Counting and Aritmetic -- basics

The Egyptian counting system was decimal. Though nonpositional, it could deal with numbers of great scale. Yet, there is no apparent way to construct numbers arbitrarily large. (Compare that with modern systems, which is positional, which by its nature allows and economy for expressing huge numbers.)

The number system was decimal with special symbols for 1, 10, 100, 1,000, 10,000, 100,000, and 1,000,000. Addition was accomplished by grouping and regrouping. Multiplication and division were essentially based on binary multiples. Fractions were ubiquitous but only unit fractions, with two exceptions, were allowed. All other fractions were required to be written as a sum of unit fractions. Geometry was limited to areas, volumes, and similarity. Curiously, though, volume measures for the fractional portions of the hekat a volume measuring about 4.8 liters, were symbolically expressed differently from others.

Simple algebraic equations were solvable, even systems of equations in two dimensions could be solved.

Symbolic notation for numbers.

1	=	vertical stroke
10	=	heal bone
100	=	a snare
1,000	=	lotus flower
10,000	=	a bent finger
100,000	=	a burbot fish
1,000,000	=	a kneeling figure

Note though that there are numerous interpretations of what these hieroglyphs might represent.

Numbers are formed by grouping.

Addition is formed by grouping

Note alternate forms for these numbers.

Multiplication is basically binary.

Example: Multiply: $47\times 24$

	47	$\times$	24
	47		1	doubling process
	94		2
	188		4
	376		8	*
	752		16	*

Selecting 8 and 16 (i.e. $8+16=24$), we have

$$\begin{eqnarray*} 24&=& 16+8\\ 47\times 24&=& 47\times (16+8)\\ &=&752+376\\ &=&1128 \end{eqnarray*}$$

Division is also basically binary.

Example: Divide: $329\div 12$

	329	$\div$	12
	12		1	doubling	329
	24		2		-192
	48		4		137
	96		8		-96
	192		16		41
	384		32		-24
					17
					-12
					5

Now

$$\begin{eqnarray*} 329&=& 16\times12+8\times12+2\times 12+1\times 12 +5\\ &=&(16+8+2+1)\times 12 +5 \end{eqnarray*}$$

So,

$$329\div12=27\,{5\over12}=27+{1\over3}+{1\over12}.$$

Obviously, the distributive laws for multiplication and division were well understood.

Fractions It seems that the Egyptians allowed only unit fractions, with just two exceptions, ${2\over3}$ and ${3\over4}$. All other fractions must be converted to unit fractions. The symbol for unit fractions was a flattened oval above the denominator. In fact, this oval was the dign used by the Egyptians for the ``mouth." In the case of the volume measure hekat, the commonly used fractional parts of ( $\frac{1}{2},\>\frac{1}{4},\>\frac{1}{8},\>\frac{1}{16},\>\frac{1}{32}$,and $\frac{1}{64}$, were denoted by parts of the symbol for the ``Horus-eye."⁴ For ordinary fractions, we have the following in modern notation.

All other fractions must be converted to unit fractions. For example:

$${2\over15}={1\over 10}+{1\over 30}$$

Ahmes gives a table of unit fractions.

$2\over n$	1/p	+	1/q	+	1/r+$\dots$
5	3		15
7	4		28
9	6		18
11	6		66
13	8		52		104
15	10		30
$\vdots$

Fractions

• Decompositions are not necessarily unique. The Egyptians did favor certain fractions and attempt to use them when possible. For example, they seems to prefer taking halves when possible. Thus the representation for $2/15$ as
  
  $$\frac{2}{15}= 1/30 +1/10$$
  
  

• The exact algorithm for determination for the decomposition is unknown and this is an active topic of research today. However, in other papyrii, there is some indication of the application of the formula
  
  $$\frac{2}{p\cdot q}=\frac{1}{p\cdot \frac{p+q}{2}}+ \frac{1}{q\cdot \frac{p+q}{2}}$$
  
  
  being used. It gives some, but not all, of the table, and certainly does not give decompositions into three or more fractions.
• It seems certain that the Egyptians understood general rules for handling fractions.