broken numbers
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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 - PowerPoint PPT PresentationTRANSCRIPT
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