The early Egyptians
settled along the fertile Nile valley as early as about 6000 BC, and
they began to record the patterns of lunar phases and the seasons, both
for agricultural and religious reasons. The Pharaoh’s surveyors used
measurements based on body parts (a palm was the width of the hand, a
cubit the measurement from elbow to fingertips) to measure land and
buildings very early in Egyptian history, and a decimal numeric system
was developed based on our ten fingers. The oldest mathematical text
from ancient Egypt discovered so far, though, is the Moscow Papyrus,
which dates from the Egyptian Middle Kingdom around 2000 - 1800 BC
.
It is thought that the Egyptians introduced the earliest fully-developed base 10 numeration system at least as early as 2700 BC (and probably much early). Written numbers used a stroke for units, a heel-bone symbol for tens, a coil of rope for hundreds and a lotus plant for thousands, as well as other hieroglyphic symbols for higher powers of ten up to a million. However, there was no concept of place value, so larger numbers were rather unwieldy (although a million required just one character, a million minus one required fifty-four characters).
The Rhind Papyrus, dating from around 1650 BC, is a kind of
instruction manual in arithmetic and geometry, and it gives us explicit
demonstrations of how multiplication and division was carried out at
that time. It also contains evidence of other mathematical knowledge,
including unit fractions, composite and prime numbers, arithmetic,
geometric and harmonic means, and how to solve first order linear
equations as well as arithmetic and geometric series. The Berlin
Papyrus, which dates from around 1300 BC, shows that ancient Egyptians
could solve second-order algebraic (quadratic) equations.
Multiplication, for example, was achieved by a process of repeated doubling of the number to be multiplied on one side and of one on the other, essentially a kind of multiplication of binary
factors similar to that used by modern computers (see the example at
right). These corresponding blocks of counters could then be used as a
kind of multiplication reference table: first, the combination of powers
of two which add up to the number to be multiplied by was isolated, and
then the corresponding blocks of counters on the other side yielded the
answer. This effectively made use of the concept of binary numbers,
over 3,000 years before Leibniz introduced it into the west, and many more years before the development of the computer was to fully explore its potential.
Practical problems of trade and the market
led to the development of a notation for fractions. The papyri which
have come down to us demonstrate the use of unit fractions based on the
symbol of the Eye of Horus, where each part of the eye represented a
different fraction, each half of the previous one (i.e. half, quarter,
eighth, sixteenth, thirty-second, sixty-fourth), so that the total was
one-sixty-fourth short of a whole, the first known example of a
geometric series.
Unit fractions could also be used for simple division sums. For
example, if they needed to divide 3 loaves among 5 people, they would
first divide two of the loaves into thirds and the third loaf into
fifths, then they would divide the left over third from the second loaf
into five pieces. Thus, each person would receive one-third plus
one-fifth plus one-fifteenth (which totals three-fifths, as we would
expect).
The Egyptians approximated the area of a circle by using shapes whose area they did know. They observed that the area of a circle of diameter 9 units, for example, was very close to the area of a square with sides of 8 units, so that the area of circles of other diameters could be obtained by multiplying the diameter by 8⁄9 and then squaring it. This gives an effective approximation of π accurate to within less than one percent.
The pyramids themselves are another indication of the sophistication of Egyptian mathematics. Setting aside claims that the pyramids are first known structures to observe the golden ratio of 1 : 1.618 (which may have occurred for purely aesthetic, and not mathematical, reasons), there is certainly evidence that they knew the formula for the volume of a pyramid - 1⁄3 times the height times the length times the width - as well as of a truncated or clipped pyramid. They were also aware, long before Pythagoras, of the rule that a triangle with sides 3, 4 and 5 units yields a perfect right angle, and Egyptian builders used ropes knotted at intervals of 3, 4 and 5 units in order to ensure exact right angles for their stonework (in fact, the 3-4-5 right triangle is often called "Egyptian")
.
It is thought that the Egyptians introduced the earliest fully-developed base 10 numeration system at least as early as 2700 BC (and probably much early). Written numbers used a stroke for units, a heel-bone symbol for tens, a coil of rope for hundreds and a lotus plant for thousands, as well as other hieroglyphic symbols for higher powers of ten up to a million. However, there was no concept of place value, so larger numbers were rather unwieldy (although a million required just one character, a million minus one required fifty-four characters).
 |
 Ancient Egyptian method of multiplication |
Multiplication, for example, was achieved by a process of repeated doubling of the number to be multiplied on one side and of one on the other, essentially a kind of multiplication of binary
factors similar to that used by modern computers (see the example at
right). These corresponding blocks of counters could then be used as a
kind of multiplication reference table: first, the combination of powers
of two which add up to the number to be multiplied by was isolated, and
then the corresponding blocks of counters on the other side yielded the
answer. This effectively made use of the concept of binary numbers,
over 3,000 years before Leibniz introduced it into the west, and many more years before the development of the computer was to fully explore its potential.Practical problems of trade and the market
led to the development of a notation for fractions. The papyri which
have come down to us demonstrate the use of unit fractions based on the
symbol of the Eye of Horus, where each part of the eye represented a
different fraction, each half of the previous one (i.e. half, quarter,
eighth, sixteenth, thirty-second, sixty-fourth), so that the total was
one-sixty-fourth short of a whole, the first known example of a
geometric series.
 |
|
Ancient Egyptian method of division |
The Egyptians approximated the area of a circle by using shapes whose area they did know. They observed that the area of a circle of diameter 9 units, for example, was very close to the area of a square with sides of 8 units, so that the area of circles of other diameters could be obtained by multiplying the diameter by 8⁄9 and then squaring it. This gives an effective approximation of π accurate to within less than one percent.
The pyramids themselves are another indication of the sophistication of Egyptian mathematics. Setting aside claims that the pyramids are first known structures to observe the golden ratio of 1 : 1.618 (which may have occurred for purely aesthetic, and not mathematical, reasons), there is certainly evidence that they knew the formula for the volume of a pyramid - 1⁄3 times the height times the length times the width - as well as of a truncated or clipped pyramid. They were also aware, long before Pythagoras, of the rule that a triangle with sides 3, 4 and 5 units yields a perfect right angle, and Egyptian builders used ropes knotted at intervals of 3, 4 and 5 units in order to ensure exact right angles for their stonework (in fact, the 3-4-5 right triangle is often called "Egyptian")
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