History of Mathematics Test 2 and more

Find two numbers such that their difference and also the difference of their cubes are given numbers; say their difference is 6 and the difference of their cubes is 504. (Use approach of Diophantus, letting the numbers be x-3 and x+3.)

a-b = 6 a = x+3 b = x-3 (x+3)-(x-3) = 6 504 = ((x+3)^3)-((x-3)^3) = (x+3)((x^2)+6x+9)-(x-3)((x^2)-6x+9) = (x^3)+6(x^2)+9x+3(x^2)+18x+27-(x^3)+6(x^2)-9x+3(x^2)-18x+27 = 18(x^2)+54 450 = 18(x^2) 25 = (x^2) x = 5 a=8, b=2

hallmarks of 19th century mathematics

abstraction and generalization

Important accomplishments Newton

analysis of white light (optics), universal law of gravitation (physics), calculus (math)

Eratosthenes

approx circumference of Earth

Which of the following is noteworthy about Leonardo of Pisa (Fibonacci)?

attempting to popularize the use of the Hindu Arabic numbers in Europe, giving a clever way to produce Pythagorean triples in his book on squares (Liber Quadratorum, introducing the problem that led to the Fibonacci sequence

Explain why Euclid was an important mathematician and author

axiomatic method, wrote The Elements (math bible), super about geometry, number theory (if b/a, then number c that b=ac), proved infinitely many primes

Which of the following does apply to the mathematical accomplishments of Cardano

beginning to use negative numbers, showing how to reduce any cubic to a cubic lacking the square term, recognizing that a given equation may have more than one root, extending algebraic methods to solve all cubics

the discovery by johann heiberg in the early 1900s of a palimpsest was important for

being a copy of The Method, by Archimedes, showing how close he was to developing calculus

Gauss should be noted for

clearly written mathematics, his insistence on rigor, generalizing the notion of equality through congruences, not publishing nearly as much as he knew

In homework, we used term-by-term integration of an infinite series to develop other infinite series. given that cos(x) = 1-((x^2)/(2!))+((x^4)/(4!))-((x^6)/(6!))+..., use term-by-term differentiation to give an infinite series representation of sin(x). (Hint: Recall that the derivative of cos(x) is -sin(x))

cos(x) = 1-((x^2)/(2!))+((x^4)/(4!))-((x^6)/(6!))+... -sin(x) = 0-(2x/(2!))+((4x^3)/(4!))-((6x^5)/(6!))+... sin(x) = (2x/(2!))-((4x^3)/(4!))+((6x^5)/(6!))-... sin(x) = (x/(1!))-((x^3)/(3!))+((x^5)/(5!))-...

George Cantor

created more found questions, absolutes, and generalizations included notations of infinity act of counting involves "one-to-one" correspondence created transfinite cardinal number system proved power set stuff transcendental numbers

Eratosthenes was important for

devising a fairly accurate measurement of the circumference of the earth

Describe and discuss some causes for the decline of Greek mathematics which were internal to the discipline

difficulty of geometric algebra and difficulty of written tradition (lack of oral tradition)

Crises in Foundations of Math

discovery of irrational numbers - resolved by Eudoxian theory of proportion invention of calculus with reckless use of limits and continuity - Cauchy, Weierstrass and others solve Parallel postulate - invention of non-Euclidean geometry Paradoxes from Cantor's theory of sets - Hilbert created and agenda to solve

Euler contributions to notation

e for base of natural logarithms, pi, f(x) for function, backwards E for summation, i for sq rt of -1 and complex numbers

characterization of 20th century math

emphasis on axiomatics

The importance of the work of Father Marin Mersenne is that he

encouraged the earliest noted gathering of mathematicians into a kind of professional society

Leibniz is noted for his work on the calculus, but is also important for

envisioning a system of logic in which questions could be settled by computation

Johns Hopkins

established to merge research and teaching Mary Elizabeth Garrett + wealth women = coed and funding

Ptolemy's Theorem ("In a cyclic quadrilateral, the product of the diagonals equals the sum of the products of the opposite sides.") The theorem was noteworthy in which of the following:

finding the chords of circles used in the development of trigonometry

Three men, each having money, found a purse containing 30 dollars. The first man said to the second, "If I take this purse, I will have two times as many dollars as you." The second man said to the third, "If I take this purse, I will have twenty-five more dollars than you." The third man said to the first, "If I take this purse, I will have three times as much as you." How many dollars did each man have?

first man amount = a second man amount = b third man amount = c a = 2b-30 b = c-5 c = 3a-30 a = 2(c-5)-30 a = 2(3a-30-5)-30 a = 6a-70-30 -5a = -100 a = 20 c = 3(20)-30 c = 30 b = 30-5 b = 25

R.H. Moore

hard ass Texan math teacher didn't allow students to get help or move on until they understood the material competition and discovery method

In the work by Archimedes discovered in the early 1900s as a palimpset, he was trying to show

his method of discovery o fsome of his mathematical theorems

conditions in 12th century Europe that indicated a need for Fibonacci's Liber Abacci were

increased economic activity and changing bookkeeping practices

The importance of the work of Galileo was that it

increased the reliance of science on math increased the reliance of science on experimentation adapted the telescope as an instrument of investigation in astronomy mathematically formulating explanations of motion

Kepler's Astronomia Nova (New Astronomy), published in 1609, was important for

investigating the orbit of Mars and showing the planets move in elliptical orbits with the sun at one focus containing his three laws of planetary motion

Use the methods of Diophantus to solve. Find three numbers such that when any two of them are added, the sum is one of three given numbers. Say the given sums are 43, 54, and 75.

let x = the sum of the three numbers x = (x-43)+(x-54)+(x-75) x = 3x-172 2x = 172 x = 86 (x-43) = 86-43 = 43 (x-54) = 86-54 = 32 (x-75) = 86-75 = 11

Find three numbers such that when any two of them are added, the sum is one of three given numbers. Say the given sums are 99, 77, and 80. (Use methods of Diophantus)

let x be the sum of the three numbers (x-99)+(x-77)+(x-80) = 3x-256 3x-256 = x 2x = 256 x = 128

This course

math and its importance in philosophy and civilization give basic equipment to pursue math intellectual arguments for why math is important

tables that were essentially values of sines were

measurements of chords of circles

Ptolemy eighth

most responsible for fall of Greek

why do we teach algebra?

no other course (except music) encourages symbolic thinking repeatedly solving equations leads to learning there is a sensible solution to all equations

two noteworthy things about Fibonacci's Liber Abacci were his

promotion of the Hindu Arabic number system in Europe and introduction of the Fibonacci sequence

The importance of the work of Descartes was that it

provided a new philosophy that was tied closely to mathematics

The work of Al Khwarizmi is, in part, notable for

providing a fresh start for algebra

which of the following is significant about Al-Khwarizmi

providing a name for algebra and algorithm, treating quadratics in a systematic general manner, writing on Hindu-Arabic numbers, through his work later Europeans learned algebra

During the seventeenth century, Descartes changed the direction of scientific inquiry by

providing a philosophy based on mathematical reasoning

The publication of Ars Magna by Cardano is important for

recognizing that a given equation may have more than one root, reducing any cubic to a cubic lacking the square term, beginning to use negative numbers, extending algebraic methods to solve all cubics, realizing complex numbers existed(not developing them)

The invention of non-Euclidean geometry was prompted by

repeated attempts to resolve the controversy over Euclid's parallel postulate

Ptolemy's Theorem states that "In a cyclic quadrilateral, the product of the diagonals equals the sum of the products of the opposite sides." In terms of algebraic notation, this theorem is stated

rhetorically

Euler should be noted for

serving as a counterexample to the stereotypical mathematician as a bad teacher, being one of the most prolific producers of first-rate mathematicians ever, continuing to do math in the face of physical barriers the many symbols he left to improve notation, contributing to virtually every branch of mathematics as well as many areas of science, being a clear expositor of mathematics, and being a caring teacher

Given that an infinite series representation of sin(x) is sin(x) = x-((x^3)/3!)+((x^5)/5!)-((x^7)/7)+..., find an infinite series representation of cos(x). Be clear about your approach. (Hint: Recall that the derivative of sin(x) is cos(x))

sin(x) = x - ((x^3)/3!)+((x^5)/5!)-((x^7)/7)+... cos(x) = 1-((3(x^2))/(3*2*1))+((5(x^4))/(5*4*3*2*1))-((7(x^6))/(7*6*5*4*3*2*1))+... cos(x) = 1-((x^2)/2!)+((x^4)/4!)-((x^6)/6!)+...

Prove that if a_=b(modn) and c is an integer, then a-b_=b-c(modn) (Recall that x_=y(modn) means x-y is a multiple of n)

since a_=b(modn), a-b=k(n) for some integer k therefore a-b+c-c_=k(n) therefore (a-c)-(b-c)_=k(n) therefore (a-c)_=(b-c)(modn)

Ptolemy's work, Almagest, was notable for containing astronomy, but also contained

tablese of chords of circles which can be interpreted as sines

Transmission to the west of the knowledge and mathematics of the Arabs could be attributed to:

the Crusades, the translations of scholars such as Adelard, the need for improved bookkeeping practices of merchants and traders

In connection with Gauss, the name "M. Leblanc" was significant because it was

the alias used by a mathematician, Sophie Germain, to conceal her identity as a woman

Which of the following was a cause for the decline of Greek mathematics?

the difficulty of geometric algebra, the rise of the Roman Empire, the loss of the oral tradition, expelling the scholars from Alexandria

At the start of the seventeenth century, scientific inquiry in Europe was dominated by

the ideas of Aristotle

The resolution of the parallel postulate controversy was

the invention of non-Euclidean geometry

one result of the crusades was

the transmission to the west of scientific knowledge and mathematical methods of the Arabs

In working with quadratic and cubic equations, why did mathematicians, such as Al-Khowarizmi and Omar Khayaam, consider so many different types?

they did not understand how to work with negative numbers

Let ABCD be a cyclic quadrilateral with AB a diameter of the circle. Then angle CAD has the same measure as angle DBC because:

they subtend the same arc of the circle

Which of the following would be a reason for the formation of scholarly organizations

to promote academic freedom, to provide opportunities to place discoveries before the academic world, to confer distinction and reputation to the members

Which of the following would not be a reason for the formation of scholarly organizations

to protect the secrecy of individual discoveries (e.g. Tartaglia's solution to the cubic)

One reason for the importance of the Arab period was their

translations preserved many of the earlier Greek works

In the algebraic work of Omar Khayyam it is important to note that he:

understood the need to move away from considering only geometric quantities

During the seventeenth century, Galileo changed the direction of scientific inquiry by:

using experimentation

Claudius Ptolemy

wrote definitive Greek work on Astronomy (Syntaxis Mathematica) in cyclic quadrilateral, product of diagonals is equal to sum of products of two pairs of opposite sides

Find three numbers such that when any two of them are added, the sum is one of the three given numbers. Say the given sums are 93, 69, and 60. (Use the methods of Diophantus to solve)

x = (x-93)+(x-69)+(x-60) x = 3x-222 -2x = -222 x = 111 x-93 = 111-93 = 18 x-69 = 111-69 = 42 x-60 = 111 - 60 = 51 18+42 = 60 18+51 =69 42+51 = 93

Solve using the geometric method of completing the square. (Be sure to show the geometry.) X^2 + 12x = 28

x^2 + 12x + 4(12/4)^2 = 28 +4(12/4)^2 x^2 + 12x + 36 = 28+36 (x+6)^2 =64 x+6 = 8 x = 2

Solve using the geometric method (completing the square) of Al Khwarizmi. x^2+12x=45

x^2+12x+4(12/4)^2 = 45+4(12/4)^2 (x+6)^2 = 45+36 (x+6)^2 = 81 x+6 = 9 x = 3 ___________________________ []______________________[] || x || || x || || X || ||__________x___________|| []______________________[]

Foundations of Calculus

"Age of Rigor" Cauchy wrote Cours d-Analyse, attempted rigorous treatment of limits Weierstrass gave definition of limit

Charles Babbage

"On the Influence of Signs in Math Reasoning" symbols simplify and clarify credited with labeling triangles in a way that makes trig laws make sense

Newton Square root of 7

(7^(1/2)) = (9(7/9))^(1/2) = 3((7/9)^(1/2)) = 3((1-(2/9))^(1/2)) = 3(1-(1/2)(2/9)-(1/8)((2/9)^2)-(1/16)((2/9)^3)-(5/128)((2/9)^4)-... = 3(1-(1/9)-(1/162)-(1/1458)-(5/52483)-...) = 2.645729... or f(x) = x²-7 f'(x) = 2x x-(f(x)/f'(x)) guess x=2 2-(-3/4) = 2.75 guess x=2.75 2.75-(.5625/5.5) = 2.6477... guess x=2.6477 2.6477-( and so on till number needed

General formula for (F(2n))^2

(F(2n))^2 = (F(2n+1)F(2n-1))-1

Expand (1-x)^(-3) using the binomial formula. The resulting coefficients are

(a+b)^n = (a^(n))(b^0)+(n/1!)(a^(n-1))(b^1)+(n(n-1)/2!)(a^(n-2))(b^2)+... (1-x)^(-3) = (1^(-3))+(-3)(1^-4)(-x)+(-3(-4)/2!)(1^-5)(-x^2)+... = 1+3x-6(x^2)+10(x^3)+... triangular numbers

use the binomial theorem to find the infinite series expansion of (1-x)^(-2) (Hint: Show your work, but the answer is: (1-x)^(-2) = 1+2x+3(x^2)+4(x^3)+...+(n+1)(x^n)+...)

(a+b)^n = (a^n)(b^0)+(n/(1!))(a^(n-1))(b^(1))+...+((n!)/(n!))(a^0)(b^n) (1^(-2)) + ((-2/(1!))(1^-3))(-x) + ((-2(-3))/2!)(1^-4)(-x^2)) + ((-2(-3)(-4))/3!)(1^(-5))(-x^3)+... 1+2x+3x^2+4x^3+...

Use either long division or the binomial theorem to write an infinite series for ((1-x)^(-2)) (write the first four terms)

(a+b)^n = (a^n)(b^0)+(n/1!)(a^(n-1))b+((n(n-1))/2!)(a^(n-2))(b^2)+((n(n-1)(n-2))/2)(a^(n-3))(b^3)+... ((1-x)^(-2)) = (1^(-2))+(-2/1!)(1^(-3))(-x)+((-2)(-3)/2!)(1^(-4))((-x)^2)+(-2(-3)(-4)/3!)(1^(-5))((-x)^3)+... = 1+2x+3(x^2)+4(x^3)+...

Solve using the geometric method (completing the square) of Al Khwarizmi: (x^2)+10x=96. (show a diagram)

(x^2)+5x+5x+(5^2) = 96+(5^2) (x^2)+10x+25 = 121 (x+5)^2 = 121 x+5 = 11 x = 6 x 5 -------------------------- x | | | |___| | | | | | 5 | | |_______________________________|

Find all three roots by first reducing to a cubic lacking a square term (x^3)+3(x^2) = 13x+15

(x^3)+3(x^2)-13x-15=0 sub x=y-(3/3) ((y-1)^3)+3((y-1)^2)-13(y-1)-15=0 (y-1)((y^2)-2y+1)+3(y^2)-6y+3-13y+13-15=0 (y^3)-2(y^2)+y-(y^2)+2y-1+3(y^2)-19y+3-2=0 (y^3)-16y=0 y((y^2)-16)=0 y(y-4)(y+4)=0 y=0,4,-4 x=-1,3,-5

Find all three roots by first reducing to a cubic lacking a square term (x^3)+3(x^2)=2

(x^3)+3(x^2)-2=0 sub x=y-1 ((y-1)^3)+3((y-1)^2)-2=0 (y-1)((y^2)-2y+1)+3(y^2)-6y+3-2=0 (y^3)-2(y^2)+y-(y^2)+2y-1+3(y^2)-6y+1=0 (y^3)-3y=0 y((y^2)-3)=0 y=0,sqrt3,-sqrt3 x=-1, sqrt(3)-1, -sqrt(3)-1

Use the method of Fibonacci, shown in class, to find a Pythagorean triple by starting with the odd square 49

49 = 7² 1+3+5+7+9+11+...47 = 24² or 49+1 = 50/2 = 25-1 = 24 7²+24² = 25² (7,24,25)

Hilbert's 3 questions

1. Is math complete (all valid theorems about system are provable) ? Godel says no 2. Is math consistent (not possible to deduce two contradictory statements) ? Godel says no 3. Is math decidable (can you know/not know whether an equation will be solvable) ? Turing says no

important consequences of algebraic system

1. attention on operations rather than answers (leads to generalization) 2. symbolic solution draws attention to the structure of solution and original equation (classify and narrow possibilities quickens path to solution)

In Liber Quadratorum, Fibonacci gave the identity ((a^2)+(b^2))((c^2)+(d^2)) = ((ac+bd)^2) + ((bc-ad)^2) = ((ad+bc)^2) + ((ac-bd)^2) Use the identity to write 1170 as a sum of squares in two different ways (Hint: 1170 = (26)(45))

1170 = (26)(45) = ((5^2)+(1^2))((6^2)(3^2)) = ((30+3)^2)+((6-15)^2) = (33^2)+((9)^2) = ((15+6)^2)+((30-3)^2) = (21^2)+(27^2)

Three men, each having denarii, found a purse having 33 denarii. The first man said to the second, "If I take this purse, I will have three times as much as you." The second man said to the third, "If I take this purse, I will have four times as much as you." The third man said to the first, "If I take this purse, I will have as much as you." How many denarri did each man have?

3b = a+33 4c = b+33 a = c+33 3b = (c+33)+33 3(4c-33) = c+66 12c-99 = c+66 11c = 165 c = 15 a = 15+33 a = 48 b = (48+33)/3 b = 27

Even though calculus was not developed prior to 1400, which of the following formulated important ideas closest to calculus

Archimedes

Diophantus

Arithmetica number theory no neg numbers, only one solution, no attempt at generalization abbreviated notation (syncopated) diophantine equations ((x^n)+(y^n))=(z^n) inspired European mathematicians

Explain why the algebraic solution to the cubic by del Ferro, Tartaglia, and Cardano was important

At the time, the prevailing thought was that math was finished evolving and could not move past what the Greeks had accomplished. The solution of the cubic showed that mathematics could continue expanding.

non-Euclidean Geometry

Both Lobachevsky and Bolyai denied the recognition they deserved gain insight into why mathematicians need attention to foundations and rigor learn to be willing to question already formed math

Let ABC be an equilateral triangle inscribed in circle. If BD is a diameter of the circle, use Ptolemy's Theorem to prove that CD=BD-AD

By Pt's Thrm: (AC)(BD)=(AB)(CD)+(BC)(AD) Since triangle ABC is equilateral, AB=BC=CA therefore (AC)(BD)=(AC)(CD)+(AC)(AD) therefore BD=CD+AD therefore CD=BD-AD

The invention of analytic geometry

Descartes, Cartesian coordinate system

Recall that the Fibonacci sequence is defined by: F(0)=1, F(1)=1, and F(n)=F(n-1)+F(n-2) for n(greater than or equal to)2. Show that F(n)<F(n+1)<2F(n). Conclude that F(n+1)/F(n) is always between 1 and 2

F(n)<F(n+1)<2F(n) F(n) < F(n)+F(n-1)=F(n+1) < F(n)+F(n)=2F(n) F(n)<F(n+1)<2F(n) F(n)/F(n) < F(n+1)/F(n) < 2F(n)/F(n) 1 < F(n+1)/F(n) < 2

Which of the following could be a general formula for F1 + F2 + F3 + ... + Fn

F1+F2+F3+...+Fn = F(n+2)-1

The Fibonacci sequence is: F1=1, F2=2, and F(n+1)=Fn+F(n-1) for n>2. Compute the first ten terms of the sequence.

F1=1, F2=1, F3=2, F4=3, F5=5, F6=8, F7=13, F8=21, F9=34, F10=55

David Hilbert

Grundlagen der Geometric - reworked axioms of Euclidean geometry 23 problems to reinvigorate math - Hilbert's honor class

Briefly describe what is meant by the "oral tradition" discussed by Van der Waerden

He placed an importance on the importance of having a teacher and passing down the knowledge of mathematics with the added benefit of verbal instruction.

briefly describe the program of translation carried out at the House of Wisdom and explain why it was significant

It translated all Greek work to Arabic, and without it all the Greek work would have been lost

Solve the following problem from Liber Abaci. A merchant doing business in Lucca doubled his money there and then spent 12 denarii. On leaving, he went to Florence, where he doubled his money and spent 12 denarii. Returning home to Pisa, he there doubled his money and spent 12 denarii, nothing remaining. How much did he have in the beginning?

L = 2x -12 F = 2L -12 P = 2F -12 = 0 2F = 12 F = 6 2L-12 = 6 2L = 18 L = 9 2x-12 = 9 2x = 21 x = 21/2

The dy/dx notation of Leibniz is now favored over Newton's fluxions. This indicates that:

Leibniz understood the importance of good notation in advancing mathematics

post-Hitler timeline

Manhattan Project, computers, The GI Bill, National Science Foundation, Sputnik, space program, Vietnam, Break of Soviet Union

The fall of Constantinople in 1453 had what effect on the revival of learning in Europe?

Manuscripts taken from the library by fleeing intellectuals made their way to Italy

The development of calculus

Newton and Leibniz created at same time and drama ensues

Prove the identity: ((a^2)+(b^2))((c^2)+(d^2)) = ((ad+bc)^2)+((ac-bd)^2). Use the identity to write 2813 as the sum of squares. (Hint: 2813 = (29)(97))

RHS = ((ad+bc)^2)+((ac-bd)^2) = (ad+bc)(ad+bc)+(ac-bd)(ac-bd) = (a^2)(d^2)+abcd+abcd+(b^2)(c^2)+(a^2)(c^2)-abcd-abcd+(b^2)(d^2) = (a^2)(d^2)+(b^2)(c^2)+(a^2)(c^2)+(b^2)(d^2) = ((a^2)+(b^2))((c^2)+(d^2)) = LHS 2813 = (29)(97) = (25+4)(81+16) = ((5^2)+(2^2))((9^2)+(4^2)) = ((5(4)+2(9))^2)+((5(9)-2(4))^2) = (38^2)+(37^2)

Archimedes showed that the surface area of a sphere is 4(pi)(r^2), where r is the radius. Show that if the sphere is inscribed in a cylinder, then the surface area of the cylinder (sides plus bases) is 3/2 the surface area of the sphere. (You will need to find the surface area of the cylinder.)

SAcyl = Top area + bottom area + rectangle area SAcyl = (pi)r^2 + (pi)r^2 + 4(pi)r^2 SAcyl = 6(pi)r^2 SAsph = 4(pi)r^2 SAcyl/SAsph = 6/4 = 3/2

The Cubic Controversy

Scipione del Ferro of Biligna soved (x^3)+bx=c, taught to Antonio Fiore, Tartaglia figured out (x^3)+b(x^2)=c and del Ferro, Cardano figured out cubic without without square term

Which of the following are noteworthy aspects of Gauss

Showed astonishing ability at mathematics as a child, insisted on rigor in mathematics, an early discoverer of non-Euclidean geometry

Explain the significance of the alias "M. LeBlanc" used by Sophie Germain in her communication with Gauss. Comment on his response and its relevance to our time

Sophie Germain published math articles under the alias "M. Leblanc" because it was thought that a woman could not understand math. Gauss promoted M. Leblank's work, and when he discovered her true identity, he praised her work and declared that mathematical understanding is not related to gender.

Apollonius

The Conics

Analytic Geometry connects algebra and geometry. Briefly describe what is involved and mention the advantages and disadvantages

When you take a geometric figure and translate it into an algebraic expression, and, when solved, can be translated back into geometric form. Advantage - easy to see and explain Disadvantage - foreboding algebra

Find two numbers such that their difference and also the difference of their squares are given numbers; say their difference is 6, and the difference of their squares is 348. (Use the approach of Diophantus, letting the numbers be x +3 and x - 3.)

a = x+3 b = x-3 (x+3)-(x-3) = 6 ((x+3)^2)-((x-3)^2) = 348 ((x^2)+6x+9)-((x^2)-6x+9) = 348 12x = 348 x = 29 a = 32 b = 26

syncopation of of algebra

a movement away from rhetorical algebra and using abreviations

Compute (F(2n+1)F(2n-1)) and (F(2n))^2 for: a) n=1 b) n=2 c)n=3 d)n=4

a) (F3)(F1) = (2)(1) = 2 (F(2))^2 = 1 b) F(5)F(3) = 5(2) = 10 (F4)^2 = 3^2 = 9 c) (F7)(F5) = 13(5) = 65 (F6)^2 = 8^2 = 64 d) (F9)(F7) = 34(13) = 442 (F8)^2 = 21^2 = 441

The Fibonacci sequence is: 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,...,Fn,... Compute F1+F2+F3+...+Fn for a) n=3 b) n=4 c) n=5 d) n=6

a) F1+F2+F3 1+1+2 = 4 b) F1+F2+F3+F4 1+1+2+3 = 7 c) F1+F2+F3+F4+F5 1+1+2+3+5 = 12 d) F1+F2+F3+F4+F5+F6 1+1+2+3+5+8 = 20

Describe the period -300 to 1200 relative to a) the development of notation b) the development of algebra

a) lack of notation, Diophantus syncopated method, arabic number system b) geometric algebra, omar defined of algebra and generalized, greeks dead end due to lack of notation and Greeks only used algebra geometrically, arabs and al-karwiza (adding and subtracting from both sides) got generalization, cardano gave multiple solutions to cubics and better notation with negative numbers

Describe the importance to the development of notation of: a) Descartes b) Leibniz

a) started the trend of using a,b,c for known quantities and x,y,z for unknown quantities in Discourse on Method (La Geometry) b) started dy/dx which allowed more advanced problem solving than Newton's fluxions

Recall that a _=b(modn) means that a-b is a multiple of n. Fill in the blank with the smallest possible prime answer a) [blank]_=2(mod11) b) 12_=[blank](mod11) c)231_=0(mod[blank])

a) x-2 = 11 x=13 13/11 = (13/2) b) 12-x = 11 x=1 12/11 = 1 c) 231-0=x x = 231 = 3*7*11