Chapter 1 Primitive Man Hunter/gatherers Counted Simple - PDF document

The Saga of Mathematics A Brief History Chapter 1 Primitive Man • Hunter/gatherers • Counted • Simple • Notches on wolf bone • Groups of pebbles and stones • Development of a simple grouping system Oh, So Mysterious Egyptian Mathematics! Lewinter and Widulski The Saga of Mathematics 1 Lewinter and Widulski The Saga of Mathematics 2 Early Civilizations Egyptian Civilization • Civilization reached a high point in Egypt at a • Humans discovered agriculture very early time, 3000 B.C. • Need for a calendar • By 3000 BC, Egypt had developed agriculture • Trading or bartering of services and goods making use of the wet and dry periods of the year • The Nile flooded during the rainy season • Production of goods • Knowing when the flooding was going to arrive • An ability to observe the universe was extremely important • Mathematics is required • The study of astronomy was developed to provide this calendar information Lewinter and Widulski The Saga of Mathematics 3 Lewinter and Widulski The Saga of Mathematics 4 Egyptian Civilization Egyptian Society • Egyptian civilization required administration, a • Established a writing system for words and system of taxes, and armies to support it numerals– hieroglyphics. • As the society became more complex, • Kept written records – papyrus. – Written records were required – The Rhind/Ahmes papyrus – Computations needed to be done as the people bartered – The Moscow papyrus their goods • Developed a calendar and watched the skies • A need for counting arose, then writing and numerals were needed to record transactions for astrological events – astronomy. Lewinter and Widulski The Saga of Mathematics 5 Lewinter and Widulski The Saga of Mathematics 6 Lewinter and Widulski 1

The Saga of Mathematics A Brief History Egyptian Society Egyptian Mathematics • Simple grouping system • Built complex structures – pyramids, (hieroglyphics) sphinx, etc. • The Egyptians used the stick for 1, the heel bone • For example, the Great Pyramid at Giza was for 10, the scroll for 100, built around 2650 BC and it is truly an the lotus flower for 1,000, the bent finger or snake extraordinary feat of engineering. for 10,000, the burbot fish • All of these things required mathematics. or tadpole for 100,000 and the astonished man for 1,000,000. Lewinter and Widulski The Saga of Mathematics 7 Lewinter and Widulski The Saga of Mathematics 8 Egyptian Numerals Addition and Subtraction • When adding, ten of • Using these symbols we can write large numbers any symbol would be simply by grouping them appropriately replaced by one of the • For example, the number 243,526 would be next higher symbol written as: • When subtracting, if you need to borrow, simply replace one of the next higher symbol by ten of the necessary symbols Lewinter and Widulski The Saga of Mathematics 9 Lewinter and Widulski The Saga of Mathematics 10 Egyptian Multiplication Egyptian Multiplication • Unique method which they correctly viewed as • Egyptians figured out is that any integer can repeated addition. be written as a sum of the powers of two • Based on doubling and is also known as the without repeating any of them didactic method . • For example, • Starting with one and doubling, they obtained a never-ending sequence of numbers: 1, 2, 4, 8, 16, • 11 = 8 + 2 + 1 32, 64, 128, ... • 23 = 16 + 4 + 2 + 1 • These numbers are the powers of two: 2 0 , 2 1 , 2 2 , • 44 = 32 + 8 + 4 2 3 , 2 4 , 2 5 , 2 6 , 2 7 , … • 158 = 128 + 16 + 8 + 4 + 2 Lewinter and Widulski The Saga of Mathematics 11 Lewinter and Widulski The Saga of Mathematics 12 Lewinter and Widulski 2

The Saga of Mathematics A Brief History Egyptian Multiplication Egyptian Multiplication • Subtract the left side • Suppose we want to numbers from 12 until multiply 12 x 17. 1 17 1 17 you reach 0. • Start with 1 and 17. • Star the left side 2 34 2 34 • Keep doubling both numbers that are being numbers until the left subtracted. 4 68 * 4 68 side gets as close as • In this case, possible to, but not � 12 – 8 = 4 8 136 * 8 136 larger than 12. � 4 – 4 = 0 Lewinter and Widulski The Saga of Mathematics 13 Lewinter and Widulski The Saga of Mathematics 14 Egyptian Multiplication Why it works? • To obtain the answer, • This ingenious method relies on the ( ) add the corresponding × + = × + × distributive law a b c a b a c 1 17 right side numbers of • Since 12 = 4 + 8, we can write the starred positions. 2 34 ( ) × = × + = × + × = + = 17 12 17 4 8 17 4 17 8 68 136 204 • In this case, * 4 68 • Not bad for thousands of years ago! � 136 + 68 = 204 • So, 12 x 17 = 204. * 8 136 • Neat! Lewinter and Widulski The Saga of Mathematics 15 Lewinter and Widulski The Saga of Mathematics 16 Egyptian Fractions Egyptian Fractions • Egyptians recognized that fractions begin • Egyptians denoted unit with the so-called reciprocals of whole fractions by placing an numbers, like 1/3 or 1/8. eye over them, e.g., to the right we see the • Egyptians used only fractions whose fractions 1/10 and numerator was 1, like 1/3 or 1/8 (with the 1/123. exception of the fraction 2/3.) • Two exceptions • A fraction whose numerator is one is called existed one for 1/2 and a unit fraction . the other for 2/3. Lewinter and Widulski The Saga of Mathematics 17 Lewinter and Widulski The Saga of Mathematics 18 Lewinter and Widulski 3

The Saga of Mathematics A Brief History Egyptian Fractions Egyptian fractions • These two fractions had their own symbols: • Egyptians insisted on writing fractions such as 3/4 or 7/8 as sums of unique unit fractions – 1/2 had a sign of its own ( ), and – 3/4 = 1/2 + 1/4 – 2/3 had its own symbol ( ). – 7/8 = 4/8 + 2/8 + 1/8 = 1/2 + 1/4 + 1/8 • All other fractions were written as the sum • It is indeed a fact that all fractions can be written of progressively smaller unit fractions. as the sum of unique unit fractions • It is interesting that Egyptian fractions were • This fact has intrigued mathematicians for used well into the middle ages, in Europe. millennia. Lewinter and Widulski The Saga of Mathematics 19 Lewinter and Widulski The Saga of Mathematics 20 Unit Fractions The Egyptian Method • This method consists of multiplying the • There are several methods for writing a denominator by unit fractions (1/2, 1/3, 1/4, fraction as the sum of unit fractions. 1/5, …) to obtain numbers that will add up – The Egyptian method to the numerator. – Decomposition using proper divisors • For example, if the fraction is 5/6, we – Sylvester’s method would take ½ x 6 = 3 and 1/3 x 6 = 2 – The Modern method – The Splitting method 5 1 1 • Since 3 + 2 = 5 (the numerator), = + 6 2 3 Lewinter and Widulski The Saga of Mathematics 21 Lewinter and Widulski The Saga of Mathematics 22 Write 7/18 Using Unit Fractions Unit Fraction Rule (The Egyptian Method) 1 • If you need , Denominator = 18 n ½ 9 (too big) use 1 1/3 6 (need 1 more) × n denominato r 1/18 1 7 Lewinter and Widulski The Saga of Mathematics 23 Lewinter and Widulski The Saga of Mathematics 24 Lewinter and Widulski 4

The Saga of Mathematics A Brief History Using Proper Divisors Using Proper Divisors 11 9 2 1 1 = + = + • This method consists of examining the • After reducing, we have 18 18 18 2 9 divisors of the denominator for factors that • Suppose, on the other hand, we want to will sum to the numerator. write 11/15 as the sum of unit fractions • For example, suppose we want to write – The factors of 15 are 1, 3, 5, and 15. 11/18 as the sum of unit fractions – It appears to be impossible! – The factors of 18 are 1, 2, 3, 6, 9, and 18. • In this case we can rename the fraction 11 9 2 – Since 11 = 9 + 2, we can write 11/15 as 22/30. = + 18 18 18 Lewinter and Widulski The Saga of Mathematics 25 Lewinter and Widulski The Saga of Mathematics 26 Using Proper Divisors Sylvester’s Method • The factors of 30 are 1, 2, 3, 5, 6, 10, 15, • Originally, developed by Fibonacci (1175-1250). and 30. • Rediscovered by J.J. Sylvester (1814-1897) in 1880. • And, 22 = 15 + 5 + 2, so we can write • Subtract from the given fraction the largest unit fraction possible. 11 22 15 5 2 1 1 1 = = + + = + + • If the result is not a unit fraction, repeat the 15 30 30 30 30 2 6 15 procedure as many times as necessary to obtain all unit fractions. Lewinter and Widulski The Saga of Mathematics 27 Lewinter and Widulski The Saga of Mathematics 28 Sylvester’s Method The Modern Method − a − 1 ac b • Note: = • Similar to Sylvester’s method. b c bc • Use the multiplier of the numerator that yields the smallest result larger than the denominator. • Therefore, ca > b. • Then set up the equation: ( M )( N ) = D + C where N = numerator of the given fraction, D = • Use the multiplier of the numerator that yields the denominator of the given fraction, M = multiplier smallest result larger than the denominator. that is chosen, C = constant that must be used to • Then, the multiplier becomes the denominator for create the equation. the unit fraction to be subtracted. Lewinter and Widulski The Saga of Mathematics 29 Lewinter and Widulski The Saga of Mathematics 30 Lewinter and Widulski 5