Broken Numbers

1
Broken Numbers

• History of Writing Fractions Sketch 4

2
A Brief Overview of Whats To Come

• Early developments
• Egyptians
• Babylonians
• Chinese
• Indians
• Hindus
• Recent developments

3
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

4
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

5
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?

6
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 Fibonaccis
  Liber Abbaci 1202 A.D.

7
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

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

1/4
\ 1/52
1/8
\ 1/104
9
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 integers 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 )

10
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 1,1230
  meaning 1 x 60 12 30 x 1/60

11
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 surveying
• i.e. 1/3 of 1/2 of 1/2 of 1/2 of 1/2 of 1/2
  1/96

12
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

13
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

14
A Closer Look

• 5/6 3/4
• (5 x 4) / 6 (3 x 6) / 4
• 38 / 24
• 1 14/24

15
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

16
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)

17
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

18
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 Stevins ( A Flemish
  mathematician and engineer) 1585 book, The Tenth
• Many representations of the decimal were used
• Apostrophe, small wedge, left parenthesis, comma,
  raised dot

19
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

20
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 Stevins book The Tenth

21
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