# Broken Numbers

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Broken Numbers. History of Writing Fractions Sketch 4. A Brief Overview of Whats 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 - PowerPoint PPT Presentation

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• Broken NumbersHistory of Writing Fractions Sketch 4

• A Brief Overview of Whats To ComeEarly developmentsEgyptiansBabyloniansChineseIndiansHindusRecent developments

• Early DevelopmentsFractions have been around for about 4000 years but have been modernized sinceInfluential cultures that aided with this modernization are: Egyptians, Babylonians, Chinese, HindusSame basic ideas but refined to fit their own system

• Notion of Partsfraction fracture fragment: suggest breaking something upObjects broken down then countedUnderlying 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 present21st century: instead of using the pint and a cup of milk for a cooking recipe, we use 3 cupsUnit fractions

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

• Resources from each cultureEgyptians used PapyriBabylonians used cuneiform tabletsChinese 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 Fibonaccis Liber Abbaci 1202 A.D.

• Egyptians Papyri1800-1600 BCThe 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

• How the Heck Did Ya Get That Table?Leading term in LH col. Is 1, RH col. 13Repeated halves carried out until # in RH col. Is less than dividend 2Fractions are then entered in RH col. to make fraction up to 2The 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 fractionsAnswer: 13(1/8 + 1/52 + 1/104)=21/8\ 1/1041/4\ 1/52

• Babylonians Clay Tablets and the Sexagesimal Place-Value System1800-1600 BCOnly used integersDivision of two integers, say m and n,was performed by multiplying one integer ,m, and another integers 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 )

• Mesopotamian ScribesAround same time as BabyloniansUsed 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

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

• Chinese100 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

• Rules from the Nine ChaptersThe rules for fraction operations was found in this bookReduce fractionsAdd fractionsMultiply fractionsExample: 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

• A Closer Look5/6 +3/4(5 x 4) / 6 + (3 x 6) / 438 / 241 14/24

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

• Hindus7th 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)

• Interesting AdditionsArabs inserted the horizontal bar in the 12th centuryLatin 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 1850The term percent began with commercial arithmetic of the 15th and 16th centuriesThe percent symbol evolved from: per 100 (1450), per 0/0 (1650), then 0/0, then % sign we use today

• Decimal On the Back-burnerChinese 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 centuryMade popular by Simon Stevins ( A Flemish mathematician and engineer) 1585 book, The TenthMany representations of the decimal were used:Apostrophe, small wedge, left parenthesis, comma, raised dot

• A Brief Timeline1800-1600 B.C. Notion of parts and the unit fraction are found in Egyptian Papryi and Babylonian clay tablets/sexagesimal system1800-1600 B.C. Mesopotamian scribes extended sexagesimal system 100 B.C. Chinese The Nine Chapter of Mathematical Art7th century Correct Astronomical System of Brahma written by Brahmagupta.7th century Hindu system modeled after Chinese850 A.D. Mahavira developed the invert and multiply rule for division of fractions

• Not So Brief of a Timeline12th century Arabs introduce horizontal bar15th and 16th century evolution of the percent sign16th century decimal fractions and the decimal introduced to European culture1585 Simon Stevins book The Tenth

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