Sophie Germain 1776-1831 HISTORICAL CONNECTIONS IN MATHEMATICS 83 2012 AIMS Education Foundation
SOPHIE GERMAIN MATHEMATICS IN A MAN S WORLD Biographical Information: Sophie Germain (zhair-man) was a French mathematician born in 1776, in Paris. She was the daughter of a wealthy, upper-class family, but lived an austere personal life, and never married. Because she was a woman, Germain was denied entrance into the educational and scientific institutions that would have nurtured her interests in mathematics and science. However, she gained a significant education through extensive correspondence with some of the finest mathematicians of her time. Germain died of cancer in 1831 at the age of 55. spear through the great thinker. Archimedes was oblivious, so absorbed in his problem that he ignored the threat. What could Archimedes have been working on? Germain determined to find out; when she learned it was mathematics, she dedicated her life to the same absorbing pursuit. Contributions: Sometimes referred to as the Hypatia of the 19th century, Germain made notable discoveries in number theory, acoustics and the theory of elasticity. She is one of the few women whose work is recorded in early mathematics history. Quotation by Germain: Algebra is but written geometry and geometry is but figured algebra. Quotation about Germain: When a person of the sex which, according to our customs and prejudices, must encounter infinitely more difficulties than men to familiarize herself with these thorny researches, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, then without doubt she must have the noblest courage, quite extraordinary talents and a superior genius. Carl Gauss Anecdotes: Inspired by Archimedes One day while reading in her father s fine library, Germain stumbled upon the story of Archimedes death. In her imagination, the words painted a vivid picture. There was Archimedes, working on a problem in the sand. Above him a Roman soldier stood poised, ready to run his Determined Disobedience Although Germain s parents provided her with a good education, they became alarmed when she preferred doing mathematics instead of more traditionally feminine pursuits like cooking and dancing. They agreed with the popular theory that brain work could be a dangerous strain on young girls. Germain often stayed up most of the night reading mathematics books and trying to solve problems. To keep her from this, Germain s parents devised a plan: they would extinguish the heat in her room and remove all sources of light. The child would have no alternative but to crawl under the covers and go to sleep. The plan backfired, however, as Germain smuggled candles into her room, hiding them in her shoes until the others in the household were sound asleep. Then she would sneak out of bed, wrap a comforter around her, and do mathematics by candlelight. HISTORICAL CONNECTIONS IN MATHEMATICS 84 2012 AIMS Education Foundation
Sorry, No Girls Allowed When Germain was 18, the historic École Polytechnique was founded in Paris. This school quickly became the leading center for mathematicians and scientists in France; many important advances can be traced to the students and professors who worked there. But women were not allowed. Although Germain was disappointed, she was not discouraged. She contacted some of the male students and arranged to borrow their lecture notes. When it was time to submit projects, she sent in her work too, but signed a man s name: Monsieur LeBlanc. The famous professor, Lagrange, was so impressed with LeBlanc s project that he wanted to meet him. Imagine his surprise when he found that LeBlanc was a woman! Germain feared his disapproval, but Lagrange was open-minded and encouraging; he even helped her establish a correspondence with other scholars working in number theory. The Gauss Connection In 1804, Germain wrote to Carl Gauss, the foremost German mathematician. She had read his work in number theory, and she offered suggestions based on her own research. Afraid that her ideas would be rejected if Gauss knew her sex, Germain again signed her name M. LeBlanc. Gauss encouraged LeBlanc to send more samples of his work. The two corresponded for more than three years before Germain finally disclosed her true identity. Gauss became one of Germain s greatest fans. He recommended that she be granted an honorary doctorate by the University of Göttingen, where he worked. Unfortunately, Germain died a month before she was to receive the degree. She and Gauss never met. Grand Prize Winner Some of Germain s most significant work was on the mathematics of elasticity. In 1816, the French Academy offered a prize for research in this area, and Germain submitted a paper describing her findings. The prize a one kilogram gold medal worth 3000 francs was not as important as Germain s distinction of being the first woman to receive the prestigious award. Unfortunately, the scientific community still refused to take Germain seriously. They admired her as a phenomenon, rather than accepting her as a colleague. She was an excellent mathematician, but with training and support, she might have been far more productive than she was. HISTORICAL CONNECTIONS IN MATHEMATICS 85 2012 AIMS Education Foundation
FOUR FOURS Using exactly four 4s and the operations +, -, x, and, make equations that equal 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. 0 = 1 = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 9 = HISTORICAL CONNECTIONS IN MATHEMATICS 86 2012 AIMS Education Foundation
$1.00 WORDS A B C D E F G H I J K L M 26 25 24 23 22 21 20 19 18 17 16 15 14 N O P Q R S T U V W X Y Z 13 12 11 10 9 8 7 6 5 4 3 2 1 Suppose each letter of the alphabet is assigned a value as in the above chart. Now, each name or word can be assigned a value. Examples: JOE: 17 + 12 + 22 = 51 CHALLENGES: HOLIDAY: 19 + 12 + 15+ 18 + 23 + 26 + 2 = 115 1. Which day of the week is a $1.00 word? 2. Which American made automobile is a $1.00 word? 3. Which Asian country is a $1.00 word? 4. Each of these sets of letters can be used to spell a $1.00 word. Arrange the letters to form a word. (a) L, C, K, S, B, and O. (b) T, C, N, A, I, and O. (c) O, R, V, E, F, R, and E. 5. Find as many words as you can that are worth exactly $1.00. HISTORICAL CONNECTIONS IN MATHEMATICS 87 2012 AIMS Education Foundation
COUNTING DIVISORS Sophie Germain (1776-1831) was a French woman who made significant discoveries in number theory. In this activity, you will make a number theory discovery dealing with divisors. Number Number Factored Into Primes Number of Divisors 2 2 1 2 3 3 1 2 Prime numbers often play a key role in making discoveries about numbers. There is an easy process, using primes, for finding the number of divisors for any number. For example, the divisors of the number 12 are 1, 2, 3, 4, 6, and 12. Thus, 12 has 6 divisors. Factored into primes, 12 = 2 2 3 1. The exponents, namely 2 and 1, can be used to get 6, the number of divisors for 12. Complete the table to help you discover how exponents can be used to determine the number of divisors for any number. EXERCISES 1. Find a number, in factored form, that has exactly 100 divisors. 2. Find a number, in factored form, that has exactly 1,000,000 divisors. 3. Find four numbers, each with exactly 6 divisors. 4 2 2 3 5 5 1 2 6 2 1 3 1 4 7 7 1 2 8 2 3 4 10 2 1 5 1 4 24 2 3 3 1 8 144 2 4 3 2 15 300 2 2 3 1 5 2 18 400 2 4 5 2 675 1,000 5,000 1,000,000 4. How many numbers have exactly 6 divisors? HISTORICAL CONNECTIONS IN MATHEMATICS 89 2012 AIMS Education Foundation
HAPPY NUMBERS Number theory is a fascinating branch of mathematics that studies the properties of the counting numbers. This activity highlights some interesting facts about a set of numbers called happy numbers. A happy number is a counting number for which the sum of the squares of the digits eventually ends in 1. Is 19 a happy number? Compute the sum of the squares of the digits of 19. 1 2 + 9 2 =1 + 81 = 82 Stop if the result is 1; otherwise repeat the process. Compute the sum of the squares of the digits of 82 8 2 + 2 2 =64 + 4 = 68 6 2 + 8 2 =36 + 64 = 100 1 2 + 0 2 + 0 2 = 1 + 0 + 0 = 1 Stop; the result is 1. 19 is a happy number! Is 11 a happy number? 1 2 + 1 2 =1 + 1 = 2 2 2 = 4 4 2 = 16 1 2 + 6 2 =1 + 36 = 37 3 2 + 7 2 =9 + 49 = 58 5 2 + 8 2 =25 + 64 = 89 8 2 + 9 2 =64 + 81 = 145 1 2 + 4 2 + 5 2 =1 + 16 + 25 = 42 4 2 + 2 2 = 16 + 4 = 20 2 2 + 0 2 = 4 We have returned to 4. The pattern above will repeat indefinitely, never ending in 1. Thus, 11 is not a happy number. Every number that is not happy will enter this repeating cycle of eight numbers. When a number enters this cycle, it cannot be a happy number because it will never end in 1. EXERCISES: 1. Twenty of the first 100 counting numbers are happy numbers. Find them; use any shortcuts you discover. 2. Is a happy number plus another happy number always a happy number? 3. Is the product of two happy numbers always a happy number? 4. Is 1776 a happy number year? 5. Find the point in the cycle at which each of these numbers enters. (a) 33 (b) 15 (c) 154 (d) 80 (e) 38 HISTORICAL CONNECTIONS IN MATHEMATICS 90 2012 AIMS Education Foundation
MIDNIGHT MATH Crossword Puzzle ACROSS 2 Germain s native country. 5 Germain s first name. 7 One source of light. 8 Germain often studied at this hour. 11 Germain used a false. 12 Someone who gives money or blood. 13 What Germain wrote to other mathematicians. 15 Noise made by lions. 16 Germain used this to write. 18 City where Germain lived. DOWN 1 Famous mathematician who inspired Germain. 3 These were not allowed in the university. 4 The part of your body that does the thinking. 6 Germain first studied in her father s. 9 Many people thought it was for girls to study math; hazardous. 10 In mathematics, a proven statement. 13 What you need for studying in the dark. 14 To reason; use one s mind. 17 Paris is famous for its many galleries. HISTORICAL CONNECTIONS IN MATHEMATICS 91 2012 AIMS Education Foundation