Egyptian Arithmetic

PURPOSE: The purpose of this blog is to define Egyptian fraction arithmetic from the Middle Kingdom by its four arithmetic operations: addition, subtraction, multiplication and division.

To simplify the task two background issues will be covered. First, our modern base 10 decimal system connects to the Egyptian Old Kingdom binary decimal fraction system, a methodology that Ahmes used to prove the correction of final Egyptian fraction answers. The second covers the Middle Kingdom Egyptian fraction numeration system, and its four arithmetic operations.

BACKGROUND: Issue # 1, modern base 10 decimals.

Before any numeration system, from any time period, can be chosen for any culture's use, a set of counting numbers must be selected. Concerning our modern base 10 numeration system, the numerals arrived from India around 800 AD. Arab and Islamic mathematics replaced Greek ciphered numerals, much as Greeks had replaced Egyptian ciphered numerals. Around 1485 AD the Hindu-Vedi 1-9 counting numbers were used by by Simon Stevin within an algorithm and the binominal theorem, adding a zero as a place-holder to write two books. The Paris Academy approved both books, one for use by the business community and one for the science community.

In summary, zero, as a mathematical idea, had existed in the ANE, in Egypt and Babylon from as early as 2,000 BC. Neither culture integrated the idea of zero as a place holder in their infinite and finite numeration systems.

1. The four modern base 10 arithmetic operations.

a. European addition: a + b = c, where a, b, and c could be any known counting number, even rational numbers.

b. Subtraction: a - b = d, with b being allowed to be larger than a, creating the domain of negative numbers. Egyptians, Babylonians and Greeks had difficulty in accepting negative numbers. Hindu and Vedic mathematicians accepted negative numbers, and passed this property of modern numeration to Arabs in 800 AD, along with its 1-9 counting numbers..

The issue of rational numbers a and b being fractions created another problem, remainders d became increasingly small, for convenience, d was rounded off by Babylonians and Egyptians prior to 2050 BCE.

Egyptian round off prior to 2050 BCE had followed another rule, using only the first 6-terms, from its binary decimal system, that will define shortly.

Before and after 2,050 Babylonians rounded of rational numbers 1/91 to a base 60 number, 1/90.

Scholars have not fairly reported scribal subtraction methods, especially in terms of the Hultsch-Bruins method reported in the RMP 2/nth table and the Liber Abaci.

c. Multiplication: Europeans saw repeated addition as defining multipication. For example 5 x 7 = 1 x 7, added together five times, or 5 x 1, added together 7 times. Prime numbers became associated with memorizing multiplication tables, reducing the number of additions that were required. Scholars have fairly reported the scribal form of Egyptian fraction arithmetic.

d. Division: Many Europeans saw division, as an inverse process to multiplication. When the two numbers, a/b involved are prime, there are only two possible results. When a > b, a rational number is defined in the Akhmm Wooden Tablet and the EMLR.