Date: Oct 17, 2012 2:02 PM
Author: Milo Gardner
Subject: Re: Why study Egyptian fraction math?
Why Study Egyptian Fraction Mathematics
BACKGROUND: Old Kingdom (OK) hieroglyphic numeration and math systems were written in base 10. Numerals were mapped many-to-one by a binary cursive algorithm onto number symbols. The OK numeration system rounded-off rational numbers by throwing away 1/64 units in arithmetic, algebra, geometry and weights and measures problems. Hieroglyphic math solved spiritual and secular math problems in infinite series statements from about 3,000 BCE to 2050 BCE.
MIDDLE KINGDOM MATH: After 2050 BCE Egyptian Middle Kingdom (MK) math system was written in a finite system. The hieratic arithmetic and math systems were developed in the OK that solved the OK round off issue. Hieratic numerals mapped numbers one-to-one onto sound symbols. The ciphered numeration system exactly recorded secular arithmetic, algebra, algebraic geometry, and weights and measure solution were recorded whenever possible. The new finite math system only solved secular math problems.
Hieroglyphic infinite series math continued to be used to solve spiritual math problems in the MK that considered the limit between zero and one. One was an ideal unity.
The primary purpose of the hieratic system corrected OK round-off errors whenever possible. Hieratic geometry problems corrected round-off errors that added-back rounded-off OK 1/64 units. MK scribes replaced radius R by semi-diameter D/2, and pi by 256/81 in area and cylinder volume problems and 1/64 and 1/320 hekat units. Round-off errors seemed uncorrected related to the poor pi approximation* (3.16). In other cases MK arithmetic, algebraic geometry and weights and measures methods exactly scaled hekat units (of grain) and smaller 1/320 units of beer, bread, domesticated fowl, and grain based wage payments.
A secondary purpose assisted scribes by decentralizing the Egyptian economy in a new absentee landlord system. Exact grain, gold and silver weights units defined the monetary system that paid Pharaoh taxes at the rate of 1/3 of 1/3 percent of net profits. Egyptian finite arithmetic scaled rational numbers n/p by LCM m to a mn/mp in a multiplication context before recording weights and measure units in concise unit fraction series.
Three texts, the Moscow Mathematical Papyrus (MMP), the Kahun Papyrus(KP) and the Rhind Mathematical Papyrus (RMP) encoded areas a circle (A), and volumes (V) of a circle, by naming height (H) of circle granaries by the same meta formulas:
A = pi(R)(R) = (2567/81)(D/2)(D/2)
A = (8/9)(D((8/9)(D) cubits (algebraic algebra formula 1.0)
replaced radius (R) with diameter (D/2) and pi by 256/81 (an easy to manipulate number) in
V = (H)(8/9)(D)(8/9)(D) cubits (algebraic algebra formula 2.0)
V = (3/2)(H)(8/9)(D)(8/9)(D) Khar (algebraic algebra formula 2.1)
V = (2/3)H(4/3)(D((4/3((D) Khar (algebraic algebra formula 2.2)
derived from scaling algebraic formula 2.1 by 3/2 considering
(3/2)V =(3/2)(3/2)(H)(8/9)(D)(8/9)(D) = (H)(4/3)(D)(4/3)D) and multiplying both sides by 2/3
in a finite economic context.
CLASSICAL GREEK, ARAB AND MEDIEVAL MATH: Greek texts, like the Hibeh Papyrus, used closely related ciphered numeration and finite arithmetic systems. Arabs after 800 AD modified the Egyptian and Greek numeration and finite arithmetic systems. Arabs and Fibonacci (1202 AD) scaled n/p by LCM m in a subtraction
(n/p - 1/m) = (mn -p)/mp,
context that applied an algorithm often set numerator (mn -p) unity (1).
MODERN TRANSLATIONS: Many 20th century translations of Middle Kingdom finite arithmetic texts have been incomplete, misleading and in error in major ways. Ahmes' actual Middle Kingdom finite notation used LCM m, a number theory concept, a fact not mentioned by scholars. Egyptian scribes scaled rational number by LCM m within (n/p)(m/m) = mn/mp to record concise unit fraction series (in a multiplication context). Egyptian scribes selected the best divisors of denominator mp (a GCD) that best summed to numerator mn by following the implicit algebraic context:
a. n/p = n/p(m/m) = mn/mq
example: 4/13 = 4/13(4/4) = 16/52 = (13 + 2 + 1)/52 = 1/4 + 1/26 + 1/52
with the divisors of mp often recorded in red that best summed to numerator mn created concise unit fraction series.
A second algebraic context was recorded in RMP 37. The subtraction context was emulated by Arabs and Fibonacci per:
b. (n/pq - 1/m)= (mn -pq)/mp
example: (4/13 - 1/4)= (16 - 13)/52 = (2 + 1)/52 = 1/26 + 1/52
also meant 4/13 = 1/4 + 1/26 + 1/52
MEDIEVAL UNIT FRACTION TEXTS: Arab and Fibonacci's finite notations applied LCM m in a subtraction context. Rational number n/p was encoded to unit fraction series by seven rules within
(n/p - 1/m)= (mn - p)/mp
The seventh rule (distinction reported by L.E. Sigler in 2002 AD) demonstrated
4/13 = 1/4 + 1/18 + 1/468 that considered
(4/13 - 1/4) = ([16 - 13)/52 - 1/18] = (54 - 52)/936 = 1/468
Arab and medieval statements employed the same LCM m that Ahmes employed 2850 years earlier. Ahmes employed two rational number conversion methods. The first method was recorded in RMP 36 by a multiplication use of LCM m recorded as (m/m).