ancient egyptian multiplication, division, root extraction

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Contents

Multiplication by Doubling and Halving (Mediation & Duplation)

Multiplication Examples

Decomposition into Powers of 2

How the Doubles Are Chosen

Written Work

Multiplication by Decomposition

Division by Repeated Subtraction and by Sum of Identified "Doubles"

Unit Fraction and Negative Powers of Two

References

Ancient Egyptian Multiplication

In the 1850s, a Scottsman named A. H. Rhind purchased a papyrus of Egyptianmathematics and a leather roll. The papyrus is collection of completed math problems and

is called the Rhind Papyrus or the Ahmes Papyrus. Ahmes was the scribe who in 1650

BCE copied the math from a much older document. The Egyptian Mathematical Leather

Roll (EMLR) contains methods for simplifying a series (a sum) of unit fractions to a single

unit fraction. Much of the Rhind Papyrus deals with fraction computation, area problems,

and "solving equations" -- finding the value of a heap.

On this web page multiplication, division, and some fractions ancient Egyptian style are

discussed.

The multiplication method is sometimes called doubling and halving or called Russianpeasant multiplication. In modern terms it employs the decomposition of one number into

its binary components, addition to produce "doubles" or multiples of the other number, and

computing a total of doubles identified by odd divisors or the powers of two. Ancient

Egyptians did not use this vocabulary and neither need you, but, the method will is

examined below using a modern version of the algroithm and examples.

Multiplication by Doubling and Halving (Mediation & Duplation)

ient Egyptian Multiplication, Division, Root Extraction http://www.mathnstuff.com/math/spoken/here/2class/60/egyptm.ht

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See this

Done step-by-step with calcui! Done with animated calculi!

To multiply two numbers by doubling and halving:

1st: Place each number at the top of one of two side-by-side columns. One column's

entries are doubled and the others are divided by two. To shorten the computationand format the problem as shown in these examples, put the smaller number at the

top of the left column and use dividing to produce other entries in that column as

directed below.

2nd: Create entries in the left column by halving the number in the row above it and

discarding the remainder. Stop at 1 or 0.

3rd: Create entries in the right column by doubling the number in the row above it.

Stop when you have matched all the rows of the left column. Note: Rather than

double the number to obtain the next number, reference and internet sources use

the phrase "add it to itself" to get the next number.

4th: Use the odd numbers in the left column to identify which matching numbers in the

right column to add.

5th: Add the "doubles" in the right column which match the identified powers of two

from the left column.

6th: The sum is the desired product.


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Decomposition into Powers of 2

Earlier "the decomposition of one number into its binary components" and "the powers oftwo" were mentioned as important in the multiplication process. Consider now the powers of

two.

The numbers 1, 2, 4, 8, 16, 32, 64, ... have been arranged in ascending and descending

order and as simple constants and as powers. They are "colored" so one may easily track and

follow the patterns below.

Earlier the numbers 92, 25, and 15 were used as multipliers. For this reason they were

chosen to help explain decomposition, ordered breaking-down, into the powers of 2.

Three different representations of 92, 25, and 15 are shown.

On the left, repeated subtraction of the largest possible power of 2 is presented.

On the right, the numbers are represented in modern binary notation and below that as the


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sum of the powers of two.

The sum of the powers of two is required in ancient Egyptian multiplication by havling and

doubling.

How the Doubles Are Chosen - The Odd Halves Match the Powers of 2

Expressing a number as the sum of powers of two is important.

The idea of multiple is important.

To multiply 10 by 3, find the 3rd multiple of 10. The multiples of 10 are 10, 20, 30,40, The 3rd multiple is 30.

To multiply 3 by 10, find the 10th multiple of 3. The multiples of 3 are 3, 6, 9, 12,

15, 18, , 27, 30, 33, The 10th multiple of 3 is 30.

Because 92 may be expressed as the sum of powers of two, the product of 92 and

45671 may be expressed as the sum of the multiples of 45671 which match the required

powers of two.


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Written Work

The Rhind Papyrus was not written in more ancient hieroglyphics but in Hieratic notation. Just for

x 25 problem is shown on the left in Hieratic numerals. Hieratic is an additive system. The scribe didn't

with check marks or the totals the teaching model displays. Can you read it?

Multiplication by Decomposition

To multiply two numbers using powers of 2:

1st: Use two side-by-side columns of numbers. The left column will contain the multiplier and

powers of two. The right column will contain multiples of the multiplicand and the product.

Additional columns of checks or marks and of identified "doubles" may be placed as

desired by the scribe.

2nd: Begin at the top of the left column and fill the column in order with the powers of two until

you have reached the multiplier.

3rd: If the multiplier is a power of two write this row twice. If the multiplier is not a power of

two, write the multiplier only once at the bottom of the left column.


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4th: On the top row of the right column place the multiplicand.

5th: Create the remaining entries in the right column by doubling the number in the row above

it. Stop when you have matched all the rows of the left column except for the multiplier.

6th: Check or mark the powers of two required to decompose the multiplier.

In the example above the multiplier is 25. The numbers 16, 8, and 1 are identified

because 25 = 16 + 8 + 1, the decomposition of 25.

7th: Add the "doubles" in the right column which match the identified powers of two from the

left column. The scribe may create a new column or use the existing right column.

8th: The sum is the desired product.

Division


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Yes, division may be completed by repeated subtraction. The following algorithm willalso complete division. It may be difficult to find the sum of "doubles."

To divide:

1st: Use two side-by-side columns of numbers. The left column will contain powers of

two (and eventually the quotient). The right column will contain multiples of the

divisor. Additional columns of checks or marks and of identified "doubles" may be

placed as desired by the scribe.

2nd: Begin at the top of the left column and fill the column in order with the powers of

two until you have reached the dividend.

3rd: On the top row of the right column place the divisor.

4th: Create the remaining entries in the right column by doubling the number in the row

above it. Stop when the "double" is as large as the dividend.

5th: This is the tough task. Find and check or mark "doubles" in the right column so that

their sum is the dividend.

6th: The sum of the indicated left-column powers of two is the quotient.

Unit Fractions and Negative Powers of Two


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The Egyptians used unit fractions long before calculus teachers taught students Taylor

and McLauren and other power series.

They use a third and a fifth and any other unit fraction needed and symbolized it by an

open mouth written above the number. The most important fractions were those above. The

most famous idea was the one below.

ReferencesAzzolino, Agnes. "multiple"

2005 at www.mathnstuff.com/math/spoken/here/1words/m/m29.htm

visited 10 April 2009.

Betr , Maria Carmela. "THE EYE OF HORUS"

at http://www.amun.com/eng/horus_e.html

visited 12 August 2009

Gardner, Milo. "Egyptian Mathematical Leather Roll." 1999-2009 Wolfram Research, Inc.at http://mathworld.wolfram.com/EgyptianMathematicalLeatherRoll.html

visited 12 August 2009.

Gnaedinger, Franz "Rhind Mathematical Papyrus" 1979 - 2003



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Newman, James Roy. The World of Mathematics

through a Google books search for "text of the rhind papyrus" at http://books.google.com


O'Connor, J J & Robertson, E F. "Egyptian Mathematics" 2006

at http://www.math.utep.edu/Faculty/lvaliente/Math1319/Egyptian.doc

visited 9 August 2009

Smith, Jr., Frank D. (Tony) "Hieroglyphics: Egyptian, Mayan, and Chinese Characters"

at http://www.valdostamuseum.org/hamsmith/eghier.html


Stone Design Corp. "The Eye of Horus "

at http://www.stone.com/Qouija/Eye_of_Horus.html


Wikipedia, the free encyclopedia "Eye of Horus"

at http://simple.wikipedia.org/wiki/Eye_of_Horus


2009, 2010 Agnes Azzolino

www.mathnstuff.com/math/spoken/here/2class/60/egyptm.htm


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