Transcript Broken Numbers

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
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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
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Notion of “Parts”
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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?
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Resources from each culture
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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
13
1/2
6 1/2
1/4
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?
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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
13
1/2
6 1/2
1/4
3 1/4
\ 1/8
1 1/2 1/8
\ 1/52
1/4
\ 1/104
1/8
Babylonians Clay Tablets and the
Sexagesimal Place-Value System
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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
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Yet Another System
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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
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
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Rules from the Nine Chapters
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The rules for fraction operations was found in this
book
– Reduce fractions
– Add fractions
– Multiply fractions
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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) / 4
38 / 24
1 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
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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)
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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
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– 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
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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
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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
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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