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Broken Numbers

Broken Numbers. History of Writing Fractions Sketch 4. A Brief Overview of What’s To Come. Early developments Egyptians Babylonians Chinese Indians Hindus Recent developments. Early Developments. Fractions have been around for about 4000 years but have been modernized since

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Broken Numbers

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  1. Broken Numbers History of Writing Fractions Sketch 4

  2. A Brief Overview of What’s To Come • Early developments • Egyptians • Babylonians • Chinese • Indians • Hindus • Recent developments

  3. Early Developments • Fractions have been around for about 4000 years but have been modernized since • Influential cultures that aided with this modernization are: Egyptians, Babylonians, Chinese, Hindus • Same basic ideas but refined to fit their own system

  4. Notion of “Parts” • fraction  fracture  fragment: suggest breaking something up • Objects broken down then counted • Underlying principle different from 21st century: Fractions were looked at in earlier days like: find the largest unit possible and take one of those and repeatedly do that until the amount you need is present 21st century: instead of using the pint and a cup of milk for a cooking recipe, we use 3 cups • Unit fractions

  5. But what about two-fifths? • Take the fifth and double it • What do you get? • The third and the fifteenth since you must express the fraction as a sum of unit fractions, Right? • But how?

  6. Resources from each culture • Egyptians used Papyri • Babylonians used cuneiform tablets • Chinese and The Nine Chapters of Mathematical Art 100 A.D. • Indian culture used a book called Correct Astronomical System of Brahma, 7th century A.D. • Europeans in the 13th century used Fibonacci’s Liber Abbaci 1202 A.D.

  7. Egyptians Papyri • 1800-1600 BC • The result of a division of two integers was expressed as an integer plus the sum of a sequence of unit fractions • Example: the division of 2 by 13

  8. 1 13 1/2 6 1/2 1/4 3 1/4 \ 1/8 1 1/2 1/8 How the Heck Did Ya Get That Table? • Leading term in LH col. Is 1, RH col. 13 • Repeated halves carried out until # in RH col. Is less than dividend 2 • Fractions are then entered in RH col. to make fraction up to 2 • The fractions added are divided by 13 and the result is recorded in the LH col. • Backslashes indicate which ones are the sum of the sequence of unit fractions • Answer: 13(1/8 + 1/52 + 1/104)=2 \ 1/52 1/4 \ 1/104 1/8

  9. Babylonians Clay Tablets and the Sexagesimal Place-Value System • 1800-1600 BC • Only used integers • Division of two integers, say m and n,was performed by multiplying one integer ,m, and another integer’s inverse, 1/n (m ∙ 1/n) • m ∙ 1/n was to be looked up in a table which only contained invertible numbers whose inverses in base 60 may be written with a finite number of digits (using the elements of the form 2p3q5r )

  10. Mesopotamian Scribes • Around same time as Babylonians • Used the base-sixty as well but had a unique representation of numbers. • Take the number 72. They would write “1,12” meaning 1 x 60 + 12. If they had a fractional part like 72 1/2, they would write “1,12;30” meaning 1 x 60 +12 + 30 x 1/60

  11. Yet Another System • Still based on the notion of parts, there is another system but only multiplicative • The idea was a part of a part of a part… • Example: the fifth of two thirds parts and the fourth • (1/5 x 2/3) + 1/4 = 23/60 • In the 17th century the Russians used this in some of the manuscripts on surveying i.e. 1/3 of 1/2 of 1/2 of 1/2 of 1/2 of 1/2 = 1/96

  12. Chinese • 100 B.C. • Notion of fractions is very similar to ours (counting a multiple of smaller units than finding largest unit and adding until the amount is reached) • One difference is Chinese avoided using improper fractions, they used mixed fractions

  13. Rules from the Nine Chapters • The rules for fraction operations was found in this book • Reduce fractions • Add fractions • Multiply fractions • Example: rule for addition Each numerator is multiplied by the denominators of the other fractions. Add them as the dividend, multiply the denominators as the divisor. Divide; if there is a remainder let it be the numerator and the divisor be the denominator

  14. A Closer Look 5/6 +3/4 (5 x 4) / 6 + (3 x 6) / 4 38 / 24 1 14/24

  15. Indian Culture and the System of Brahma • Correct Astronomical System of Brahma written by Brahmagupta in 7th century A.D. • Presented standard arithmetical rules for calculating fractions and also dealing with negatives • Also addressed the rules dealing with division by zero

  16. Hindus • 7th century A.D. • Similar approach as Chinese (maybe even learned from that particular culture) • Wrote the two numbers one over the other with the size of the part below the number of times to be counted (no horizontal bar) • The invert and multiply rule was used by the Hindu mathematician Mahavira around 850 A.D. (not part of western arithmetic until 16th century)

  17. Interesting Additions • Arabs inserted the horizontal bar in the 12th century • Latin writers of the Middle Ages were the first to use the terms numerator and denominator (“counter”, how many, and “namer”, of what size, respectively) • The slash did not appear until about 1850 • The term “percent” began with commercial arithmetic of the 15th and 16th centuries • The percent symbol evolved from: per 100 (1450), per 0/0 (1650), then 0/0, then % sign we use today

  18. Decimal On the Back-burner • Chinese and Arabic Cultures had used decimal fractions fairly early in mathematics but in European cultures the first use of the decimal was in the 16th century • Made popular by Simon Stevin’s ( A Flemish mathematician and engineer) 1585 book, The Tenth • Many representations of the decimal were used: • Apostrophe, small wedge, left parenthesis, comma, raised dot

  19. A Brief Timeline • 1800-1600 B.C. Notion of parts and the unit fraction are found in Egyptian Papryi and Babylonian clay tablets/sexagesimal system • 1800-1600 B.C. Mesopotamian scribes extended sexagesimal system • 100 B.C. Chinese The Nine Chapter of Mathematical Art • 7th century Correct Astronomical System of Brahma written by Brahmagupta. • 7th century Hindu system modeled after Chinese • 850 A.D. Mahavira developed the invert and multiply rule for division of fractions

  20. Not So Brief of a Timeline • 12th century Arabs introduce horizontal bar • 15th and 16th century evolution of the percent sign • 16th century decimal fractions and the decimal introduced to European culture • 1585 Simon Stevin’s book The Tenth

  21. Resources Used • Belinghoff, William P. and Fernando Q. Gouvea. Math Through the Ages: a gentle history for teachers and others :Oxton House Publishers, 2002 • Grattan-Guinness, I. Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences : Routledge, 1994 • Victor J. Katz. A History of Mathematics, Pearson/Addison Wesley, 2004

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