# History and Numbers

## Introduction

The civilization of Ancient Egypt spans some 3000 years of history. In ancient times the country was first unified under a king named Narmer - sometimes called Menes - in ca 3000 BCE. The ruling families of ancient egypt were roughly divided into dynasties. There are all together some 30 dynasties that span the history of this ancient civilization. The fourth dynasty was the ruling family responsible for the construction of the famous pyramids at Giza. The Great Pyramid of Khufu is the topic of many scholarly (and not so scholarly) books. The architects of the time must have had knowledge of mathematics - and more specifically of geometry - in order to construct not only their pyramids, but also the temples they constructed.

We have no mathematical papyri from the time of the pyramids of Giza. There are however some temple reliefs predating that time period by several decades showing rituals associated with planning the temples that accompanied the pyramids. The reliefs show the King marking off predetermined lengths with rope to create the correct dimensions for the temples.

Our present day knowledge of much of the mathematics of the Ancient Egyptians comes from the 12th dynasty. This family ruled Egypt some 1000 years after the pyramids at Giza were constructed. There are two major sources: the Moscow Papyrus - which dates to ca. 1800 BCE - and the Rhind or Ahmes papyrus - which dates to ca 1900 BCE, but is likely a copy from an original dating to ca 1800 BCE.

In addition to these two papyri there are several other smaller fragments as well as tomb inscriptions and notes written on pottery shards (so called ostraca) that give us a glimpse into the mind of the Egyptian scientists.

Experts have cautioned us against drawing too wide ranging conclusions about what the Egyptians may or may not have known. We only have a handful of ancient sources. There is no way to know if we have a reasonable cross-section of Egyptian knowledge of mathematics, or if we only have a glimpse.

## The Egyptian Number System and Mathematical Notation

The Ancient Egyptians used a base 10 number system. The number one was depicted by a simple stroke, the number 2 was represented by two stokes, etc. The numbers 10, 100, 1000, 10,000 and 1,000,000 had their own hieroglyphs.

Higher numbers were assigned their own hieroglyphs. Number 10 is a hobble for cattle, number 100 is represented by a coiled rope, the number 1000 is represented by a lotus flower, the number 10,000 is represented by a finger, the number 100,000 is represented by a frog and a million was represented by a god with his hands raised in adoration.

The number system is shown is for instance cattle counts and in reliefs depicting offering scenes.

Cattle Count, Ancient Egypt. From Lepsiius Denkmahler.

The scene above was copied by the German Egyptologist Lepsius. The scene depicts a cattle count. In the middle register we see 835 horned cattle on the left, right behind them are some 220 animals (cows?) and on the right 2235 goats. In the bottom register we see 760 donkeys on the left and 974 goats on the right. These scenes would have depicted the wealth of the tomb owner and the depiction on the tomb wall also meant the tomb owner would have these animals with him in the afterlife.

King's Son Wepemnefret. Phoebe A. Hearst Museum of Anthropology. Photograph by Bruce White. From: Slab Stelae of the Giza Necropolis By Peter Der Manuelian - 2003

In this stela of the King's Son Wepemnefret we see the deceased sitting before a table full of offerings. The items on the table are meant to depict bread. Notice the numbers that occur everywhere in the text. This is a depiction of all the items Wepemnefret wanted access to in the afterlife. Next to his knee for instance you see what looks like an inverted Y and loop of some sort with the lotus flowers below them. The left symbol represents linnen and the symbol next to it represents alabaster vessels. The numerals below them mean that he has offerings of 1000 pieces of linnen and 1000 alabaster vessels. On the rigth below the table we see offerings of 1000 pieces of bread, 1000 jars of beer, 1000 antelopes and 1000 oxen. The rest of the text refers to even more offerings.

## Multiplication and Division

### Multiplication

The Egyptians used a technique called "repeated doubling" to compute their products. The technique is most easily explained through an example. Suppose we wish to compute $11 \times 15$. We first find several easy multiples of 15:

$1 X 15 = 15$ √
$2 \times 15 = 30$ √
$4 \times 15 = 60$
$8 \times 15 = 120$ √

Note that $11=1+2+8$