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# Broken Numbers - PowerPoint PPT Presentation

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

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## PowerPoint Slideshow about 'Broken Numbers' - koto

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### Broken Numbers

History of Writing Fractions Sketch 4

• Early developments

• Egyptians

• Babylonians

• Chinese

• Indians

• Hindus

• Recent 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

• 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

• 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?

• 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.

• 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

• 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

• The rules for fraction operations was found in this book

• Reduce fractions

• Multiply fractions

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 System

5/6 +3/4

(5 x 4) / 6 + (3 x 6) / 4

38 / 24

1 14/24

• 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)

• 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

• 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

• 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