Multiplication by Doubling and Halving (Mediation & Duplation)
Decomposition into Powers of 2
How the Doubles Are Chosen
Multiplication by Decomposition
Division by Repeated Subtraction and by Sum of Identified "Doubles"
Unit Fraction and Negative Powers of Two
Ancient Egyptian Multiplication
In the 1850s, a Scottsman named A. H. Rhind purchased a papyrus of Egyptian mathematics 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 Russian peasant 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)|
To multiply two numbers by doubling and halving:
Decomposition into Powers of 2|
Earlier "the decomposition of one number into its binary components" and "the powers of two" 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 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.
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.
The Rhind Papyrus was not written in more ancient hieroglyphics but in Hieratic notation. Just for fun, the 12 x 25 problem is shown on the left in Hieratic numerals. Hieratic is an additive system. The scribe didn't bother 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:
Yes, division may be completed by repeated subtraction. The following algorithm will also complete division. It may be difficult to find the sum of "doubles."
Unit Fractions and Negative Powers of Two|
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.
© 2009, 2010 Agnes Azzolino