Old Kingdom Egyptian mathematics was quite different from our present day view of mathematics (https://en.wikipedia.org/wiki/Ancient_Egyptian_mathematics). They used only positive numbers and used only unit fractions (e.g. 1/n). Egyptians pre-1740 BCE, like the later per-Hellenistic Greeks and Romans, had no “zero” character (https://en.wikipedia.org/wiki/Egyptian_numerals#Zero_and_negative_numbers). For these, and other reasons, the Egyptian system of numbers is often seen as inferior to our own. Yet, the Egyptians were able to construct some of the largest and most pleasing architectural structures known to man. We need to better understand and appreciate their system, both for its complexity and for its philosophical basis.
The Egyptians, and later Greeks and Romans, used characters to represent values in decimal systems:
The Roman system addresses the task of tallying, i.e. counting. It uses numerals made up of simple lines and strokes (http://youtu.be/Ik4yloCszYo). In contrast, the Egyptians used a much more complex system of numerals, particularly for characters higher than 100. Their use of a water lily for 1,000, a bent finger for 10,000, a tadpole for 100,000 and a kneeling man with both hands raised (perhaps the neter Heh) for 1 million are much more complex than the simple lines and curves of the Romans. Both the high values and the complexity of the characters of the Egyptian system seem beyond the requirements for simple tallying from which our system is said to have evolved. To even have a character for 1,000,000 is amazing! How long would it take a person to tally a million? At one count per second, it would take 277 straight hours. Why would the Egyptian need such a number?
In regards to the use of a numeral for zero, the Egyptians did use a character for zero in accounting after 1740 BCE: nfr -
The Ancient Greeks avoided the use of a character to represent “nothing”. Early Greek studies in mathematics, prior to the works of Euclid circa 300 BCE, involved both philosophical and mystical beliefs (https://en.wikipedia.org/wiki/Arithmetic). It seems that the Ancient Egyptians, and the Greeks who followed, couldn’t see a role for zero or “nothing” in their number system which probably reflects the strong linkage between their use of a number system and its necessary representation of their philosophy and world view. It wasn’t until the time of Ptolemy circa 70 CE that the Greeks began using zero as true numeral in their astronomy: (https://en.wikipedia.org/wiki/Greek_numerals#Zero).
For characters to represent number values the Greeks simply used their alphabet employing equivalence between letters and characters up to the value of 9,000. Our system shows many similarities to the Greek system that provides many numerals for efficient use in arithmetic calculations.
Modern day Western culture uses a base-10 system of numbers for calculations. As a result we use and need to memorize multiplication tables for the numbers 2 through 9. The Egyptians used a methodology based on the doubling of numbers to complete multiplication and division (https://en.wikipedia.org/wiki/Ancient_Egyptian_mathematics#Multiplication_and_division). A system based on doublings is a powerful and relatively easy system to use as it requires only the 2-times table for all multiplication and division. It has been suggested that a similar system was used for multiplication of Roman Numerals millennia later. Egyptologist R.A. Schwaller de Lubicz was the first to explore the power of this system in addressing a number of complex algebraic equations with his French publication “Le Temple de l'homme” (Paris: Caractères, 1957) that is now available in English under the title “The Temple of Man (Schwaller de Lubicz. 1998). A system of doubling is reflected in our present day use of the binary system in computer technology.
Geometrically, architecturally and artistically, the Egyptians recognized the importance of using a triangle with sides measuring 3:4:5 to generate 90-degree right angles. Tied to this knowledge of the 3:4:5 ratios, the Egyptians essentially solved quadratic equations (https://en.wikipedia.org/wiki/Ancient_Egyptian_mathematics#Quadratic_equations).
Representations of irrational functions are found throughout Egyptian architecture and art. There are numerous examples of the use of the both Pi (π ) and Phi (Φ )in Ancient Egypt (https://en.wikipedia.org/wiki/Golden_ratio). Whereas Pi is taught to all school-age children for the practical calculation of the dimensions of a circle, the lesser-known ratio of Phi, also known as the Golden Section, is not so well recognized in Western culture. This ratio is found in many natural phenomena from biological structures to atomic-scale crystals (https://en.wikipedia.org/wiki/Golden_ratio#Nature). Phi is related to the fibonacci series that is often found in nature: http://vimeo.com/9953368.
Human endeavors by many artists, musicians, historians, architects, psychologists, and mystics have explored the use of Phi in their works.
The Golden Section is the ratio built on two quantities “a” and “b” such that the ratio of the smaller (a) to the larger (b) equals the ratio of the larger (b) to their sum (a+b).
Mathematically, the ratio is irrational with a continuing non-repeating series of numbers to the right of the decimal point: 1.68033 . . . It is thus difficult to deal with arithmetically. Geometrically it is easily dealt with; represented as the ratios of lines, squares and volumes (Lawlor, R. 1982).
As an irrational number, Phi is tied to the concept of creation and generation in Ancient Egypt (Schwaller de Lubicz 1998). Schwaller de Lubicz (1998 page 125) states, “Phi is a function and not a number.” This is an important distinction between the Ancient Egyptian and our present day modern Western systems, where we are interested in calculating particular values, whereas the Egyptian system seems more intent on capturing the nature of the broader functioning of the world around us.
Both the 3:4:5 and Phi functions in Egyptian structures and art were used throughout the 3-millenium duration of the culture starting at least 2,000 years before the work attributed to the Greeks such as Pythagoras that began only circa 600 BCE (Lawlor, R. 1982).
It is hard to believe that the Ancient Egyptians didn’t have an appropriately sophisticated system of math, geometry and algebra when seeing the size, precision and beauty of their constructions. We customarily regard what is early as likely to be primitive and inferior. It is difficult to avoid a pre-judgment of their different, earlier system as being somehow inferior to our later “development”. Thus we highly value our present use of numbers as concrete tools for calculating in an “objective” world. The Egyptian’s developed and maintained a number system for thousands of years that seems to contain subtler and broader meaning for numbers. For example, at one level in their use of Phi that is so wildly seen in the natural living biological world, we can see their attempts to capture a mysterious distinction between the existence of physical matter and the creation of the life-force with its new emergent properties (Schwaller de Lubicz 1998).
As with mysticism, there are suggestions that the Egyptians influenced the later development of the Greek mathematics that are so highly valued by the present-day Western World (https://en.wikipedia.org/wiki/Greek_mathematics#cite_note-LH-2). Thus, Greek mathematics are said to have begun with Thales, who was trained by an Egyptian priest!
In conclusion, it is not a question of whether any one mathematical system is inferior or superior to any other. It is rather a question of recognizing the seeming distance between our calculation orientation and the efforts of the Egyptians to connect with the natural and spiritual world that they considered important. Schwaller de Lubicz (1998) makes the case that in spite of their building prowess, the Ancient Egyptians were not primarily interested in engineering and the concrete physical world. Rather, their aim seems to indicate a desire to capture in their numerical system the broader nature of the creative and enlivening forces that make up us and our world. It is difficult not to agree with him.
Lawlor, R. 1982. Sacred Geometry: Philosophy and Practice. Thames and Hudson.
Schwaller de Lubicz, R.A., 2011. Le Temple de l'homme.
Schwaller de Lubicz, R.A., 1998. The Temple of Man. Inner Traditions.
Schwaller de Lubicz, R.A., 1957. Le Temple de l'Homme, (3 vol. en coffret) Édition Caractère, Paris, 1957. Réédition Dervy Livres.
Schwaller de Lubicz, R.A., 1998. Sacred Science: The King of Pharaonic Theocracy. Inner Traditions.