math test 2

Cubic Equations

Strategy of solving: Convert cubic to quadratic by substitution

Ptolemaic model

The stars all moved on a sphere that contained the orbits of the planets and the sun around the earth.

Condensed book on the Calculation of al-Jabr and al-Mugabala

"al-Jabr" and "al-Muqabala" refer to operations used Islamic empire to solve equations. (algebra)

Fermat's Last Theorem

It is impossible to write a cube as a sum of two cubes, a fourth power as a sum of two fourth powers, and, in general, any power beyond the second as a sum of two similar powers. For this, I have discovered a truly wonderful proof, but the margin is too small to contain it.

Problem from Diophantus Arithmetica: To find two numbers, having given their sum and the sum of their squares. Condition. Doubling the sum of the squares must exceed the square of their sum by a square.

Sum 20, sum of squares 208. 2x the difference. Therefore, the numbers are 10+x and 10−x. Thus, 200+2x2=208. Hence, x=2, and the numbers are 12 and 8.

Copernicus model

Sun is at the center of the universe with the earth and other planets rotating around the sun.

Golden ratio / Golden segment

The sequence of ratios of consecutive terms (Fsub(n+1)/Fsub(n)) is 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, . . . . The limit of this sequence is known as the golden ratio.

Translation Movement

First: The movement to translate the world's knowledge into Arabic. Second: translate the products of the Arabic translation movement into Latin.

Contributors to the symbolic stage

Francois Viete: Used capital letters to represent algebraic quantities: Vowels for unknowns and consonants for constants. Descarte: Invented Cartesian coordinate system. Used letters at end of alphabet for unknowns, letters at beginning for constants (y = ax^2+bx+c). Fermat: Also had the idea of a coordinate system

Cardano and Ars Magna

Gave a complete account of how to solve cubic equations as well as quartic (fourth-degree) equations. Some solutions contain complex numbers, which Cardano did not understand and ignored.

Fermat's Approach to the Problem of the Points

He first determined the minimum number of points that had to be played to guarantee a winner. Fermat's strategy was to list all the possible outcomes of playing 4 more points. If we let a be a point that player A wins and b be a point that player B wins, we have these 16 different possibilities.

Pascal's Approach to the Problem of the Points

He first determined the minimum number of points that had to be played to guarantee a winner. He used his triangle.

Pierre de Fermat

He invented a coordinate system at approximately the same time as Descartes. His work with maxima, minima, and tangents were important in the development of calculus. His work in number theory cannot be underestimated.

René Descartes

He starts with the simplest elements and proceeds using the process of deduction. The starting point was to discover the simplest ideas that could not be doubted.

Book of Addition and Subtraction after the Method of the Indians

Hindu Arabic numerals.

Fibonacci

His real name was Leonardo of Pisa. He wrote books and proved number theory theorems. Wrote Liber abaci, Practica geometriae (8 chapter of geometry problems based on Euclids Elements, precise proofs), Liber quadratorum ("book of squares" number theory with methods for finding Pythagorean triples). He marks beginning of mathematical renaissance in western Europe

Problem of the Points

How would you divide the stakes in a game of chance between two equally skilled players A and B, where player A needs 2 more points to win and player B needs 3 more points to win? In our example, player A could win the first point and player B could win the next 2 points. We still don't have a winner, but when the next point is played one of them has to win the point and win the game. That would be 4 points.

Binomial Expansion

In Treatise on the Arithmetical Triangle how to use the triangle to determine the coefficients of the terms in a binomial expansion. (x+1)0=1(x+1)0=1 (x+1)1=1x+1y(x+1)1=1x+1y (x+y)2=1x2+2xy+1y2(x+y)2=1x2+2xy+1y2 The numbers on each row of the triangle correspond to the coefficients of the terms in a binomial expansion.

Treatise on the Arithmetical Triangle

Pascal's book

Binomial Probability

Pascal's triangle can be used to solve any binomial probability problem—that is, any problem with two equally likely independent outcomes.

French Triumvirate

Pascal, Descartes, Fermat

Descartes Line Segments

Step #1: Open the compass to measure the length aa. Put the point of the compass on B and mark the point on the slanted line, where the compass hits the line D. Connect the two points, B and D. The line segment, line-segment BD, should have length a. Step #2: Draw a line parallel to line-segment BD that goes through C. Mark the point on the slanted line, where the parallel intersects it as point E. Step #3: Show that the length of line-segment CE is a^2. Triangles ABD and ACE are similar triangles. (Why? They share a common angle at A. Since line-segment BD and line-segment CE are parallel, ∠BAD and ∠EAC, are corresponding angles formed when a transversal crosses parallel lines and are congruent. Since two out of the three pairs of angles are congruent, the third pair must be congruent, and the triangles are similar.)

La Geometrie

Written by René Descartes. Had the most impact. Descartes lays out the foundations of analytic geometry with the demonstration of his coordinate system.

Galileo

Wrote the "Dialogue Concerning the Two Chief World Systems", which was banned by the church ofc.

Discriminant

b^2 - 4ac (can't be done if numbers are negative)

Fermat's pseudo-equality method

f(x + e) ~= f(x)

Development of algebra

from Arabic word "al-jabr" from Al-Khwarizmi's work. algebra was still in the rhetorical stage (in words). Coefficients a and b had to be positive since negative quantities weren't accepted.

Diophantus Arithmetica problems: To find four numbers such that the sums of all sets of three are given.Condition. One-third of the sum of all must be greater than any one singly. Sums of three 22, 24, 27, 20.

x is the sum of all four. Therefore, the numbers are x−22,x−24,x−27,x−20. Therefore, 4x−93=x, x=31, and the numbers are 9,7,4,11. Method of solving: a+b+c+d = 22 ...

Cogito Ergo Sum

Descartes philosophy. It means "I think, therefore I am."

Kepler's three laws

1. The planets move about the sun in elliptical orbits with the sun at one of the foci. 2. The radius vector joining a planet to the sun sweeps over equal areas in equal intervals of time. 3. "It is absolutely certain and exact that the ratio which exists between the periodic times of any two planets is precisely the ratio of the (3/2)th power of the mean distances [of the planet to the sun]." distance ^3 / time ^2

Liber abaci: Rabbit problem

A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also. Because the abovewritten pair in the first month bore, you will double it; there will be two pairs in one month. One of these, namely the first, bears in the second month, and thus there are in the second month 3 pairs; of these in one month two are pregnant, and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month; . . . there will be 377 pairs, and this many pairs are produced from the abovewritten pair in the mentioned place at the end of the one year. You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth, and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the abovewritten sum of rabbits, namely 377, and thus you can in order find it for an unending number of months.

Golden rectangle

A rectangle whose side lengths are in the ratio of the golden ratio is called a golden rectangle.

Pascal's Triangle

A right triangle with the top row and the first column containing all ones.

Hindu-Arabic numerals

Base 10 positional numeration system. scholars and traders from the Islamic Empire used it first and introduced it to Europe. Form of numerals varied across regions and changed over time. Different rules for writing and computation for a long period of time.

Al-Khwarizmi

Book of Addition and Subtraction According to the Hindu Calculation. used geometrical demonstrations to justify his procedures for solving equations

Baghdad

Commercial, cultural, and intellectual center of the Islamic world. House of Wisdom was an academy comparable to the Museum at Alexandria.

Solving quadtratic equations (Islamic Empire)

Completing the Square: Ex, x^2 + 12x = 64. Draw one square with sides x and two rectangles with dimensions 6x. Now complete the square, lower square corner area is 6^2 = 36. So the square is (x+6)^2 = 64 + 36.

Brahe

Danish astronomer. Compiled data from 38 years of observing planetary motion. Made observations from his own observatories.

Kepler

Kepler was a deeply religious man. He had a very Pythagorean outlook on the world and believed that God had made the universe according to a mathematical plan. In 1612, during the Counter Reformation, the Catholic church excommunicated him due to his stance on the heliocentric theory of the universe.

Blaise Pascal

Known as the "greatest might have been"

Fermet

Known as the "prince of amateurs"

Problem from Diophantus Arithmetica: To divide a square number into two squares.

Let the square number be 16. X^2 [is] one of the required squares. Therefore, 16−x^2 must be equal to a square. Take a square of the form (mx−4)2, 4 being taken as the absolute term because the square of 4=16, that is, take (say) (2x−4)^2 and equate it to 16−x^2. Therefore, 4x^2−16x=−x^2. Therefore, x=16/5, and the squares are 256/25, 144/25.

Napier's logarithms

Logarithms make computations simpler by replacing multiplication and division with addition and subtraction. This simplification can be seen in the following formula: sinA⋅sinB=12[cos(A−B)−cos(A+B)] This formula was well known in Napier's time and may have led him to initially restrict logarithms to the sines of angles. Napier's approach in defining a logarithm was geometrical. Consider the line segment line-segment AB and the ray line-segment DE.

Mersenne primes and Mersenne numbers

Mersenne claimed that 2^(n)−1 is prime for prime numbers n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257 and is composite for all other prime numbers less than 257. These numbers that are also prime are called Mersenne prime numbers.

Liber abaci (The Book of Calculations)

Published in 1202 upon Fibonacci's return to Pisa after travelling around the Mediterranean. Introduced the Hindu-Arabic decimal system and the use of Arabic numerals into Europe.

Stages of algebraic development

Rhetorical: Equations and operations expressed in words and written out entirely. (Eqyptians) Syncopated: Abbreviations for many words, like p for "plus" (Diophantus) Symbolic: using + and - and symbols to represent constants and unknowns

Descartes Quadratic Equation

Step #1. Draw the line ll and mark a point, L, on the line. Step #2. Construct a line perpendicular to l at the point L. Step #3. Using L as one endpoint, mark the point N on the perpendicular line so that LN has length a^2. Step #4. Mark the point MM on line l so that LM has length b. Step #5. Draw the line MN and mark the two points where MN intersects the circle as O and P. According to Descartes, z=OM is the solution of our equation.

Probability Problems

Suppose you toss a coin three times (or toss three coins once). There is only 1 way to get 3 heads (HHH). There are 3 ways to get 2 heads and 1 tail (HHT, HTH, THH). There are 3 ways to get 1 head and 2 tails (TTH, THT, HTT). There is only 1 way to get 3 tails (TTT). This number pattern—1, 3, 3, 1—is one of the rows in Pascal's triangle. Look at the row: 1 6 15 20 15 6 1. These numbers tell us the number of different outcomes when we toss a coin 6 times. First of all, since there are two possible outcomes on each of the tosses, there will be a total of 2^6=64 outcomes. There is 1 way to get 6 heads; 6 ways to get 5 heads and 1 tail; 15 ways to get 4 heads and 2 tails; 20 ways to get 3 heads and 3 tails; 15 ways to get 2 heads and 4 tails; 6 ways to get 1 head and 5 tails; and only 1 way to get 6 tails. The sum of these numbers is 64 1+6+15+20+15+6+1=64 What is the probability that when you toss a coin 6 times you will get 3 heads and 3 tails? Since there are 20 ways to get 3 heads and 3 tails, and there are 64 total possible outcomes, the probability will be 20/64=5/16.

Fibonacci sequence

The number sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . . This sequence, in which each number is the sum of the two preceding numbers, has proved useful and appears in mathematics and science.

Ellipse

The orbit of a planet is an ellipse with the sun at one focus.

Napier's rods

The set consisted of 10 rectangular strips of bone. Inscribed with the multiples of the numbers on the top row. Ex. 1615 x 365 = 589475 Use rods with the multiples of 1, 6, 1, and 5. --8075 -9690- 4845-- 589475

Perfect Numbers

Those numbers that are equal to the sum of their proper factors. Ex. 28 = 1 + 2 + 4 + 7 + 14