Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Euler’s Elegant Equation The five most important constants in mathematics, and the amazing equation that unites them. 0011 0010 1010 1101 0001 0100 1011 Michelle Manes Assistant Professor University of Hawaii at Manoa [email protected] 23 1 45 iπ e +1=0 0011 0010 1010 1101 0001 0100 1011 • Each fundamental algebraic operation appears exactly once. • Each of five fundamental mathematical constants appears exactly once. 23 1 45 iπ e +1=0 0011 0010 1010 1101 0001 0100 1011 23 1 45 iπ e +1=0 0011 0010 1010 1101 0001 0100 1011 • The first number. • At least 20,000 years ago, people were counting by adding up ones. • Story of one: PBS documentary. 23 1 45 Ishango bone 0011 0010 1010 1101 0001 0100 1011 23 1 45 Sumerian tokens 0011 0010 1010 1101 0001 0100 1011 23 1 45 One what? 0011 0010 1010 1101 0001 0100 1011 23 1 45 1 in Modern Mathematics 0011 0010 1010 1101 0001 0100 1011 • “Successor”: whole number after n is n+1. • Multiplicative identity: a × 1 = 1 × a = a for any number a. 23 1 45 • Well ordering principle: every nonempty subset of (positive whole) numbers has a smallest element. • Suppose is 0100 a number smaller than 1: 0011 0010 1010 there 1101 0001 1011 0 < r < 1. • Then r × r < r × 1 = r < 1. • Continue with that reasoning: …r5 < r4 < r3 < r2 < r < 1. • The set of numbers less than one has no smallest element! • If there’s no number less than 1, there’s no number between n and n+1. 23 1 45 iπ e +1=0 0011 0010 1010 1101 0001 0100 1011 23 1 45 iπ e +1=0 0011 0010 1010 1101 0001 0100 1011 • “Natural number”? • Digit in positional number system. (India ~500 BC, Yucatan peninsula) • As a quantity. (Not until much later!) • The Nothing That Is: A natural history of zero (by Kaplan and Kaplan). • Zero: The biography of a dangerous idea (by Seife). 23 1 45 Mayan numbers 0011 0010 1010 1101 0001 0100 1011 • Base 20 positional system. 23 1 45 Additive number systems 0011 0010 1010 1101 0001 0100 1011 • Repeat a symbol to indicate bigger numbers. • No need for “0,” just omit that symbol. 23 1 45 Ancient Greece 0011 0010 1010 1101 0001 0100 1011 “How could he have missed it? To what heights science would have risen by now, if only he had made that discovery!” - Gauss, about Archimedes 23 1 45 0 in positional number system 0011 0010 1010 1101 0001 0100 1011 • “Arabic” numbers brought to Europe by Fibonacci (12th century). 23 1 45 0 as a quantity Brahmagupta (560 AD) 0011 0010 1010 1101 0001 0100 1011 • Sum of zero and positive is positive, sum of zero and zero is zero. • A number multiplied by 0 is 0. • A number remains unchanged when 0 is subtracted from it. • Zero divided by zero is zero. • Widespread use in western world not until 17th century! 23 1 45 0 in Modern Mathematics 0011 0010 1010 1101 0001 0100 1011 • Additive identity: a + 0 = 0 + a = a for any number a. • Multiplicative behavior: a × 0 = 0 × a = 0 for any number a. 23 1 45 0011 0010 1010 1101 0001 0100 1011 • 0 + 0 = 0 • Multiply by some number a: a × (0 + 0) = a × 0 • Distributive law: (a × 0) + (a × 0) = a × 0 • Subtract a × 0 from each side: a×0=0 23 1 45 iπ e +1=0 0011 0010 1010 1101 0001 0100 1011 23 1 45 i π e +1=0 0011 0010 1010 1101 0001 0100 1011 23 • “Imaginary number”. • i2 = –1. • An Imaginary Tale: The story of i (by Nahin). 1 45 Number Systems 0011 0010 1010 1101 0001 0100 1011 i 23 1 45 Algebraic numbers 0011 0010 1010 1101 0001 0100 1011 • Take a polynomial equation with integer coefficients. Its solutions are algebraic numbers. • All integers are algebraic: 5 is a solution of x = 5. • All rational numbers are algebraic: ½ is a solution of 2x = 1. • i is algebraic: It is a solution of x2 = –1. • A number that is not algebraic is transcendental. 23 1 45 Number Systems Complex numbers 0011 0010 1010 1101 0001 0100 1011 Real numbers Algebraic numbers Rational numbers 23 1 45 Transcendental numbers i in History 0011 0010 1010 1101 0001 0100 1011 1530s: Tartaglia discovers cubic formula. For this equation: 3 x = 15x + 4 his method yields a root of 3 € € 3 23 1 45 x = 2 + −121 + 2 − −121 . i in History 0011 0010 1010 1101 0001 0100 1011 Imagine € 2 + −121 = a + b −1 3 2 − −121 = a − b −1 ( ) ( ) Some clever algebra yields x = 4. 23 1 45 so x = a + b −1 + a − b −2 = 2a. € € 3 i in History 0011 0010 1010 1101 0001 0100 1011 • 1600s: Descartes argues the physical impossibility of complex numbers. • 1600s: Wallis tries to picture them as a vertical motion. • 1700s: Wessel describes the complex plane and says multiplying by i is the same as rotating 90 degrees. 23 1 45 Multiply by i 0011 0010 1010 1101 0001 0100 1011 4i i 4 23 1 45 Multiply by i 0011 0010 1010 1101 0001 0100 1011 ai i a 23 1 45 iπ e +1=0 0011 0010 1010 1101 0001 0100 1011 23 1 45 π i e +1=0 0011 0010 1010 1101 0001 0100 1011 • 3.1415926535897932384626433… • Ratio of circumference to diameter of any circle. • Irrational (Lambert, 1761). • Transcendental (von Lindemann, 1882). • A History of π (by Beckman). 23 1 45 0011 0010 1010 1101 0001 0100 1011 23 1 45 0011 0010 1010 1101 0001 0100 1011 23 1 45 π in History 0011 0010 1010 1101 0001 0100 1011 • “And he made a molten sea, ten cubits from one brim to the other; it was round all about, and his height was five cubits, and a line of thirty cubits did encompass it all around.” - Kings 7:23 23 1 45 circumference 30 π= = =3 diameter 10 π in History 0011 0010 1010 1101 0001 0100 1011 10 1 • 200 BC: Archimedes found 3 < π < 3 . 71 7 € 23 1 45 π in History 0011 0010 1010 1101 0001 0100 1011 • 16th century: Ludolph van Ceulen calculated π to 35 decimal places and had the result carved on his tombstone. • Germans still refer to π as die Ludolphsche Zahl. 23 1 45 π in History 0011 0010 1010 1101 0001 0100 1011 • 1706: First appearance of the symbol π. • 1873: Shanks spent 20 years calculating π to 707 decimal places. Mistake (found in 1945) in the 528th decimal place. • 1897: Indiana bill #246. • 1949: Computer took 70 hours to calculate π to 2,000 decimal places. • Current record: 5 trillion digits. 23 1 45 Approximations to π π = 4 0011 0010 1010 1101 0001 0100 1011 2 1 1 1 1 1 1 1 1 1 = × + × + + × π 2 2 2 2 2 2 2 2 2 2 2×2 4 ×4 6×6 8×8 = × × × × π 1× 3 3 × 5 5 × 7 7 × 9 € 32 2+ 52 2+ 72 2+ 2 + 23 1 45 π 1 1 1 1 = 1− + − + − 4 3 5 7 9 π 2 1 1 1 €1 1 = 2 + 2 + 2 + 2 + 2 + 6 1 2 3 4 5 € 1+ 12 12 π i e +1=0 0011 0010 1010 1101 0001 0100 1011 23 1 45 e +1=0 1 iπ 0011 0010 1010 1101 0001 0100 1011 • • • • 2.718281828459045235360287471… Irrational (Euler, 1737). Transcendental (Hermite, 1873). e: The story of a number (by Maor). 23 45 e in History 0011 0010 1010 1101 0001 0100 1011 • 1661: Huygens investigates the area under the curve y = 1/x. 23 1 45 e in History 0011 0010 1010 1101 0001 0100 1011 • 1690: Leibniz working on the calculus investigates the function f(x) = ex. • Writes to Huygens about it, naming the constant b. • 1854: Shanks calculates 205 decimal places of e. 23 1 45 So what is e? 0011 0010 1010 1101 0001 0100 1011 • Base of the natural logarithm. 23 1 45 So what is e? 0011 0010 1010 1101 0001 0100 1011 • A limit. 1 e = lim1+ n n →∞ n 23 1 n (1+1/n)n 1 € (2)1 = 2 10 (1.1)10 = 2.5937424601 100 (1.01)100 1000 (1.001)1000 = 2.716923932235893… 45 = 2.704813829421526… Formulas for e 0011 0010 1010 1101 0001 0100 1011 1 1 1 1 1 e = 1+ + + + + + 1 1× 2 1× 2 × 3 1× 2 × 3 × 4 1× 2 × 3 × 4 × 5 23 1 1 1 1 1 1 1 = 1− + − + − + e 1 1× 2 1× 2 × 3 1× 2 × 3 × 4 1× 2 × 3 × 4 × 5 1 e =2+ 1 1+ 2 2+ 3+ 3 4 4+ 5 + 45 iπ e +1=0 0011 0010 1010 1101 0001 0100 1011 • “The most beautiful theorem in mathematics”. • History is unclear. • Certainly known to Euler (1707-1783). • Stigler’s Law: “No scientific discovery is named after its original discoverer.” 23 1 45 Making sense of iπ e 0011 0010 1010 1101 0001 0100 1011 n e = lim1+ n →∞ 1 n e = lim1+ n →∞ 2 n 2 1 = lim1+ n n →∞ n 2 n 2 1 2 e = lim1+ + 2 = lim1+ n n n →∞ n →∞ n 2 n 23 1 1 = lim1+ n n →∞ 2 n 1 n 45 Making sense of iπ e 1+iπ 0011 0010 1010 1101 0001 0100 1011 x e = lim1+ n n →∞ n x iπ e = lim1+ n n →∞ iπ n 23 1 45 Making sense of iπ e 0011 0010 1010 1101 0001 0100 1011 2 (1+iπ/2) x e = lim1+ n n →∞ n x iπ e = lim1+ n n →∞ iπ n 23 1 45 1+iπ/2 Making sense of iπ e 0011 0010 1010 1101 0001 0100 1011 x e = lim1+ n n →∞ n iπ e = lim1+ n n →∞ 23 1 x n iπ (1+iπ/5)5 45 1+iπ/5 Making sense of iπ e 0011 0010 1010 1101 0001 0100 1011 x e = lim1+ n n →∞ 23 n 1 x iπ e = lim1+ n n →∞ n iπ (1+iπ/10)10 45 1+iπ/10 Making sense of iπ e 0011 0010 1010 1101 0001 0100 1011 iπ e = lim1+ n n →∞ iπ n 23 1 45 0011 0010 1010 1101 0001 0100 1011 23 1 45