History of Mathematics
|
|
- Alaina Walsh
- 5 years ago
- Views:
Transcription
1 History of Mathematics Paul Yiu Department of Mathematics Florida Atlantic University Spring A: Newton s binomial series
2 Issac Newton ( ) obtained the binomial theorem and the area of a segment of the circle by interpolation. 1
3 Newton to Oldenberg, 1676 At the beginning of my mathematical studies, when I had met with the works of our celebrated Wallis, on considering the series by the intercalation of which he himself exhibits the area of the circle and the hyperbola, the fact that, in the series of curves whose common base or axis is x and the ordinates (1 x 2 ) 0 2, (1 x 2 ) 1 2, (1 x 2 ) 2 2, (1 x 2 ) 3 2, (1 x 2 ) 4 2, (1 x 2 ) 5 2, etc., if the areas of every other of them, namely, x, x 1 3 x3, x 2 3 x x5, x 2 3 x x5 1 7 x7, etc., could be interpolated, we should have the areas of the intermediate ones, of which the first (1 x 2 ) 1 2 is the circle: 2
4 Newton s discovery of the area of a segment of a circle in order to interpolate these series I noted that in all of them the first term was x, and that the second terms 0 3 x3, 1 3 x3, 2 3 x3, 3 3 x3, etc., were in arithmetical progression, and hence that the first two terms of the series to be intercalated ought to be x 1 3 (1 2 x3 ), x 1 3 (3 2 x3 ), x 1 3 (5 2 x3 ), etc. To intercalate the rest I began to reflect that the denominators 1, 3, 5, 7, etc. were in arithmetic progression, so that the numerical coefficients of the numerators only were still in need of investigation. But in the alternately given areas these were the figures of powers of the number 11, namely of these, 11 0, 11 1, 11 2, 11 3, 11 4, that is, first 1, then 1,1; thirdly, 1,2,1; fourthly 1,3,3,1; fifthly 1,4,6,4,1,etc. 3
5 Newton s discovery of the area of a segment of a circle And so I began to inquire how the remaining figures in these series could be derived from the first two given figures, and I found that on putting m for the second figure, the rest would be produced by continual multiplication of the terms of this series, m 0 m 1 m 2 m 3 m 4, etc For example, let m =4, and (m 1), that is 6 will be the third term, and (m 2), that is 4 the fourth, and 4 1 4m 3, that is 1 the fifth, and (m 4), that is 0 the sixth, at which term in this the case the series stops. 4
6 Newton s discovery of the binomial series Accordingly, I applied this rule for interposing series among series, and since, for the circle, the second terms was 1 3 (1 2 x3 ), I put m = 1 2, and the terms arising were , or 1 1 8, , or , , or 5 128, and so to infinity. Whence I came to understand that the area of the circular segment which I wanted was 1 2 x x x x x9 etc. 9 5
7 Newton s discovery of the area of a segment of a circle Accordingly, I applied this rule for interposing series among series, and since, for the circle, the second terms was 1 3 (1 2 x3 ), I put m = 1 2, and the terms arising were , or 1 1 8, , or , , or 5 128, and so to infinity. Whence I came to understand that the area of the circular segment which I wanted was 1 2 x x x x x9 etc x2 dx = x x x x x
8 Newton discovery of the binomial series And by the same reasoning the areas of the remaining curves, which were to be inserted, were likewise obtained: as also the area of the hyperbola and of the other alternate curves in this series (1 + x 2 ) 0 2, (1 + x 2 ) 1 2, (1 + x 2 ) 2 2, (1 + x 2 ) 3 2, etc. And the same theory serves to intercalate other series, and that through intervals of two or more terms when they are absent at the same time. This was my first entry upon these studies, and it had certainly escaped my memory, had I not a few weeks ago cast my eye back on some notes. 7
9 Newton s discovery of the binomial series But when I had learnt this, I immediately began to consider that the terms that is to say, (1 x 2 ) 0 2, (1 x 2 ) 1 2, (1 x 2 ) 2 2, (1 x 2 ) 3 2, etc., 1, 1 x 2, 1 2x 2 + x 4, 1 3x 2 +3x 4 x 6, etc. could be interpolated in the same way as the areas generated by them: and that nothing else was required for this purpose but to omit the denominators 1,3,5,7. etc., which are in the terms expressing the areas; 8
10 Newton s discovery of the binomial series this means that the coefficients of the terms of the quantity to be intercalated (1 x 2 ) 1 2, (1 x 2 ) 3 2, or in general, (1 x 2 ) m, arise by the continued multiplication of the terms of this series m m 1 2 m 2 3 m 3, etc., 4 (1 x 2 ) m = ( ) m ( x 2 ) k. k 9
11 Newton s discovery of the binomial series this means that the coefficients of the terms of the quantity to be intercalated (1 x 2 ) 1 2, (1 x 2 ) 3 2, or in general, (1 x 2 ) m, arise by the continued multiplication of the terms of this series so that (for example) m m 1 2 m 2 3 m 3, etc., 4 (1 x 2 ) m = ( ) m ( x 2 ) k. k (1 x 2 ) 1 2 was the value of x2 1 8 x x6, etc., (1 x 2 ) 3 2 of x x x6, etc., and (1 x 2 ) 1 3 was the value of x2 1 9 x x6, etc. 10
12 Newton s discovery of the binomial series So then the general reduction of radicals into infinite series by that rule, which I laid down at the beginning of my earlier letter, became known to me, and that before I was acquainted with the extraction of roots. But once this was known, that other could not long remain hidden from me. For in order to test these processes, I multiplied x2 1 8 x x6, etc. into itself; and it became 1 x 2, the remaining terms vanishing by the continuation of the series to infinity. And even so x2 1 9 x x6, etc. multiplied twice into itself also produced 1 x 2. 11
13 Newton s discovery of the binomial series And as this was not only sure proof of these conclusions so too it guided me to try whether, conversely, these series, which it thus affirmed to be roots of the quantity 1 x 2, might not be extracted out of it in an arithmetical manner. And the matter turned out well. 1 1 D.J.Struik, A Source Book of Mathematics, Princeton,
14 Newton s derivation of the exponential series ( ) Start with 1 1+t =1 t + t2 t 3 + +( 1) n t n +. (2) Integration yields x 1 log(1 + x) = 0 1+t dt = x x2 2 + x3 3 x ( 1)n x n+1 n +1 (3) Put y =log(1+x) so that x = e y 1, and try to solve for x in terms of y from the infinite series y = x x2 2 + x ( 1)n 1 x n n ( ) 13
15 Newton s derivation of the exponential series (3) Put y =log(1+x) so that x = e y 1, and try to solve for x in terms of y from the infinite series y = x x2 2 + x ( 1)n 1 x n +. ( ) n Put x = a 1 y + a 2 y 2 + a 3 y a n y n +. Note that in ( ) only the term x on the right hand side contains y. From this, a 1 =1. Again, in ( ), only the first two terms on the right hand side contain y 2. By comparing coefficients of y 2,wehave 0=a a2 1 = a = a 2 =
16 Newton s derivation of the exponential series (3) Put y =log(1+x) so that x = e y 1, and try to solve for x in terms of y from the infinite series y = x x2 2 + x ( 1)n 1 x n +. ( ) n Put x = a 1 y + a 2 y 2 + a 3 y a n y n +. Note that in ( ) only the term x on the right hand side contains y. From this, a 1 =1. Again, in ( ), only the first two terms on the right hand side contain y 2. By comparing coefficients of y 2,wehave 0=a a2 1 = a = a 2 = 1 2. Once more, in ( ), only the first three terms on the right hand side contain y 3. 0=a a 1a a3 1 = a = a = a 3 = 1 6. Exercise. Proceed one step further to show that a 4 =
17 Newton s derivation of the exponential series (4) Newton found the first few terms x = y y y y y5 + and confidently concluded that a n = 1 n! after the roots have been extracted to a suitable period, they may sometimes be extended at pleasure by observing the analogy of the series. Exercise. Write down the series for 1 1 t 2 =(1 t 2 ) 1 2 using the binomial theorem. Integrate to obtain the series for arcsin x: arcsin x = x x x x7 7 + Then invert to form the series sin y = y 1 3! y ! y5 1 7! y7 + + ( 1)n (2n +1)! y2n
18 History of Mathematics Paul Yiu Department of Mathematics Florida Atlantic University Spring B: Jacob Bernoulli s summation of the powers of natural numbers
19 Jakob Bernoulli Ars Conjectandi, 1713 Jakob Bernoulli arranged the binomial coefficients in the table below and made use of them to sum the powers of natural numbers Let the series of natural numbers 1, 2, 3, 4, 5, etc. up to n be given, and let it be required to find their sum, the sum of the squares, cubes, etc. Bernoulli then gave a simple derivation of the formula n = 1 2 n2 + n, for the sum of the first n natural numbers. He then continued. 1
20 Sums of squares of integers A term in the third column is generally taken to be (n 1)(n 2) 1 2 = n2 3n +2, 2 and the sum of all terms (that is, of all n2 3n+2 2 )is n(n 1)(n 2) = n3 3n 2 +2n, 6 and 1 2 n2 = n3 3n 2 +2n n 1; 2
21 Sums of squares of integers 1 but 3 2 n = n 2 n2 = n3 3n 2 +2n + 1; 6 n = 3 4 n n and 1=n. Substituting, we have 1 2 n2 = n3 3n 2 +2n 6 = 3n2 +3n 4 n = 1 6 n n n, of which the double n 2 (the sum of the squares of all n) 2 = 1 3 n n n. 2 More generally, ( ) ( k k + k+1 ) ( k + + n ( k) = n+1 k+1). This identity can be established by considering the number of (k +1) element subsets of {1, 2, 3,...,n+1}, noting that the greatest element m of each subset must be one of k +1, k +2,...,n +1, and that there are exactly ( ) m 1 k subsets with greatest element m. 3
22 Sum of cubes of integers A term of the fourth column is generally (n 1)(n 2)(n 3) and the sum of all terms is n(n 1)(n 2)(n 3) it must certainly be that n3 n n = n3 6n 2 +11n 6, 6 = n4 6n 3 +11n 2 6n. 24 1= n4 6n 2 +11n 2 6n. 24 Hence, 1 6 n3 = n4 6n 3 +11n 2 6n + 24 n n
23 Sums of cubes of integers 1 6 n3 = n4 6n 3 +11n 2 6n 24 + n 2... When all substitutions are made, the following results: n 3 = 1 4 n n n n
24 Thus, we can step by step reach higher and higher powers and with slight effort form the following table: n 1 = 1 2 n n, n 2 = 1 3 n n n, n 3 = 1 4 n n n2 n 4 = 1 5 n n n n n 5 = 1 6 n n n n2 n 6 = 1 7 n n n5 1 6 n n n 7 = 1 8 n n n n n2 n 8 = 1 9 n n n n n n n 9 = 1 10 n n n n n n2 n 10 = 1 11 n n n9 n 7 +n n n. Whoever will examine the series as to their regularity may be able to continue the table. 6
25 Sums of powers of integers Taking c to be the power of any exponent, the sum of all n c,or n c 1 = c +1 nc nc + c c(c 1)(c 2) 2 Anc 1 + Bn c c(c 1)(c 2)(c 3)(c 4) + Cn c c(c 1)(c 2)(c 3)(c 4)(c 5)(c 6) + Dn c 7 +, the exponents of n continually decreasing by 2 until n or n 2 is reached. The capital letters A, B, C, D denote in order the coefficients of the last terms in the expressions for n 2, n 4, n 6, n 8 etc., namely, A = 1 6, B = 1 30, C = 1 42, D = The coefficients are such that each one completes the others in the same expression to unity. Thus, D must have the value 1 30, because =1. 7
26 Reorganization Let S k (n) :=1 k +2 k + + n k. We make use of the series expansion of e z : e z =1+z + z2 zk + + 2! k! + and analogous series for e 2z, e 3z,...,e nz : e z = 1+z + z2 zk + + 2! k! + e 2z = 1+2z + 22 z k z k + 2! k!. e (n 1)z = 1+(n 1)z + (n 1)2 z 2 2! Combining these (with 1=1), we have n + S 1 (n 1)z + S 2(n 1)z 2 for the series expansion of 2! + + (n 1)k z k k! + + S k(n 1)z k k! 1+e z + e 2z + + e (n 1)z = enz 1 e z
27 The Bernoulli numbers and sums of powers of integers The series expansion of the function on the right hand side can be found from e nz 1 e z 1 = enz 1 z z e z 1. (1) The first factor clearly has series expansion e nz 1 z = n + n2 nk+1 z + + 2! (k +1)! zk + (2) Suppose we write z e z 1 = B 0 + B 1 1! z + B 2 2! z2 + + B k k! zk + These coefficients B 0, B 1,...,B k,... are called the Bernoulli numbers. The product of these two series has beginning term n, and subsequently, for each k =1, 2,..., the coefficient of z k equal to B i i! n j+1 (j +1)! = 1 k ( ) k +1 B i n k+1 i (k +1)! i i+j=k Now comparison gives S k (n 1) = 1 k +1 i=0 i=0 k ( ) k +1 B i n k+1 i. i 9
28 10
29 Bernoulli numbers and sum of powers of integers (1) If we agree, in the binomial expansion of (B +1) k+1, to replace every B i by the number B i, then the above formula can be simply written as S k (n 1) = (B + n)k+1 B k+1. ( ) k +1 (2) This symbolic device can further help computing S k (n 1) easily, without actually calculating the Bernoulli numbers. Comparing ( ) with k S k 1 (n 1) = (B + n) k B k, we note that formally S k (n 1) = ks k 1 (n 1)dn + cn where the constant c so chosen that the sum of the coefficients is equal to 0. 11
30 Sums of powers of integers (3) An analogous expression holds for S k (n): S k (n) = ks k 1 (n)dn + cn where the constant c so chosen that the sum of the coefficients is equal to 1. Examples: Since S 2 (n) = 1 3 n n n,wehave S 3 (n) = (n n n)dn + c 3n (c 3 appropriately chosen) = 1 4 n n n2 ; (c 3 =0) S 4 (n) = (n 4 +2n 3 + n 2 )dn + c 4 n (c 4 appropriately chosen) = 1 5 n n n n; S 5 (n) = (n n n3 16 ) n dn + c 5 n (c 5 appropriately chosen) = 1 6 n n n n2 ; (c 5 =0). 12
31 The Bernoulli numbers The Bernoulli numbers B n, n =0, 1,...,are, by definition, the coefficients in the series expansion of z e z 1 : z e z 1 = B 0 + B 1 z + B 2 2! z2 + B 3 3! z3 + + B k k! zk +. Clearly, B 0 =1. It is quite easy to check that B 1 = 1 2 (exercise). One important observation is that all subsequent odd-indexed Bernoulli numbers are zero. In terms of the even-indexed Bernoulli numbers, we have z e z 1 =1 z 2 + B 2k (2k)! z2k. k=1 Here are the beginning even-indexed Bernoulli numbers: B 2 = 1 6, B 4 = 1 30, B 6 = 1 42, B 8 = 1 30, B 10 = 5 66, B 12 =
32 History of Mathematics Paul Yiu Department of Mathematics Florida Atlantic University Spring C: Euler s first calculation of k k k k +
33 Euler s 1734 on sums of reciprocals of even powers Paper 41: De Summis Serierum Reciprocarum. Euler begins with the series for sine and cosine. y = s s3 3! + s5 5! s7 7! +, x = 1 s2 2! + s4 4! s6 6! +. If, for a fixed y, the roots of the equation 0=1 s y + s3 3!y s5 5!y + are A, B, C, D, E etc., then the infinite polynomial factors into the product of 1 s A, 1 s B, 1 s C, 1 s D,... 1
34 1 s (1 y + s3 3!y s5 5!y + = s )( 1 s ) ( 1 s ). A B D By comparing coefficients, 1 y = 1 A + 1 B + 1 C + 1 D + 2
35 1 s (1 y + s3 3!y s5 5!y + = s )( 1 s ) ( 1 s ). A B D By comparing coefficients, 1 y = 1 A + 1 B + 1 C + 1 D + If A is an acute angle (smallest arc) with sin A = y, then all the angles with sine equal to y are A, ±π A, ±2π + A, ±3π A, ±4π + A,... The sums of these reciprocals is 1 y. Sum of these reciprocals taking two at a time is 0, taking three at a time is 1 3!y. etc. 3
36 In general, if a + b + c + d + e + f + = α, ab + ac + bc + = β, abc + abd + bcd + = γ, then a 2 + b 2 + c 2 + = α 2 2β, a 3 + b 3 + c 3 + = α 3 3αβ +3γ, and a 4 + b 4 + c 4 + = α 4 4α 2 β +4αγ +2β 2 4δ. 4
37 Euler denotes by P the sum of the numbers, Q the sum of the squares, R the sum of cubes, S the sum of fourth powers etc., and writes Now he applies these to P = α, Q = Pα 2β, R = Qα Pβ +3γ, S = Rα Qβ + Pγ 4δ,. α = 1 1, β =0, γ = y 3!y, δ =0, ɛ = 1 5!y,... and obtains P = 1 y, Q = 1 y 2, R = Q y 1 2!y, S = R y P 3!y,... 5
38 The Leibziz series Now Euler takes y =1. For the equation 1=s s3 3! + s5 5! s7 7! +, all the angles with sines equal to 1 are From these, π 2, π 2, 3π 2, 3π 2, 5π 2, 5π 2, 7π 2, 7π 2, 9π 2, 9π 2,... ( π ) 11 + =1. This gives Leibniz s famous series = π 4. 6
39 Noting that P = α =1, β =0, so that Q = P =1, Euler proceeded to obtain π2 + = From this, he deduced that π2 + =
40 Continuing with R = 1 2,S= 1 3,T= 5 24,V= 2 15,W= 6 17,X= ,... Euler obtained π3 = , π4 + = , π4 + = , π5 = , π6 + = , π6 + = ,. 8
41 Euler gave an alternative method of determining these sums. This time, he considered y =0and made use of 0=s s3 3! + s5 5! s7 7! +. Removing the obvious factor s, he noted that the roots of are 0=1 s2 3! + s4 5! s6 7! + ±π, ±2π, ±3π,... so that ) 1 (1 s2 3! +s4 5! s6 7! + = )(1 s2 )(1 s2 )(1 s2 s2 π 2 4π 2 9π 2 16π 2 From these, the sums n=1 1 n 2k can be computed successively for k = 1, 2, 3,... 9
42 Euler then summarized the formulas which made him famous throughout Europe (for the first time): π2 + = , π4 + = , π6 + = , π8 + = , π10 + = , = 691π ,. 10
43 On the other hand, nothing is known about n=1 1 n k when the exponent k is odd. In 1978, L. Apery proved that n=1 1 n is an irrational number. 3 11
On the general Term of hypergeometric Series *
On the general Term of hypergeometric Series * Leonhard Euler 1 Here, following Wallis I call series hypergeometric, whose general terms proceed according to continuously multiplied factors, while the
More informationSUMS OF POWERS AND BERNOULLI NUMBERS
SUMS OF POWERS AND BERNOULLI NUMBERS TOM RIKE OAKLAND HIGH SCHOOL Fermat and Pascal On September 22, 636 Fermat claimed in a letter that he could find the area under any higher parabola and Roberval wrote
More informationMathematics 324 Riemann Zeta Function August 5, 2005
Mathematics 324 Riemann Zeta Function August 5, 25 In this note we give an introduction to the Riemann zeta function, which connects the ideas of real analysis with the arithmetic of the integers. Define
More informationIntroduction to Series and Sequences Math 121 Calculus II Spring 2015
Introduction to Series and Sequences Math Calculus II Spring 05 The goal. The main purpose of our study of series and sequences is to understand power series. A power series is like a polynomial of infinite
More informationMath 200 University of Connecticut
IRRATIONALITY OF π AND e KEITH CONRAD Math 2 University of Connecticut Date: Aug. 3, 25. Contents. Introduction 2. Irrationality of π 2 3. Irrationality of e 3 4. General Ideas 4 5. Irrationality of rational
More informationB.Sc. MATHEMATICS I YEAR
B.Sc. MATHEMATICS I YEAR DJMB : ALGEBRA AND SEQUENCES AND SERIES SYLLABUS Unit I: Theory of equation: Every equation f(x) = 0 of n th degree has n roots, Symmetric functions of the roots in terms of the
More informationThe Not-Formula Book for C2 Everything you need to know for Core 2 that won t be in the formula book Examination Board: AQA
Not The Not-Formula Book for C Everything you need to know for Core that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) A.J.Hobson
JUST THE MATHS UNIT NUMBER.5 DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) by A.J.Hobson.5. Maclaurin s series.5. Standard series.5.3 Taylor s series.5.4 Exercises.5.5 Answers to exercises
More informationSERIES
SERIES.... This chapter revisits sequences arithmetic then geometric to see how these ideas can be extended, and how they occur in other contexts. A sequence is a list of ordered numbers, whereas a series
More informationSTUDY GUIDE Math 20. To accompany Intermediate Algebra for College Students By Robert Blitzer, Third Edition
STUDY GUIDE Math 0 To the students: To accompany Intermediate Algebra for College Students By Robert Blitzer, Third Edition When you study Algebra, the material is presented to you in a logical sequence.
More informationTAYLOR AND MACLAURIN SERIES
TAYLOR AND MACLAURIN SERIES. Introduction Last time, we were able to represent a certain restricted class of functions as power series. This leads us to the question: can we represent more general functions
More informationPower series and Taylor series
Power series and Taylor series D. DeTurck University of Pennsylvania March 29, 2018 D. DeTurck Math 104 002 2018A: Series 1 / 42 Series First... a review of what we have done so far: 1 We examined series
More informationC. Complex Numbers. 1. Complex arithmetic.
C. Complex Numbers. Complex arithmetic. Most people think that complex numbers arose from attempts to solve quadratic equations, but actually it was in connection with cubic equations they first appeared.
More informationMTS 105 LECTURE 5: SEQUENCE AND SERIES 1.0 SEQUENCE
MTS 105 LECTURE 5: SEQUENCE AND SERIES 1.0 SEQUENCE A sequence is an endless succession of numbers placed in a certain order so that there is a first number, a second and so on. Consider, for example,
More informationSeveral Remarks on Infinite Series
Several Remarks on Infinite Series Comment # 72 from Enestroemiani s Index Commentarii academiae scientarum Petropolitanae 9 (737), 744, p. 60 88 Leonard Euler The remarks I have decided to present here
More informationCoach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers
Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}
More informationContents. CHAPTER P Prerequisites 1. CHAPTER 1 Functions and Graphs 69. P.1 Real Numbers 1. P.2 Cartesian Coordinate System 14
CHAPTER P Prerequisites 1 P.1 Real Numbers 1 Representing Real Numbers ~ Order and Interval Notation ~ Basic Properties of Algebra ~ Integer Exponents ~ Scientific Notation P.2 Cartesian Coordinate System
More informationOn the remarkable Properties of the Coefficients which occur in the Expansion of the binomial raised to an arbitrary power *
On the remarkable Properties of the Coefficients which occur in the Expansion of the binomial raised to an arbitrary power * Leonhard Euler THEOREM If for the power of the binomial raised to the exponent
More informationOn Maxima and Minima *
On Maxima and Minima * Leonhard Euler 50 If a function of x was of such a nature, that while the values of x increase the function itself continuously increases or decreases, then this function will have
More informationBaltic Way 2008 Gdańsk, November 8, 2008
Baltic Way 008 Gdańsk, November 8, 008 Problems and solutions Problem 1. Determine all polynomials p(x) with real coefficients such that p((x + 1) ) = (p(x) + 1) and p(0) = 0. Answer: p(x) = x. Solution:
More informationP.1: Algebraic Expressions, Mathematical Models, and Real Numbers
Chapter P Prerequisites: Fundamental Concepts of Algebra Pre-calculus notes Date: P.1: Algebraic Expressions, Mathematical Models, and Real Numbers Algebraic expression: a combination of variables and
More informationAS1051: Mathematics. 0. Introduction
AS1051: Mathematics 0 Introduction The aim of this course is to review the basic mathematics which you have already learnt during A-level, and then develop it further You should find it almost entirely
More informationCOMPLEX ANALYSIS TOPIC V: HISTORICAL REMARKS
COMPLEX ANALYSIS TOPIC V: HISTORICAL REMARKS PAUL L. BAILEY Historical Background Reference: http://math.fullerton.edu/mathews/n2003/complexnumberorigin.html Rafael Bombelli (Italian 1526-1572) Recall
More informationAlgebra 2. Chapter 4 Exponential and Logarithmic Functions. Chapter 1 Foundations for Functions. Chapter 3 Polynomial Functions
Algebra 2 Chapter 1 Foundations for Chapter 2 Quadratic Chapter 3 Polynomial Chapter 4 Exponential and Logarithmic Chapter 5 Rational and Radical Chapter 6 Properties and Attributes of Chapter 7 Probability
More information1. The positive zero of y = x 2 + 2x 3/5 is, to the nearest tenth, equal to
SAT II - Math Level Test #0 Solution SAT II - Math Level Test No. 1. The positive zero of y = x + x 3/5 is, to the nearest tenth, equal to (A) 0.8 (B) 0.7 + 1.1i (C) 0.7 (D) 0.3 (E). 3 b b 4ac Using Quadratic
More informationFundamentals. Introduction. 1.1 Sets, inequalities, absolute value and properties of real numbers
Introduction This first chapter reviews some of the presumed knowledge for the course that is, mathematical knowledge that you must be familiar with before delving fully into the Mathematics Higher Level
More informationCHAPTER 1: Functions
CHAPTER 1: Functions 1.1: Functions 1.2: Graphs of Functions 1.3: Basic Graphs and Symmetry 1.4: Transformations 1.5: Piecewise-Defined Functions; Limits and Continuity in Calculus 1.6: Combining Functions
More informationUnit 1. Revisiting Parent Functions and Graphing
Unit 1 Revisiting Parent Functions and Graphing Precalculus Analysis Pacing Guide First Nine Weeks Understand how the algebraic properties of an equation transform the geometric properties of its graph.
More informationOn Various Ways of Approximating the Quadrature of a Circle by Numbers. 1 Author Leonh. Euler
On Various Ways of Approximating the Quadrature of a Circle by Numbers. Author Leonh. Euler Translated and Annotated by Thomas W. Polaski. Archimedes and those who followed him investigated the approximate
More informationRon Paul Curriculum Mathematics 8 Lesson List
Ron Paul Curriculum Mathematics 8 Lesson List 1 Introduction 2 Algebraic Addition 3 Algebraic Subtraction 4 Algebraic Multiplication 5 Week 1 Review 6 Algebraic Division 7 Powers and Exponents 8 Order
More informationClick here for answers. Click here for answers. f x
CHLLENGE PROBLEM Challenge Problems Chapter 3 Click here for answers.. (a) Find the domain of the function f x s s s3 x. (b) Find f x. ; (c) Check your work in parts (a) and (b) by graphing f and f on
More information80 Wyner PreCalculus Spring 2017
80 Wyner PreCalculus Spring 2017 CHAPTER NINE: DERIVATIVES Review May 16 Test May 23 Calculus begins with the study of rates of change, called derivatives. For example, the derivative of velocity is acceleration
More informationRadical Expressions, Equations, and Functions
Radical Expressions, Equations, and Functions 0 Real-World Application An observation deck near the top of the Sears Tower in Chicago is 353 ft high. How far can a tourist see to the horizon from this
More informationSums and Products. a i = a 1. i=1. a i = a i a n. n 1
Sums and Products -27-209 In this section, I ll review the notation for sums and products Addition and multiplication are binary operations: They operate on two numbers at a time If you want to add or
More information1. Find the Taylor series expansion about 0 of the following functions:
MAP 4305 Section 0642 3 Intermediate Differential Equations Assignment 1 Solutions 1. Find the Taylor series expansion about 0 of the following functions: (i) f(z) = ln 1 z 1+z (ii) g(z) = 1 cos z z 2
More informationInstructions. Do not open your test until instructed to do so!
st Annual King s College Math Competition King s College welcomes you to this year s mathematics competition and to our campus. We wish you success in this competition and in your future studies. Instructions
More information16. . Proceeding similarly, we get a 2 = 52 1 = , a 3 = 53 1 = and a 4 = 54 1 = 125
. Sequences When we first introduced a function as a special type of relation in Section.3, we did not put any restrictions on the domain of the function. All we said was that the set of x-coordinates
More informationHomework Points Notes Points
A2 NBC (13) 13.1 p 876 #1-8 13.2 p 882 #1-7 13.3 p 888 #1-11 13.4 p 897 #1-11 13.5 p 904 #2-20 even 13.1 Trig Identities 13.2 Verifying Trig Identities 13.3 Sum and Difference... 13.4 Double and Half...
More informationSequences and the Binomial Theorem
Chapter 9 Sequences and the Binomial Theorem 9. Sequences When we first introduced a function as a special type of relation in Section.3, we did not put any restrictions on the domain of the function.
More informationNumbers and their divisors
Chapter 1 Numbers and their divisors 1.1 Some number theoretic functions Theorem 1.1 (Fundamental Theorem of Arithmetic). Every positive integer > 1 is uniquely the product of distinct prime powers: n
More informationPUTNAM TRAINING NUMBER THEORY. Exercises 1. Show that the sum of two consecutive primes is never twice a prime.
PUTNAM TRAINING NUMBER THEORY (Last updated: December 11, 2017) Remark. This is a list of exercises on Number Theory. Miguel A. Lerma Exercises 1. Show that the sum of two consecutive primes is never twice
More informationChapter 1. Making algebra orderly with the order of operations and other properties Enlisting rules of exponents Focusing on factoring
In This Chapter Chapter 1 Making Advances in Algebra Making algebra orderly with the order of operations and other properties Enlisting rules of exponents Focusing on factoring Algebra is a branch of mathematics
More informationPre AP Algebra. Mathematics Standards of Learning Curriculum Framework 2009: Pre AP Algebra
Pre AP Algebra Mathematics Standards of Learning Curriculum Framework 2009: Pre AP Algebra 1 The content of the mathematics standards is intended to support the following five goals for students: becoming
More information3.4 Introduction to power series
3.4 Introduction to power series Definition 3.4.. A polynomial in the variable x is an expression of the form n a i x i = a 0 + a x + a 2 x 2 + + a n x n + a n x n i=0 or a n x n + a n x n + + a 2 x 2
More informationMATH Spring 2010 Topics per Section
MATH 101 - Spring 2010 Topics per Section Chapter 1 : These are the topics in ALEKS covered by each Section of the book. Section 1.1 : Section 1.2 : Ordering integers Plotting integers on a number line
More informationAdvanced Math. ABSOLUTE VALUE - The distance of a number from zero; the positive value of a number. < 2 indexes draw 2 lines down like the symbol>
Advanced Math ABSOLUTE VALUE - The distance of a number from zero; the positive value of a number. < 2 indexes draw 2 lines down like the symbol> ALGEBRA - A branch of mathematics in which symbols, usually
More informationExecutive Assessment. Executive Assessment Math Review. Section 1.0, Arithmetic, includes the following topics:
Executive Assessment Math Review Although the following provides a review of some of the mathematical concepts of arithmetic and algebra, it is not intended to be a textbook. You should use this chapter
More informationMATH 117 LECTURE NOTES
MATH 117 LECTURE NOTES XIN ZHOU Abstract. This is the set of lecture notes for Math 117 during Fall quarter of 2017 at UC Santa Barbara. The lectures follow closely the textbook [1]. Contents 1. The set
More informationNumerical Analysis Exam with Solutions
Numerical Analysis Exam with Solutions Richard T. Bumby Fall 000 June 13, 001 You are expected to have books, notes and calculators available, but computers of telephones are not to be used during the
More informationb = 2, c = 3, we get x = 0.3 for the positive root. Ans. (D) x 2-2x - 8 < 0, or (x - 4)(x + 2) < 0, Therefore -2 < x < 4 Ans. (C)
SAT II - Math Level 2 Test #02 Solution 1. The positive zero of y = x 2 + 2x is, to the nearest tenth, equal to (A) 0.8 (B) 0.7 + 1.1i (C) 0.7 (D) 0.3 (E) 2.2 ± Using Quadratic formula, x =, with a = 1,
More informationMATH 104, HOMEWORK #3 SOLUTIONS Due Thursday, February 4
MATH 104, HOMEWORK #3 SOLUTIONS Due Thursday, February 4 Remember, consult the Homework Guidelines for general instructions. GRADED HOMEWORK: 1. Give direct proofs for the two following its. Do not use
More informationOn the Interpolation of series *
On the Interpolation of series * Leonhard Euler 389 A series is said to be interpolated, if its terms are assigned which correspond to fractional or even surdic indices. Therefore, if the general term
More informationRadical Expressions and Graphs 8.1 Find roots of numbers. squaring square Objectives root cube roots fourth roots
8. Radical Expressions and Graphs Objectives Find roots of numbers. Find roots of numbers. The opposite (or inverse) of squaring a number is taking its square root. Find principal roots. Graph functions
More informationMTH 05. Basic Concepts of Mathematics I
MTH 05. Basic Concepts of Mathematics I Uma N. Iyer With Appendices by Sharon Persinger and Anthony Weaver Department of Mathematics and Computer Science Bronx Community College ii To my parents and teachers
More informationCatholic Central High School
Catholic Central High School Course: Basic Algebra 2 Department: Mathematics Length: One year Credit: 1 Prerequisite: Completion of Basic Algebra 1 or Algebra 1, Basic Plane Geometry or Plane Geometry,
More informationNYS Algebra II and Trigonometry Suggested Sequence of Units (P.I's within each unit are NOT in any suggested order)
1 of 6 UNIT P.I. 1 - INTEGERS 1 A2.A.1 Solve absolute value equations and inequalities involving linear expressions in one variable 1 A2.A.4 * Solve quadratic inequalities in one and two variables, algebraically
More informationChapter 3: Polynomial and Rational Functions
.1 Power and Polynomial Functions 155 Chapter : Polynomial and Rational Functions Section.1 Power Functions & Polynomial Functions... 155 Section. Quadratic Functions... 16 Section. Graphs of Polynomial
More informationChapter 3: Factors, Roots, and Powers
Chapter 3: Factors, Roots, and Powers Section 3.1 Chapter 3: Factors, Roots, and Powers Section 3.1: Factors and Multiples of Whole Numbers Terminology: Prime Numbers: Any natural number that has exactly
More informationAlgebraic. techniques1
techniques Algebraic An electrician, a bank worker, a plumber and so on all have tools of their trade. Without these tools, and a good working knowledge of how to use them, it would be impossible for them
More informationArithmetic. Integers: Any positive or negative whole number including zero
Arithmetic Integers: Any positive or negative whole number including zero Rules of integer calculations: Adding Same signs add and keep sign Different signs subtract absolute values and keep the sign of
More informationSolution Week 68 (12/29/03) Tower of circles
Solution Week 68 (/9/03) Tower of circles Let the bottom circle have radius, and let the second circle have radius r. From the following figure, we have sin β = r, where β α/. () + r α r r -r β = α/ In
More informationUnit 1. Revisiting Parent Functions and Graphing
Unit 1 Revisiting Parent Functions and Graphing Revisiting Statistics (Measures of Center and Spread, Standard Deviation, Normal Distribution, and Z-Scores Graphing abs(f(x)) and f(abs(x)) with the Definition
More informationCALC 2 CONCEPT PACKET Complete
CALC 2 CONCEPT PACKET Complete Written by Jeremy Robinson, Head Instructor Find Out More +Private Instruction +Review Sessions WWW.GRADEPEAK.COM Need Help? Online Private Instruction Anytime, Anywhere
More informationMTH 06. Basic Concepts of Mathematics II. Uma N. Iyer Department of Mathematics and Computer Science Bronx Community College
MTH 06. Basic Concepts of Mathematics II Uma N. Iyer Department of Mathematics and Computer Science Bronx Community College ii To my parents and teachers IthankAnthonyWeaverforeditingthisbook. Acknowledgements
More information1 + lim. n n+1. f(x) = x + 1, x 1. and we check that f is increasing, instead. Using the quotient rule, we easily find that. 1 (x + 1) 1 x (x + 1) 2 =
Chapter 5 Sequences and series 5. Sequences Definition 5. (Sequence). A sequence is a function which is defined on the set N of natural numbers. Since such a function is uniquely determined by its values
More informationMath Academy I Fall Study Guide. CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8
Name: Math Academy I Fall Study Guide CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8 1-A Terminology natural integer rational real complex irrational imaginary term expression argument monomial degree
More informationA video College Algebra course & 6 Enrichment videos
A video College Algebra course & 6 Enrichment videos Recorded at the University of Missouri Kansas City in 1998. All times are approximate. About 43 hours total. Available on YouTube at http://www.youtube.com/user/umkc
More informationTennessee s State Mathematics Standards Precalculus
Tennessee s State Mathematics Standards Precalculus Domain Cluster Standard Number Expressions (N-NE) Represent, interpret, compare, and simplify number expressions 1. Use the laws of exponents and logarithms
More informationAlgebra III INSTRUCTIONAL PACING GUIDE (Days Based on 90 minutes)
EA, IA, PC-1. Connect algebra and trigonometry with other branches of mathematics. EA, IA, PC-1.7 G-1. G-1.8 G-1.9 Understand how to represent algebraic and trigonometric relationships by using tools such
More informationSHOW ALL YOUR WORK IN A NEAT AND ORGANIZED FASHION
Intermediate Algebra TEST 1 Spring 014 NAME: Score /100 Please print SHOW ALL YOUR WORK IN A NEAT AND ORGANIZED FASHION Course Average No Decimals No mixed numbers No complex fractions No boxed or circled
More informationNotes for Expansions/Series and Differential Equations
Notes for Expansions/Series and Differential Equations In the last discussion, we considered perturbation methods for constructing solutions/roots of algebraic equations. Three types of problems were illustrated
More informationGeneral and Specific Learning Outcomes by Strand Applied Mathematics
General and Specific Learning Outcomes by Strand Applied Mathematics Number Develop number sense. Develop number sense. Specific Learning Outcomes Specific Learning Outcomes Specific Learning Outcomes
More informationMATHEMATICS HIGHER SECONDARY FIRST YEAR VOLUME II REVISED BASED ON THE RECOMMENDATIONS OF THE TEXT BOOK DEVELOPMENT COMMITTEE
MATHEMATICS HIGHER SECONDARY FIRST YEAR VOLUME II REVISED BASED ON THE RECOMMENDATIONS OF THE TEXT BOOK DEVELOPMENT COMMITTEE Untouchability is a sin Untouchability is a crime Untouchability is inhuman
More informationPRIME NUMBERS YANKI LEKILI
PRIME NUMBERS YANKI LEKILI We denote by N the set of natural numbers: 1,2,..., These are constructed using Peano axioms. We will not get into the philosophical questions related to this and simply assume
More informationI-2 Index. Coterminal Angles, 738 Counting numbers, 8 Cramer, Gabriel, 309 Cramer s rule, 306 Cube root, 427, 434 Cylinder, right circular, 117
Index Absolute value, 18 equations, 154, 162 inequalities, 159, 162 Absolute error, 158 Addition, 4 associative property, 19 commutative property, 18 of complex numbers, 481 of fractions, 21 of functions,
More informationPre-calculus 12 Curriculum Outcomes Framework (110 hours)
Curriculum Outcomes Framework (110 hours) Trigonometry (T) (35 40 hours) General Curriculum Outcome: Students will be expected to develop trigonometric reasoning. T01 Students will be expected to T01.01
More informationInteger (positive or negative whole numbers or zero) arithmetic
Integer (positive or negative whole numbers or zero) arithmetic The number line helps to visualize the process. The exercises below include the answers but see if you agree with them and if not try to
More informationFundamentals. Copyright Cengage Learning. All rights reserved.
Fundamentals Copyright Cengage Learning. All rights reserved. 1.2 Exponents and Radicals Copyright Cengage Learning. All rights reserved. Objectives Integer Exponents Rules for Working with Exponents Scientific
More information1 Introduction to Ramanujan theta functions
A Multisection of q-series Michael Somos 30 Jan 2017 ms639@georgetown.edu (draft version 34) 1 Introduction to Ramanujan theta functions Ramanujan used an approach to q-series which is general and is suggestive
More informationCenterville High School Curriculum Mapping Algebra II 1 st Nine Weeks
Centerville High School Curriculum Mapping Algebra II 1 st Nine Weeks Chapter/ Lesson Common Core Standard(s) 1-1 SMP1 1. How do you use a number line to graph and order real numbers? 2. How do you identify
More informationMATH 2200 Final LC Review
MATH 2200 Final LC Review Thomas Goller April 25, 2013 1 Final LC Format The final learning celebration will consist of 12-15 claims to be proven or disproven. It will take place on Wednesday, May 1, from
More informationFinding Prime Factors
Section 3.2 PRE-ACTIVITY PREPARATION Finding Prime Factors Note: While this section on fi nding prime factors does not include fraction notation, it does address an intermediate and necessary concept to
More informationPrecalculus AB Honors Pacing Guide First Nine Weeks Unit 1. Tennessee State Math Standards
Precalculus AB Honors Pacing Guide First Nine Weeks Unit 1 Revisiting Parent Functions and Graphing P.F.BF.A.1 Understand how the algebraic properties of an equation transform the geometric properties
More informationIntermediate Math Circles March 11, 2009 Sequences and Series
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Intermediate Math Circles March 11, 009 Sequences and Series Tower of Hanoi The Tower of Hanoi is a game
More informationPacing Guide. Algebra II. Robert E. Lee High School Staunton City Schools Staunton, Virginia
Pacing Guide Algebra II Robert E. Lee High School Staunton City Schools Staunton, Virginia 2010-2011 Algebra 2 - Pacing Guide 2010 2011 (SOL# s for the 2011 2012 school year are indicated in parentheses
More information1 Question related to polynomials
07-08 MATH00J Lecture 6: Taylor Series Charles Li Warning: Skip the material involving the estimation of error term Reference: APEX Calculus This lecture introduced Taylor Polynomial and Taylor Series
More informationMath Review. for the Quantitative Reasoning measure of the GRE General Test
Math Review for the Quantitative Reasoning measure of the GRE General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important for solving
More informationTime: 1 hour 30 minutes
Paper Reference(s) 6664/01 Edexcel GCE Core Mathematics C Silver Level S3 Time: 1 hour 30 minutes Materials required for examination Mathematical Formulae (Green) Items included with question papers Nil
More informationCHAPTER 1 NUMBER SYSTEMS. 1.1 Introduction
N UMBER S YSTEMS NUMBER SYSTEMS CHAPTER. Introduction In your earlier classes, you have learnt about the number line and how to represent various types of numbers on it (see Fig..). Fig.. : The number
More informationMATH141: Calculus II Exam #4 review solutions 7/20/2017 Page 1
MATH4: Calculus II Exam #4 review solutions 7/0/07 Page. The limaçon r = + sin θ came up on Quiz. Find the area inside the loop of it. Solution. The loop is the section of the graph in between its two
More informationIntroduction to Algebra
Translate verbal expressions into mathematics expressions. Write an expression containing identical factors as an expression using exponents. Understand and apply the rules for order of operations to evaluate
More informationCHAPTER 2 Review of Algebra
CHAPTER 2 Review of Algebra 2.1 Real Numbers The text assumes that you know these sets of numbers. You are asked to characterize each set in Problem 1 of Chapter 1 review. Name Symbol Set Examples Counting
More informationIn Exercises 1 12, list the all of the elements of the given set. 2. The set of all positive integers whose square roots are less than or equal to 3
APPENDIX A EXERCISES In Exercises 1 12, list the all of the elements of the given set. 1. The set of all prime numbers less than 20 2. The set of all positive integers whose square roots are less than
More informationAlgebra 2. Units of Study. Broward County Public Schools
Algebra 2 1200330 Units of Study Broward County Public Schools Mathematical Concepts - Year at a Glance Quarter 1 Quarter 2 Quarter 3 Quarter 4 Foundations for - Introduction to Parent - Exploring Transformations
More informationBernoulli Numbers and their Applications
Bernoulli Numbers and their Applications James B Silva Abstract The Bernoulli numbers are a set of numbers that were discovered by Jacob Bernoulli (654-75). This set of numbers holds a deep relationship
More informationxvi xxiii xxvi Construction of the Real Line 2 Is Every Real Number Rational? 3 Problems Algebra of the Real Numbers 7
About the Author v Preface to the Instructor xvi WileyPLUS xxii Acknowledgments xxiii Preface to the Student xxvi 1 The Real Numbers 1 1.1 The Real Line 2 Construction of the Real Line 2 Is Every Real
More information1 Chapter 2 Perform arithmetic operations with polynomial expressions containing rational coefficients 2-2, 2-3, 2-4
NYS Performance Indicators Chapter Learning Objectives Text Sections Days A.N. Perform arithmetic operations with polynomial expressions containing rational coefficients. -, -5 A.A. Solve absolute value
More information1 Numbers. exponential functions, such as x 7! a x ; where a; x 2 R; trigonometric functions, such as x 7! sin x; where x 2 R; ffiffi x ; where x 0:
Numbers In this book we study the properties of real functions defined on intervals of the real line (possibly the whole real line) and whose image also lies on the real line. In other words, they map
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
More information