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 present21st 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

1 131/2 6 1/21/4 3 1/4\ 1/8 1 1/2 1/8 \ 1/52 1/4\ 1/104 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

1/8\ 1/104

1/4\ 1/52

1 1/2 1/8 \ 1/8

3 1/41/4

6 1/21/2

131

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

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

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 surveyingi.e. 1/3 of 1/2 of 1/2 of 1/2 of 1/2 of 1/2 = 1/96

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

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

A Closer Look

5/6 +3/4(5 x 4) / 6 + (3 x 6) / 438 / 241 14/24

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

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)

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

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

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

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

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