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 | 24 | |||

47 | 1 | doubling process | ||

94 | 2 | |||

188 | 4 | |||

376 | 8 | * | ||

752 | 16 | * |

Selecting 8 and 16 (i.e. ), we have

**Division** is also basically binary.

Example: Divide:

329 | 12 | ||||

12 | 1 | doubling | 329 | ||

24 | 2 |
-192 |
|||

48 | 4 | 137 | |||

96 | 8 | -96 |
|||

192 | 16 | 41 | |||

384 | 32 | -24 |
|||

17 | |||||

-12 |
|||||

5 |

So,

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, and .
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 (
,and , 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:

Ahmes gives a table of unit fractions.

1/p | + | 1/q | + | 1/r+ | ||

5 | 3 | 15 | ||||

7 | 4 | 28 | ||||

9 | 6 | 18 | ||||

11 | 6 | 66 | ||||

13 | 8 | 52 | 104 | |||

15 | 10 | 30 | ||||

- 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 as

- 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

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.