According to the Encyclopaedia Britannica, Egyptian Hieroglyphics appeared fully formed at least about 5,000 years ago:

The most ancient hieroglyphs date from the end of the 4th millennium BC and comprise annotations to the scenes cut in relief- -found on slabs of slate in chapels or tombs- -that had been donated as votive offerings. Although by no means all of these earliest signs can be read today, it is nonetheless probable that these forms are based on the same system as the later classical hieroglyphs. In individual cases, it can be said with certainty that it is not the copied object that is designated but rather another word phonetically similar to it. This circumstance means that hieroglyphs were from the very beginning phonetic symbols. An earlier stage consisting exclusively of picture writing using actual illustrations of the intended words cannot be shown to have existed in Egypt; indeed, such a stage can with great probability be ruled out. No development from pictures to letters took place; hieroglyphic writing was never solely a system of picture writing. It can also be said with certainty that the jar marks (signs on the bottom of clay vessels) that occur at roughly the same period do not represent a primitive form of the script. Rather, these designs developed in parallel fashion to hieroglyphic writing and were influenced by it. It is not possible to prove the connection of hieroglyphs to the slightly older cuneiform characters used by the Sumerians in southern Mesopotamia. Such a relationship is improbable because the two scripts are based on entirely different systems. What is conceivable is a general tendency toward words being fixed by the use of signs, without transmission of particular systems.

Early Egyptian Hieroglyphic mathematics used decimal description, as we do now,

and binary calculation, as do our computers. Their fundamental expansion of numbers was in terms of a power series in powers of two, using both negative and positive powers: ... 1/64 1/32 1/16 1/8 1/4 1/2 1 2 4 8 16 32 64 ... Just as Fourier transforms are power series in exp(i t), where exp(i t) is a U(1) group rotation of the unit circle S1, the binary power series can represent Walsh functions, based on the 2-point group Z2 of the 0-dimensional sphere S0. (Walsh functions may be useful in Quantum Computing.) The six fractions 1/64 1/32 1/16 1/8 1/4 1/2 were enough for a lot of rough calculations, so they became known as the Horus-Eye fractions, from their identification with the six parts of the Eye of Horus:

The Egyptians generally used unit fractions, of the form 1/N, so their usual notation consisted of integers and their inverses:

Early Egyptian mathematics seems to be more sophisticated than was thought to be the case a few years ago. To quote Kevin Brown's discussion of recent work of Milo Gardner: "... the 2/n table of the Rhind Papyrus, which dates from more than a thousand years before Pythagoras, seems to show an awareness of prime and composite numbers, a crude version of the "Sieve of Eratosthenes", a knowledge of the arithmetic, geometric, and harmonic means, and of the "perfectness" of the number 6."

Terry Nevin has an interesting WWW site about the relationship between the Great Pyramid and Egyptian mathematics. The Horus-Eye image above is from his page. The images of decimal hieroglyphics and fractional hieroglyphics are from the book of Georges Ifrah, referenced below.

Milo Gardner has written a number of commentaries on Egyptian mathematics. Some of them can be found on the WWW site of Kevin Brown, and by searching the MAA History of Math Archive. St. Andrews University has a good website about early Egyptian Hieroglyphic mathematics, containing this example of multiplication:

The the Rhind papyrus recommends that multiplication be done in the following way. Assume that we want to multiply 41 by 59. Take 59 and add it to itself, then add the answer to itself and continue:- 41 59 ______________ 1 59 2 118 4 236 8 472 16 944 32 1888 ______________ Since 64 is greater than 41, there is no need to go beyond the 32 entry. Now go through a number of subtractions 41 - 32 = 9, 9 - 8 = 1, 1 - 1 = 0 to see that 41 = 32 + 8 + 1. Next check the numbers in the right hand column corresponding to 32, 8, 1 and add them. 59 ______________ 1 59 X 2 118 4 236 8 472 X 16 944 32 1888 X ______________ 2419 Notice that the multiplication is achieved with only additions, notice also that this is a very early use of binary arithmetic. Reversing the factors we have 59 41 ______________ 1 41 X 2 82 X 4 16 8 328 X 16 656 X 32 1312 X _______________ 2419

Here are some speculations that, as far as I know, are unsupported by evidence, but are fun: What if the Egyptians had used the mouth symbol by itself, not to stand for a fraction, but for the only number, 0, that has no finite reciprocal? Then, the Egyptian zero (mouth) would look a lot like the Mayan zero (eye). What if the Egyptians had used the other Eye of Horus to correspond to the six numbers 2 4 8 16 32 64 so that, together, both Eyes of Horus

could represent the 64+64 dimensional even Clifford algebra Cle(0,8) used in the D4-D5-E6-E7-E8 VoDou Physics model? What if the Egyptians, who liked unit fractions, had known that the Golden Mean PHI could be represented by the continued fraction 1 PHI = 1 + _________________________ 1 1 + __________________ 1 1 + ___________ 1 1 + _____ 1+... HOW DID ALL THIS WELL-DEVELOPED HIEROGLYPHIC LANGUAGE AND MATHEMATICS APPEAR ABOUT 5000 YEARS AGO? WHERE DID IT COME FROM? Egyptian hieroglyphics seem to have been a fully developed system of writing at least 5,000 years ago. Perhaps Egyptian and Mayan hieroglyphics are regional forms of a Global Early Language picture-writing, and Chinese characters are a simplified abstract version of the same Global Early Language picture-writing. Perhaps such languages as cuneiform Sumerian and the Sarasvati-Sindhu language of India are less sophisticated writings used in trade and commerce.

Charles Muses, in 1955, saw on the Edfu temple walls some Egyptian numerical symbols for the number 240 mentioned together with Thoth (who was associated with learning and the ogdoad number 8). Muses says the 5-pointed star was an Egyptian numerical notation for 5, and a spiral for 100, and there were 2 spirals and 8 stars.

Muses also noted that the number 240 can be represented in a 3x3 square with a double spiral in the center and 8 stars (5-pointed) around it, similar to a 3x3 Magic Square, and that similar symbols are found in a Central American codex, called Codex Fejevary-Mayer.

According to The Message of the Sphinx, by Hancock and Bauval, The Book of Two Ways was written on Egyptian coffins about 4,000 years ago. It refers to the Highland of Aker, which is the Dwelling Place of Osiris. In the part of the world that is Egypt, the Highlands might refer to the Abyssinian Highlands where the Blue Nile originates. It is also the place of the East African Rift Valleys, the only continental area on Earth where plate tectonics is now pulling a continent apart. A 4-story church is carved out of a solid rock, in the same part of the world with Rift Valley landscape:

The history of the people there is written in the Kbre Nege'st. According to Hancock and Bauval, the word Aker is similar to the word Akeru meaning ancestors of Re, and to the word Akhu meaning Venerable, Transfigured Being, Shining Being, or Astral Spirit. To me, Aker reminds me of Akira. At about the same time that the Egyptian civilization developed along the Nile River, below the Highland of Aker created by the spreading Rift Valleys, the Chinese civilization developed along the Wei River, near Xi'an, below the mountains created by compression of India's collision with Asia. As Egypt has a Great Pyramid at Giza, China has a Great Pyramid 40 miles southwest of Xi'an:

According to Childress, China's Great Pyramid is said to be at least 4,000 years old, and likely much older. It is said that its sides were originally painted black on the North, blue-gray (or faded green) on the East, red on the South, white on the West, and yellow on the top Center platform. More Chinese Pyramids exist. Another possible Chinese connection is a legendary foundation of Taoism by people from Shen Zhou, or Spirit Ships, which some people say came from a Pacific Ocean "Atlantis", but which may have come from the Nile Lakes homeland of the Sphinx-Giza Civilization. Another indication of connection is that the all Egyptian art and architecture was set out on a grid of 19 lines, just as a Chinese Wei-Qi board is a grid of 19 lines. The Mayans may have used similar grids.

References: A History of Mathematics, by Victor J. Katz (HarperCollins 1993); From One to Zero, by Georges Ifrah (Penguin 1987); The Message of the Sphinx, by Hancock and Bauval (Crown 1996); Lost Cities of China, Central Asia, and India, by David Hatcher Childress (Adventures Unlimited Press 1991).

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