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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
- 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”

- 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?

- 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

- 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

- 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

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

- 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 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 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 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 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

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 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 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 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 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 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 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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