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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

Broken Numbers

History of Writing Fractions Sketch 4


A brief overview of what s to come
A Brief Overview of What’s To Come

  • Early developments

  • Egyptians

  • Babylonians

  • Chinese

  • Indians

  • Hindus

  • Recent developments


Early developments
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


Notion of parts
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


But what about two fifths
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?


Resources from each culture
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.


Egyptians papyri
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


How the heck did ya get that table

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


Babylonians clay tablets and the sexagesimal place value system
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 )


Mesopotamian scribes
Mesopotamian Scribes System

  • 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


Yet another system
Yet Another System 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


Chinese
Chinese System

  • 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


Rules from the nine chapters
Rules from the Nine Chapters System

  • 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


A closer look
A Closer Look System

5/6 +3/4

(5 x 4) / 6 + (3 x 6) / 4

38 / 24

1 14/24


Indian culture and the system of brahma
Indian Culture and the System of Brahma System

  • 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


Hindus
Hindus System

  • 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)


Interesting additions
Interesting Additions System

  • 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


Decimal on the back burner
Decimal On the Back-burner System

  • 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


A brief timeline
A Brief Timeline System

  • 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


Not so brief of a timeline
Not So Brief of a Timeline System

  • 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


Resources used
Resources Used System

  • 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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