Egyptian MathematicsEgyptian MathematicsBy Edwin BarnesBy Edwin Barnes

7 Kolbe7 KolbeSt Joseph’s Catholic and Anglican St Joseph’s Catholic and Anglican

SchoolSchool

Egyptian Number SymbolsEgyptian Number Symbols

The Egyptians didn’t use numbers like we The Egyptians didn’t use numbers like we do, they used symbols to represent the do, they used symbols to represent the

numbers instead.numbers instead.

= 1= 1

The symbol for one may come The symbol for one may come from a finger. Everyone starts from a finger. Everyone starts off counting on their fingers! off counting on their fingers!

= 10= 10

The symbols get more The symbols get more complicated as the numbers complicated as the numbers get bigger. The symbol for ten get bigger. The symbol for ten is a piece of rope. is a piece of rope.

= 100= 100

The symbols reflect everyday The symbols reflect everyday Egyptian things. This symbol is a Egyptian things. This symbol is a coil of rope. coil of rope.

The symbol for a thousand is a The symbol for a thousand is a water lily. It shows the leaf, water lily. It shows the leaf, stem and root, but not the stem and root, but not the flower.flower.

= 1 000= 1 000= 10 000= 10 000

The symbol for ten thousand is a The symbol for ten thousand is a finger. Perhaps it is a finger ten finger. Perhaps it is a finger ten thousand times as big as the thousand times as big as the symbol for one! symbol for one!

= 100 000= 100 000

The symbol for a hundred The symbol for a hundred thousand is a frog, sometimes thousand is a frog, sometimes as a tadpole. as a tadpole.

= 1 000 000= 1 000 000

The symbol for a million is a The symbol for a million is a god called Heh. It also means god called Heh. It also means just a very large number, like just a very large number, like 'squillion'. 'squillion'.

NumbersNumbersTo make a number, the Egyptians would write To make a number, the Egyptians would write down the symbols in order of largest to smallest down the symbols in order of largest to smallest from left to right to form groups. For example:from left to right to form groups. For example:

= 7= 7= 22= 22

= 266= 266

In some ways, this system is better than ours In some ways, this system is better than ours because if we want to write the number one because if we want to write the number one million, we have seven digits to write, but the million, we have seven digits to write, but the

Egyptians only had to write one.Egyptians only had to write one.

But the downside of the system is that if they But the downside of the system is that if they wanted to write the number one less than one wanted to write the number one less than one

million, 999,999, they would have to write 9 million, 999,999, they would have to write 9 one hundred thousands, 9 ten thousands and one hundred thousands, 9 ten thousands and so on down to 9 ones, using a massive total of so on down to 9 ones, using a massive total of

54 digits!54 digits!

AdditionAddition The method of addition was very simpleThe method of addition was very simple They collected together the symbols that They collected together the symbols that

were the same from both of the numberswere the same from both of the numbers If there were ten of the same symbol, it If there were ten of the same symbol, it

could then be substituted for the symbol could then be substituted for the symbol higher up in valuehigher up in value

This is an example of Egyptian Addition. This is an example of Egyptian Addition.

SubtractionSubtraction

To subtract a number, the Egyptian would To subtract a number, the Egyptian would write down the two numbers, and then write down the two numbers, and then

take away the symbols that appeared in take away the symbols that appeared in the second one from the first one.the second one from the first one.

For example:For example:

SubtractionSubtraction

This is, however, complicated when more This is, however, complicated when more of a symbol are to be taken away than of a symbol are to be taken away than there are present, for example 63 - 48 there are present, for example 63 - 48

In this case, they would convert one of In this case, they would convert one of the tens into ten units, and use it to the tens into ten units, and use it to complete the calculation, for example:complete the calculation, for example:

MultiplicationMultiplicationThey drew a table like this:They drew a table like this:

This column is

all the POWERS OF TWO

And this column is the number in the first column multiplied by 12

The Egyptians would find The Egyptians would find the numbers in the first the numbers in the first column that add up to 17, column that add up to 17, which is 1 + 16which is 1 + 16

Then they would find the Then they would find the multiples of 12 which multiples of 12 which correspond, and add them correspond, and add them together like so:together like so:

It is easy to complete these columns because, as you are only using powers of two, each number is the number above it doubled

The Egyptians used quite a complex The Egyptians used quite a complex method of multiplication.method of multiplication.

We will use the example of 12 x 17We will use the example of 12 x 17

DivisionDivision Division in Ancient Egyptian times required Division in Ancient Egyptian times required

the use of multiplication and often involved the use of multiplication and often involved fractions. fractions.

This is because the Egyptian scribes This is because the Egyptian scribes recognised that division is the inverse to recognised that division is the inverse to multiplication, and used that fact to help multiplication, and used that fact to help them work out divisionsthem work out divisions

They didn’t divide as we would, but asked They didn’t divide as we would, but asked themselves themselves a x ? = ba x ? = b (instead of a / b = ?) (instead of a / b = ?)

We can use the example 42 divided by 3.We can use the example 42 divided by 3.

They would draw the “powers of two” table They would draw the “powers of two” table again, but this time using 3again, but this time using 3

They would keep They would keep going until 42 could going until 42 could be made in the right be made in the right columncolumn

They then found the They then found the corresponding corresponding powers of 2 and powers of 2 and added them added them together to find the together to find the answeranswer

FractionsFractions

The fractions that the Egyptians used were The fractions that the Egyptians used were not so different to ours, except that they were not so different to ours, except that they were limited to the use of limited to the use of unit fractionsunit fractions, which , which are fractions that have a numerator of 1.are fractions that have a numerator of 1.

The fractions they use do not have a The fractions they use do not have a numerator and a denominator, they are numerator and a denominator, they are made up of a number and then a “mouth” made up of a number and then a “mouth” symbol on top of it.symbol on top of it.

Here is an example of an Egyptian FractionHere is an example of an Egyptian Fraction

This symbol represents a “part”. It is like the

numerator, but always means 1

This is the denominator

This represents the fraction ½ This represents the fraction ½

FractionsFractions

If an Egyptian wanted to write a fraction If an Egyptian wanted to write a fraction with a numerator of more than one, such with a numerator of more than one, such as as ¾, ⅔ or ⅜, then they would have to ¾, ⅔ or ⅜, then they would have to express it as several unit fractions added express it as several unit fractions added together. For example:together. For example:

There are some rules about expressing fractions as There are some rules about expressing fractions as a sum in this way:a sum in this way:

1.1. When a fraction can be expressed in more than When a fraction can be expressed in more than one form, use the form which requires the least one form, use the form which requires the least number of unit fractions. number of unit fractions.

2.2. Always use the largest unit fraction possible Always use the largest unit fraction possible unless this means that the previous rule cannot unless this means that the previous rule cannot be complied with. be complied with.

3.3. No unit fraction may be used more than once in No unit fraction may be used more than once in an expression, so you can’t write an expression, so you can’t write ⅔ as ⅓ + ⅓⅔ as ⅓ + ⅓

4.4. Write the unit fractions in order of size from Write the unit fractions in order of size from largest to smallest.largest to smallest.

The Eye of HorusThe Eye of HorusThe Eye of Horus is an iconic The Eye of Horus is an iconic

symbol often associated with symbol often associated with Ancient Egypt and its Ancient Egypt and its mathematics. mathematics.

The name comes from the Egyptian The name comes from the Egyptian God of mathematics, Horus. God of mathematics, Horus.

It represents the basis of Egyptian It represents the basis of Egyptian mathematics- the unit fractions. mathematics- the unit fractions.

It also symbolizes the ancient It also symbolizes the ancient mathematical concept of infinitymathematical concept of infinity

Each part of the Eye represents a fractionEach part of the Eye represents a fraction

Each fraction is half the one before it, Each fraction is half the one before it, and the fractions keep going on, and the fractions keep going on,

getting smaller and smallergetting smaller and smaller

The idea is that, if you go on forever The idea is that, if you go on forever and then you add all the fractions and then you add all the fractions together, eventually you’ll get to 1together, eventually you’ll get to 1

The Eye of HorusThe Eye of HorusFor example:For example:

At this point, we reached a total of At this point, we reached a total of 8191/8192, or 0.9998779296875, 8191/8192, or 0.9998779296875,

which is very, very close to 1, but not which is very, very close to 1, but not quite there. The more fractions you quite there. The more fractions you add on, the closer you get to 1, but add on, the closer you get to 1, but

you will never actually reach 1 unless you will never actually reach 1 unless you carry on to infinityyou carry on to infinity

Units of MeasurementUnits of Measurement

The Egyptians used their own body to The Egyptians used their own body to measure things around, which often meant measure things around, which often meant that measurements were not very accurate that measurements were not very accurate

or consistentor consistent

1 Palm1 Palm

1 Cubit1 Cubit

1 Digit1 Digit

Thank you for Thank you for watching my presentationwatching my presentation

By Edwin Barnes By Edwin Barnes 7 Kolbe7 Kolbe

St Joseph’s Catholic and Anglican St Joseph’s Catholic and Anglican SchoolSchool