Mathematical Papyrus

Explanation

Long time ago I planned to write a fake message from an extraterrestrial civilization, where the first part of the message must teach the reader extraterrestrial mathematical symbolism and the second part includes a powerful notation for large numbers. But then I decided, instead of this, to write the same stuff in the ancient Egyptian language, since I love ancient Egypt history and because Egyptian hieroglyphic text is no more understandable for usual modern man than extraterrestrial one. Ancient math symbolism and hieroglyphic writing of words were taken mainly from "The Rhind/Ahmes Papyrus" and Alan Gardiner’s "Egyptian Grammar". So, on this page you can see a fake mathematical papyrus introducing "Imhotep’s Number". The text was written such that even without knowledge of Egyptian, attentive reader can understand mathematical symbolism and the notation. If you want, try to decode it yourself, if not, read below the translation in English with modern symbolism.

Translation in English

Book of Imhotep

The book was written by Imhotep in the 12-th regnal year of Djoser, the king of Upper and Lower Egypt.

1+2=3 3-2=1 3 >2 2>1

3+2=5 5-2=3 5 >3 3>2

4+4=8 8-4=4 8>4

0+1=1 1-1=0 1>0

A=2 and b=3 then A+b=5 and b>A

A=5 and b=3 then A+b=8 and A>b

A=4 and b=4 then A+b=8

b=0 then A+b=A and b+A=A and A+1>b

A>b and b>H then A>H

calculate s(1000000, m(0, m(1000000, 0)), 1000000)

A= 0 then k(A)=0 and m(b, H)>A

A=m(b, 0) then k(A)=A and A(i)=i

A=b+H and H>0 then k(A)=k(H) and A(i)=b+H(i)

A=m(b, H) and k(H)= m(0, 0) then k(A)=w and A(0)= m(b, H(0)) and A(i+1)=A(i)+m(b, H(0))

A=m(b, H) and k(H)=w then k(A)=w and A(i)=m(b, H(i))

A=m(b, H) and k(H)=m(t+1, 0) and b>t then k(A)=k(H) and A(i)=m(b, H(i))

A=m(b, H) and k(H)=m(t+1, 0) and t+1>b then k(A)=w and A(i)=m(b, H(x(i))) and x(0)=m(t, 0) and x(p+1)=m(t, H(x(p)))

s(A, 0, b)=A+b

k(b)= m(0, 0) then s(A, b, 1)=A

k(b)= m(0, 0) and t>0 then s(A, b, t+1)=s(A, b(0), s(A, b, t))

k(b)=w then s(A, b, t)=s(A, b(t), A)

s(5, 0, 5)=10

s(10, m(0, 0)+m(0, 0), 6)= 1000000

s(10, m(0, m(0, 0)), 3)=s(10, m(0, 0)+m(0, 0)+m(0, 0)+m(0, 0), 10)

s(4, m(0, m(0, 0))+m(0, m(0, 0))+m(0, m(0, 0)), 2) = s(4, m(0, m(0, 0))+m(0, m(0, 0))+m(0, 0)+m(0, 0)+m(0, 0), 4)

s(10, m(0, m(1, 0)), 3)=s(10, m(0, m(0, m(0, m(0, 0)))), 10)

s(1000000, m(0, m(1000000, 0)), 1000000) stars

About the "author"

Imhotep was a chancellor of the pharaoh Djoser (2667-2648 BC) and the architect of the Djoser's pyramid (historically the first Egyptian pyramid), the first who used stone columns to support a building. After death he was known in ancient Egypt as the wisest man of all times, "God of Medicine" and "Son of Ptah" (the god Ptah in Egyptian mythology was the creator of the universe). Since Djoser started to reign in 2667 BC, then the mentioned 12-th regnal year of Djoser is the year 2655 BC.

About notation for the large number

In the text we use Egyptian letters "A" (vulture), "b" (foot), "H" (wick of twisted flax), "t" (loaf), "p" (stool), "x" (placenta), "i" (flowering reed), "s" (folded cloth), "m" (owl), "k" (basket with handle), "n" (water), "a" (forearm), "w" (quail chick), "r" (mouth), triliteral signs "xpr" (scarabaeus), "nfr" (lute) and so on.

The first part, as it was said, teaches the reader mathematical symbolism:

• "ix" for "⇒" (then);
• "Hna" for "∧" (and);
• "nfr" for "0";
• "xpr:r" for "=";
• "r" for ">"(more than);
• "legs" for "+" and "−" (depending on the direction of the "legs").

Egyptians had the following signs to denote powers of ten: I (1), ⋂ (10), coil of rope (100), lotus (10^3), bent finger (10^4), tadpole (10^5) and "man with raised arms" (10^6). If they needed to express n×(10^k), where n and k are integers, 1≤n≤9 and 0≤k≤6, they just repeated n times the symbol for 10^k. For example, ⋂⋂III=23.

In the second part we can see the notation itself: it is hyper operations, extended to transfinite ordinals, with system of fundamental sequences for those ordinals, the similar system as for Buchholz’s functions ψ_ν.

Strictly saying, the rules of the system of fundamental sequences are applied only for ordinals written in the normal form, but, using this modest set of math tools, we can’t introduce the conception of transfinite ordinals and ordinal collapsing functions, can’t write an ordinal notation. So the normal form is not defined. Nevertheless, this set of rules is applicable for calculation of concrete number, which I call "Imhotep’s Number", s(10^6, m(0, m(10^6, 0)), 10^6), since "m(0, m(10^6, 0))" corresponds to the ordinal ψ_0(Ω_{10^6}) written in the normal form, and all terms, which can appear at the calculation of this number using those rules, also will correspond to ordinals written in the normal form.

Note that we use a ternary function "s" for representation of hierarchy of hyper operations; a binary function "m" corresponds to hierarchy of ordinal functions ψ_ν; spaces are used instead of commas between variables of functions; sides of cartouche – instead of round brackets; "k" plays the same role as the cofinality; "w" – as the first transfinite ordinal; m(0,0) – as 1.

In the third part several examples of working with math expressions were shown.