GRACE MISSION COLLEGE
Catiningan, Socorro, Oriental Mindoro
e-Mail: grace.missioncollege@yahoo.com
Fear of the Lord is the foundation of true knowledge, but fools despise wisdom and discipline.
-Proverbs 1:7
Module 1 in HISTORY OF MATHEMATICS
EGYPTIAN MATHEMATICS
EGYPTIAN MATHEMATICS
The earliest Egyptian settlements that
were located along the Nile date back to 6000
BCE. The use of mathematical practices in
Ancient Egypt was tightly connected with
religion and agriculture. The Egyptians were
known to record the seasons and describe the
patterns of moon phases.
System of measurement
At the dawn of their civilization, the
Egyptians developed a unique system of
measurement. It was only natural that the ancient
units of measurement were based on body parts as
the easiest way to find the size of an object is to
compare it with the length of various parts of the
body. Let’s explore some units of length for a
better understanding of the system:
The finger, the basic unit, was about 1.875 cm
Four fingers constituted the palm that was
approximately 7.5 cm
Seven palms constituted the cubit that was about 52.5 cm
Symbols
God heh or a man with both hands raised was the symbol for a million, sometimes used for infinity
also. A symbol resembling a circle was again used to represent infinity because a journey through a circle
never ends.
Multiples of these values were expressed by repeating the symbol as many times as needed. As they did not
have a concept of zero, there was no symbol for it.
Numerals
According to scientific evidence, the Egyptians were the first to develop the base 10 numeration
system. Again, using body parts, they created a relatively simple system that was based on ten figures.
Archeologists believe that the Egyptians invented the decimal numeric system around 3000 - 2700 BCE. The
Egyptians used the following hieroglyphs to represent numerals:
ANCIENT EGYPTIANS NUMERAL SYSTEM
A stroke was used for units – one stroke meant “one,” two strokes stood for “two,” etc.
An arch was used for tens – one arch was used for “ten,” two arches represented “twenty,” etc.
A coiled rope was used for hundreds – one coiled rope meant “100,” while three coiled ropes were
used to represent “three hundred,” etc.
A lotus plant was used for thousands – one flower denoted “1,000,” four flowers meant “4,000,” etc.
The number 10,000 was depicted by a finger
A frog was used to depict the number 100,000
A hieroglyph depicting a kneeled god with raised arms denoted 1,000,000.
Fractions
The development of trade led to the appearance of fractions. According to archeological evidence, the
ancient Egyptians introduced fractions as early as 1000 BCE. The reciprocal of a fraction was depicted by the
hieroglyph “mouth.” Besides, they had special hieroglyphs to write such fractions as 1/2, 2/3, and 3/4. The
use of fractions helped them to find the approximate area of a circle – they noticed that the area of a circle that
has a diameter of 9 units is equivalent to the area a square of side 8 units. Therefore, to find areas of other
circles they multiplied their diameter by 8/9 and then squared it. Thus, the ancient Egyptians obtained the
approximate value of p.
ACTIVITY 1
Direction: Convert the following Egyptians hieroglyphics into modern day numbers.
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ACTIVITY 2
Direction: Add or subtract the given Egyptians Hieroglyphics.
THE AHMES PAPYRUS
The Ahmes was written in hieratic, and probably originated from the Middle Kingdom: 2000-1800 BC. It
claims to be a ``thorough study of all things, insight into all that exists, knowledge of all obscure secrets." In
fact, it is somewhat less. It is a collection of exercises, substantially rhetorical in form, designed primarily
for students of mathematics. Included are exercises in
fractions
notation
arithmetic
algebra
geometry
mensuration
THE RHIND PAPYRUS
The Rhind Papyrus, dating from around 1650 BCE,
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 BCE, shows that ancient Egyptians could
solve second-order algebraic (quadratic) equations.
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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). Ancient Egyptian method of division
MOSCOW PAPYRUS
The Moscow Papyrus is one of the oldest written mathematical texts from
ancient Egypt around 2000 B.C to 1800 B.C. The Moscow Papyrus contains 25
mathematical problems. Some of the problems are unreadable or too damaged to
translate. Problem 14 from the Moscow Papyrus shows an illustration with an example
to find the volume of a truncated pyramid. The modern formula for a similar figure,
called frustum, is V= (a^2 + a*b + B^2)*(h/3), where a,b, and h are shown in the
diagram below.
The text is now located in the largest museum in Europe,
the Pushkin Museum of Fine Arts in Russia. This ancient piece
of mathematical history inspires awe and wonder in those
appreciative of mathematics because the Egyptians never
explained “how” their example in problem 14 worked, nor did
they show any deductive reasoning behind this problem.
Finding the volume of a truncated pyramid is very challenging if doing so by experiments alone. One
cannot just stumble to this conclusion just by trial and error. No one knows how the Egyptians derived the
formula, and considering that the author of the text remains unknown, we might never know. However, many
historians of mathematics have their theories, but even their theories are imminent. Mathematical historians
have little to no evidence to prove their theories, since the text was written over 4000 years ago.
4
Amazingly, considering the age of the Moscow Papyrus, it is still being carefully examine by
Egyptologists. This is a valuable piece of history and is worthy of being recognized in a great museum like
the Pushkin Museum. One can only wonder of the brilliance or good fortune of the Egyptians.
ACTIVITY 3
Direction: Answer the following questions:
1. How did the Ancient Egyptians contribute in the field of Mathematics today?
2. What do you think are the common problems encountered by ancient Egyptians in their numeration
system?
RESOURCES:
https://mathandmind.com/articles/egyptian-mathematics
https://igonzalez1.weebly.com/blog/the-moscow-papyrus
https://www.storyofmathematics.com/egyptian.html/
ADDITIONAL INFORMATION
Visit https://youtu.be/E0LLyYCk9yM
Prepared by:
09127648540
Jeorge Ornedo Hugno
jeorgeornedohugno14@gmail.com
SIR JEORGE O. HUGNO
Subject Teacher
“As far as the laws of mathematics refer to reality, they are not certain, and as far as
they are certain, they do not refer to reality.”
- Albert Einstein
