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14 – The Later 19th Century – Arithmetization of Analysis The student will learn about the contributions to mathematics and mathematicians of the late 19th century. 1 §14-1 Sequel to Euclid Student Discussion. 2 §14-1 Sequel to Euclid “. . . A course in this material is very desirable for every perspective teacher of high-school geometry. The material is definitely elementary, but not easy, and is extremely fascinating.” 3 §14-2 Three Famous Problems Student Discussion. 4 §14-2 Construction Limits 1. Can construct only algebraic numbers. i.e. solutions to polynomial equations with rational coefficients. Example : x2 – 2 = 0 Note: non-algebraic numbers are transcendental numbers. 2. Can not construct roots of cubic equations with rational coefficients but with no rational roots. Descartes’ rational root test. Example 8x3 – 6x –1 = 0 Possible rational roots are ± 1, ± ½, ± ¼, and ± 1/8. 5 §14-2 Quadrature of a Circle 2 Reduces to the equation – s s 2 = π r 2 or s = r π r However, π is not an algebraic number and hence s2 = r2 cannot be constructed. x2 - π = 0 6 §14-2 Duplication Problem Reduces to the equation x 3 = 2 or x 3 – 2 = 0 But this has no rational roots and hence is not possible. 7 §14-2 Angle Trisection Trig Identity cos θ = 4 cos 3 (θ/3) – 3 cos (θ/3) Let θ = 60º and x = cos (θ/3) then the identity becomes: ½ = 4 x 3 – 3x or 8x 3 – 6x – 1 = 0 But this has no rational roots and hence is not possible. 8 §14 -3 Compass or Straightedge Student Discussion. 9 §14 -3 Compass Lorenzo Mascheroni and Georg Mohr All Euclidean constructions can be done by compass alone. Need only show: 1. Intersection of two lines. 2. Intersection of one line and a circle. 10 §14 -3 Straightedge Jean Victor Poncelet All Euclidean constructions can be done by straight edge alone in the presence of one circle with center. Fully developed by Jacob Steiner later. Need only show: 1. Intersection of one line and a circle. 2. Intersection of two circles. 11 §14 -3 Compass or Straightedge Abû’l-Wefâ proposed a straightedge and a rusty compass. Yet others used a two-edged ruler with sides not necessarily parallel. 12 §14- 4 Projective Geometry Student Discussion. 13 §14- 4 Poncelet Principle of duality Two points determine a line. Two lines determine a point. Principle of continuity – from a case proven in the real plane there is a continuation into the imaginary plane. 14 14-5 Analytic Geometry Student Discussion. 15 14-5 Julius Plücker Line Coordinates • A line is defined by the negative reciprocals of its x and y intercepts. • A point now becomes a “linear” equation. • A line becomes an ordered pair of real numbers. More later. 16 §14 - 6 N-Dimensional Geometry Student Comment 17 §14 - 6 N-Dimensional Geometry Hyperspace for n dimensions and n > 3. Emerged from analysis where analytic treatment could be extended to arbitrary many variables. n dimensional space has - • Points as ordered n-tuples (x 1, x 2, . . . , x n) • Metric d (x, y) = [(x 1–y1) 2 + (x 2–y2) 2 + . . . +(x n–yn) 2] • Sphere of radius r and center (a 1, a 2, . . . , a n ) (x 1–a1) 2 + (x 2–a2) 2 + . . . +(x n–an) 2 = r2 • Line through (x 1, x 2, . . . , x n) and (y 1, y 2, . . . , y n) (k (y 1–x1) 2, k (y 2–x2) 2, . . . , k (y n–xn) 2 ) k 0. 18 §14-7 Differential Geometry Student Discussion. 19 §14 – 8 Klein and the Erlanger Program Student Discussion. 20 §14 – 9 Arithmetization of Analysis Student Discussion. 21 §14–10 Weierstrass and Riemann Student Discussion. 22 §14–11 Cantor, Kronecker, and Poincaré Student Discussion. 23 §14–12 Kovalevsky, Noether and Scott Student Discussion. 24 §14–13 Prime Numbers Student Discussion. 25 §14–13 How many Prime Numbers Is there a formula to calculate the number of primes less than some given number? Consider the following: n Number of primes < n 10 4 Confirm this. 100 14 ... ... 10 9 50,847,534 10 10 455,052,511 n ??? n / ln n 26 §14–13 2n- 1 2 n - 1 generates primes: Composite n 2n - 1 1 1 2n # digits 2 3 10 4 3 7 20 7 4 31 30 10 5 63 40 13 ... ... ... 127 39 digit prime 10k 1 + 3k 521 How many digits? 216,091 64,828 digits 27 §14–13 2n 2 1 2n Fermat thought that 2 1 generated only primes: n 2 2n 1 1 5 2 17 Composite 3 257 4 65,537 5 4,294,967,297 Also composite for n = 145 and lots of others 28 §14–13 Palindromic Primes 11, 131, 151, . . . , 345676543, . . . There are no four digit palindromic primes. WHY? 11 is the only palindromic primes with an even number of digits. There are 5,172 five digit palindromic primes. Homework – find the smallest five digit prime. 29 Functions to Generate Primes f (n) = n 2 – n + 41 yields primes for n < 41. n 1 2 3 4 5 6 ... f (n) 41 43 47 53 61 71 f (n) = n 2 – 79 n + 160 yields primes. Homework – find a polynomial that yields all primes. 30 Twin Primes 2, 3 and 5, 7 and 11, 13 and 137, 139 and 1007, 1009 and infinitely many more. My new phone number is 2 5 · 5 3 · 11 · 191 Note 2, 5, 11 and 191 are the first of twin primes. Note 2, 5, 11, and 191 are all palindromic primes. Goldbach’s Conjecture – Every even integer > 2 can be written as a sum of two prime numbers. 1000 = 3 + 997 Homework – write 2002 as the sum of two primes. Goldbach Bingo 31 Poincaré’s Model Hyperbolic Geometry Normal points m o P n l Ideal points Ultra-ideal points 32 Assignment Papers presented from Chapters 11 and 12. 33