33)! 2. The Parthenon: The length to width ratio = 1.61803 has been considered the most pleasing to the eye. This ratio was named the golden ratio by the Greeks. In the world of mathematics, the numeric value is called "phi", named for the Greek sculptor Phidias. (500 BC - 432 BC). The space between the columns form golden rectangles. There are golden rectangles throughout this structure which is found in Athens, Greece. 3. Notre Dame Cathedral-Notre Dame in Paris, which was built in between 1163 and 1250 4. The Mona Lisa-Leonardo Da Vinci has long been associated with the golden ratio The painting of the Mona Lisa was begun in about 1503 and work on it continued for years. Scholars believe that Leonardo built his portraits on the basis of a triangular construction with the spiral superimposed on it. The 'spiral' frames her face, with the rounded side on the right, and the vertical side on the left. Also, the 'spiral' winds from the tip of her nose, grazing the bottom of her chin, and all the way around to her right arm, from elbow to thumb. 5. The Great Pyramid The pyramid was built so that half the perimeter of the base divided by the height would equal to pi.
Plimpton 322, Babylonian Clay Tablet
1900-1600 BCE #322 in the GA Plimpton Collection at Columbia displays 2 sides of a right triangle indicating the knowledge of the Pythagorean Thm, shows most advanced mathematics before Greek mathematics
Roman Numerals
7 characters no symbol for zero base inconsistent additive not positional
Arabic Method of Completing the square
Al'Khwarizmi classified quadratic equations as follows using roots, squares of roots, and numbers. Squares are represented as x2 and roots as x. Squares (x2) equal to roots (x). Squares (x2) equal to numbers. Roots (x) equal to numbers. Squares (x2) and roots (x) equal to numbers. Squares (x2) and numbers equal to roots (x). Roots (x) and numbers equal to squares (x2). x2 + 8x = 33 The form of the equation in this example is Al'Khwarizmi's classification, "squares (x2) and roots (x) equal to numbers." Square completed for an example equation Square completed for the equation x squared plus 8x equals 33. The square is completed in 3 steps: Step 1 is to make a square that is x by x. Step 2 is to make rectangles off each side of the square. Each rectangle is x by 2 so that the four rectangles together have an area of 8x. Step 3 is to make 2 by 2 squares in each corner of the figure to complete the square. The length of the side of the completed square is x plus 4. The area of the square in the center of the figure is x2. The total area of the four rectangles is 4 (2x) = 8x. The total area of the four small squares is 4 (22) = 16. The area of the large square (the entire figure) is (x + 4)2. The area of the large square is equal to the sum of the area of the parts of the square: (x + 4)2 = x2 + 8x + 16 Since x2 + 8x = 33, we can replace x2 + 8x in the equation above with 33 and solve for x as follows: (x + 4)2 = 33 + 16 (x + 4)2 = 49 x + 4 = 7 or x + 4 = -7 x = 3 or x = -11
Arabic Influence on Hindu/Arabic Numbers (750-850)
Al-Khowarizmi wrote "Book of Addition and Subtraction According to the Hindu Calculation" to explain the use of the Hindu decimal system of Numerals. Al-Khowarizmi also wrote- "Hisab al-jabr w'al muqabalah" which was the origin of the word "algebra" His influence on mathematics was so great that the numerals were labeled "Arabic" even though they had Hindu origin.
Fractals
BENOIT B. MANDELBROT Mandelbrot used the term "Fractal" derived from the Latin word "fractus" which means broken or shattered glass. The Mandelbrot Set is created from functions which iterate and do not diverge, while remaining bounded. Waclaw Sierpinski Sierpinski described the Triangle in 1915. It appeared in Italian art from the 12th - 13th Century. The Sierpinski Triangle (5th Iteration) is one of the first FEA (Fractal Element Antenna) used. Fractal Antenna Array for Wireless Communication uses 3 interations of Sierpinski's carpet. Niels Fabian Helge von Koch Koch snowflake: a continuous curve which does not have a tangent at any point, infinite perimeter with a finite area, each stage of the snowflake is contained in the same size circle with a finite area Georg Ferdinand Ludwig Philipp Cantor, Cantor's comb (divide each step into 3 equal parts and discard the middle part)
Pythagorus's Proof of the Pythagorean Thm
Born 569 BC Samos, Greece Died 500-475 BC, Metrapontum, Italy
Carl Friedrich Gauss (1777-1856)
Born: April 30, 1777, Braunschwieg, Germany Died: February 23, 1855, Gottingen, Germany Regarded as one of the three greatest mathematicians of all times, along with Archimedes and Newton Gauss introduced the Gaussian or Normal Distribution, method of least squares Gaussian Elimination (row reduction, Used to solve systems of equations) first to prove the fundamental theorem of algebra pioneer in nonEuclidean geometry introduced symbol for congruency
Blaise Pascal
Born: June 19, 1623, Clermont-Ferrand France Died: August 19, 1662, Paris, France Pascaline: Mechanical Calculator (1645) Theorems: The Generation of Conic Sections (1648) Experimented with Atmospheric Pressure (1648) Experimented with a Perpetual Motion Machine (~1650) Conversations with Fermat led to mathematical theory of probability (1654) Built a Roulette Machine (1655) Began writing controversial thoughts on religion (Wrote: Pensees) ~1657
Hyperbolic Geometry
Both men developed the Geometry independently Nikolay Ivanovich Lobachevsky (1829) (Russia) János Bolyai (1831) (Hungary) Postulates 1-4 are the same as Euclidean Geometry Postulates Postulate 5: Through a point not on a line there is more than one line parallel to the given line. Model: Disk
Diophantine Equations
Diophantine Equations Diophantine equations are named for the Greek mathematician Diophantus Born: about 200 Died: about 284. He is considered to be one of the last of the great Greek mathematicians. He is also known as the "father of algebra". Some scholars disagree with this title because many of his methods for solving linear and quadratic equations go back to the Babylonians. Wrote Arithmetica which contains solutions to 130 algebraic equations some of which are determinate (have a unique solution) some indeterminate and theory of numbers. A Diophantine equation is an indeterminate equation. Indeterminate equations are equations that have more than one possible set of solutions. Although these equations are named for Diophantus, India's contribution to integral solutions of Diophantine equations can be traced back to the: Sulba Sutras, Indian mathematical texts written between 800 BC and 500 BC. Baudhayana (circa 800 BC) found two sets of positive integral solutions to a system of Diophantine equations and also worked on systems of Diophantine equations with up to four unknowns. Apastamba (circa 600 BC) attempted systems of Diophantine equations with up to five unknowns. Indian texts from the time of Aryabhata (499 AD), Bhaskara I (6th century), and Brahmagupta (628) also made significant contributions to our understanding of Diophantine equations. Use Euclidean method for finding the GCD to find first solution set. The linear Diophantine equation ax + by = c has a solution if and only if d divides c where d =GCD (a, b) Second solution set: x = x0 + (b/d)·t y = y0 - ·(a/d)t
Euxodus School
Eudoxus developed the theory of proportion, partly to account for and study the incommensurables (irrationals). He produced many theorems in plane geometry and furthered the logical organization of proof. He also introduced the notion of magnitude. He gave the first rigorous proof on the quadrature of the circle. (Construct with compass and straight edge, a square with the same area as a circle) Method of Exhaustion
Cubic Equation Controversy
Fra Luca Pacioli - 1445 - 1514: Asserted that a solution to the cubic equation was impossible to find -Tartaglia (1499 - 1557) Was 12 when he was maimed and his father was killed by an invading French force. Known as the stutterer. First translation of Euclid's "Elements" in a modern European language Claimed that he could find the solution to any equation of the type x³ + px = q -The Cubic Contest, 1535 at the University of Bologna Antonio Maria Fiore, del Ferro's protégé, challenged Tartaglia to a public problem-solving contest. Each posed 30 cubic equation problems Whoever solved the most in 50 days would win Tartaglia solve all of Fiore's in 2 hours Fiore did not solve any. -Cardano persuaded Tartaglia to reveal his formula for Cardano's book Cardano concluded that del Ferro had actually was the original founder of the cubic solution therefore he didn't think he needed to keep his oath of secrecy to Tartaglia,Cardano along with his pupil, Lodovico Ferrari, expanded and diversified Tartaglia's solution for other cubic cases over the next 6 years Scipione del Ferro- 1465 - 1526 was ultimately credited with solving the cubic equation for the special case: x³ + px = q where p and q are positive Tartaglia died penniless in 1557 and remained almost completely uncredited for his work until well past his death.
Taxi Cab Geometry
Hermann Minkowski (1864-1909) was born in Russia and was Albert Einstein's teacher in Zurich when Einstein was a young boy. He first proposed the idea of taxicab geometry in about 1900. Minkowski developed a series of metrics or functions used for measuring distances. Minkowski's metric axioms are: Given points A and B, there is a unique real number, denoted AB>0, where AB is the distance between A and B. For all points A and B, the distance AB is positive unless A = B. For all pairs of points A and B, AB = BA. Minkowski named his metrics taxicab geometry, because they form a system that mirrors the route a taxicab must take to go from one point to another. A taxicab cannot go through buildings; it must travel on streets that generally meet at right angles.
What prompted other geometries other than Euclidean?
Mathematicians could not prove Euclid's parallel lines statement using his other postulates. If Euclid's parallel statement could not be proven from the others then it must be a conjecture and perhaps not true. 19th century mathematicians investigated other possibilities to refute Euclid's parallel statement.
Practical Uses for Elliptical Geometry
Navigation, Astronomy, and, Space Exploration Example: The shortest flying distance from Norway to the Seattle is a path across Greenland. Seattle is south of Norway so it is not apparent why flying North over Greenland would be shorter. The answer is that Norway, Greenland, and Seattle are collinear locations in elliptical geometry.
Heliocentric view of the universe
Nicolas Copernicus - First astronomer to propose this, Every planet, including Earth, revolved around the Sun. The Earth rotates daily on its axis. The Earth's motion affected what people saw in the heavens. Galileo confirmed this with the telescope put on trial for not heliocentric view by the Catholic Church
Triangular Numbers
Number of dots: 1, 3, 6, 10, 15, 21, 28, 45,55, ... Rule: tn = (n(n+1))/2 t of triangular #, n is a natural number Notation: t2n+1 represents the 2n+1 triangular #
Calculus Controversy
Players: 1. Sir Isaac Newton, England Born: January 4, 1643, Died: March 31, 1727 2. Gottfried Leibniz Born: July 1, 1646, Died: November 14, 1716 Controversy: ohn Wallis, Newton's friend, urged him to take ownership of his calculus findings. Criticized Leibniz for publishing work similar to Newton's work. Nicolas Fatio de Duiller (Swiss mathematician: 1664 - 1753) accused Leibniz of plagiarizing Newton's Work. He Said: "I am now fully convinced by the evidence itself on the subject that Newton is the first inventor of this calculus, and the earliest by many years; whether Leibniz, its second inventor, may have borrowed anything from him, I should rather leave to the judgement of those wo had seen the letters of Newton, and his original manuscripts." Leibniz fires back saying he should gain credit as an independent inventor of calculus. -Enter Royal Society of London founded 1660: Leibniz reached out wanting an apology for such accusations Royal Society responded by appointing a committee of 11 to examine all documents they had on the matter. Most were personal friends of Newton. Committee took months to respond to the Royal Society Ruling in favor of Newton, questioned if Leibniz was guilty of unconscious plagiarism. Leibniz never questioned if he invented calculus first, just wanted credit for his independent findings.
Elliptical Geometry
Reimann (1826-1866) (Germany) Postulate 5: Every two lines intersect. Model: Sphere
Ptolemy's Theorem
The product of the diagonals of an inscribed quadrilateral, where angle 1 and angle 2 are congruent, is equal to the sum of the products of the opposite sides
Spiral of Theodorus
The sum of the hypotenuses of the triangles (square roots of counting numbers) are in the spiral is the length adding up to an irrational number
Pascal's Wager
To believe in God is a better bet than not believing in God, and so it makes sense to believe "just in case".
Elliptical Postulate 1
Two points can determine more than one line if the points are directly opposite each other.
Networks
a collection of points, called vertices, and a collection of lines, called arcs, connecting these points. A network is traversable if you can trace each arc exactly once by beginning at some point and not lifting your pencil from the paper. The problem of crossing each bridge exactly once reduces to one of traversing the network representing these bridges.
The Cairo Mathematical Papyrus
examined in 1962 Egyptians knew: 3-4-5 right triangle, 5, 12, 13 right triangle, 20, 21,19 right triangle from Ptolemeic dynasties contains 40 problems: 9 with Pythagorean Thm problems with rectangles and diagonals
Leonhard Euler (1707-1783)
first discovery-solution to the Basel problem (sum of reciprocal of squares) solved the 7 bridges of Königsberg using networks characteristics for polyhedrons Euler line and the 9 point circle popularized the use of e, i, f(x), sigma, trigonometric functions, and pi
Garfield's proof of the Pythagorean Thm
known in the Arab World-7th century AD 20th president of the US, 1876
Permutation
number of ways to arrange objects when order matters n!/(n-1)!
Combination
order of ways to arrange objects when order does not matter n!/((n-r)!r!)
Rhind Papyrus 1650 BC
purchased in Luxor, Egypt in 1858 by A. Henry Rhind found in Thebes written in hieratic script (cursive hieroglyphics) by a scribe named Ahmes single scroll 18 feet long and 3 inches high arrived at British museum missing a section made possible to translate bc of the Rosetta Stone contains mathematical "secrets" and exercises how to multiply and divide (85 problems) Egyptian Arithmetic Additive-reduced multiplication and division to repeated additions (multiply by doubling and summing) contained a unit fraction table for numerators as odd numbers b/t 5 and 101
Golden Ratio
~1.618033 The golden ratio occurs when a line is divided into two parts so that: the longer part of the segment (BD) divided by the shorter part of the segment (AD) the whole length of the segment (AB) divided by the longer part of the segment (BD).
