By Edwin Barnes 7 Kolbe St Joseph’s Catholic and Anglican School Egyptian Mathematics By Edwin Barnes 7 Kolbe St Joseph’s Catholic and Anglican School

Egyptian Number Symbols The Egyptians didn’t use numbers like we do, they used symbols to represent the numbers instead. = 1 000 = 10 000 = 100 = 1 000 000 = 100 000 = 10 = 1 The symbol for a hundred thousand is a frog, sometimes as a tadpole. The symbol for a million is a god called Heh. It also means just a very large number, like 'squillion'. The symbols reflect everyday Egyptian things. This symbol is a coil of rope. The symbol for one may come from a finger. Everyone starts off counting on their fingers! The symbols get more complicated as the numbers get bigger. The symbol for ten is a piece of rope. The symbol for a thousand is a water lily. It shows the leaf, stem and root, but not the flower. The symbol for ten thousand is a finger. Perhaps it is a finger ten thousand times as big as the symbol for one!

Numbers In some ways, this system is better than ours because if we want to write the number one million, we have seven digits to write, but the Egyptians only had to write one. But the downside of the system is that if they wanted to write the number one less than one million, 999,999, they would have to write 9 one hundred thousands, 9 ten thousands and so on down to 9 ones, using a massive total of 54 digits! To make a number, the Egyptians would write down the symbols in order of largest to smallest from left to right to form groups. For example: = 22 = 7 = 266

Addition 12 + 43 = 55 The method of addition was very simple They collected together the symbols that were the same from both of the numbers If there were ten of the same symbol, it could then be substituted for the symbol higher up in value This is an example of Egyptian Addition. 12 + 43 = 55

Subtraction To subtract a number, the Egyptian would write down the two numbers, and then take away the symbols that appeared in the second one from the first one. For example: - = 26 12 14

Subtraction This is, however, complicated when more of a symbol are to be taken away than there are present, for example 63 - 48 In this case, they would convert one of the tens into ten units, and use it to complete the calculation, for example: - = 15

Multiplication 12 x 17 They drew a table like this: And this column is the number in the first column multiplied by 12 The Egyptians used quite a complex method of multiplication. We will use the example of 12 x 17 The Egyptians would find the numbers in the first column that add up to 17, which is 1 + 16 Then they would find the multiples of 12 which correspond, and add them together like so: 1 12 2 24 4 48 8 96 16 192 This column is all the POWERS OF TWO It is easy to complete these columns because, as you are only using powers of two, each number is the number above it doubled 12 192 = 204 +

They would draw the “powers of two” table again, but this time using 3 Division 42 / 3 Division in Ancient Egyptian times required the use of multiplication and often involved fractions. This is because the Egyptian scribes recognised that division is the inverse to multiplication, and used that fact to help them work out divisions They didn’t divide as we would, but asked themselves a x ? = b (instead of a / b = ?) We can use the example 42 divided by 3. They would draw the “powers of two” table again, but this time using 3 They then found the corresponding powers of 2 and added them together to find the answer They would keep going until 42 could be made in the right column 1 3 2 6 4 12 8 24 6 + 12 + 24 = 42 2 4 8 + + 14 =

Fractions Here is an example of an Egyptian Fraction The fractions that the Egyptians used were not so different to ours, except that they were limited to the use of unit fractions, which are fractions that have a numerator of 1. The fractions they use do not have a numerator and a denominator, they are made up of a number and then a “mouth” symbol on top of it. Here is an example of an Egyptian Fraction This symbol represents a “part”. It is like the numerator, but always means 1 This represents the fraction ½ This is the denominator

There are some rules about expressing fractions as a sum in this way: When a fraction can be expressed in more than one form, use the form which requires the least number of unit fractions. Always use the largest unit fraction possible unless this means that the previous rule cannot be complied with. No unit fraction may be used more than once in an expression, so you can’t write ⅔ as ⅓ + ⅓ Write the unit fractions in order of size from largest to smallest. If an Egyptian wanted to write a fraction with a numerator of more than one, such as ¾, ⅔ or ⅜, then they would have to express it as several unit fractions added together. For example: + ⅔ =

The Eye of Horus + + + + + ... 1 8 - 1 4 - 1 16 - 1 2 - 1 32 - 1 64 - The Eye of Horus is an iconic symbol often associated with Ancient Egypt and its mathematics. The name comes from the Egyptian God of mathematics, Horus. It represents the basis of Egyptian mathematics- the unit fractions. It also symbolizes the ancient mathematical concept of infinity Each part of the Eye represents a fraction + + + + + ... Each fraction is half the one before it, and the fractions keep going on, getting smaller and smaller 1 8 - 1 4 - 1 16 - 1 2 - The idea is that, if you go on forever and then you add all the fractions together, eventually you’ll get to 1 1 32 - 1 64 -

The Eye of Horus For example: At this point, we reached a total of 8191/8192, or 0.9998779296875, which is very, very close to 1, but not quite there. The more fractions you add on, the closer you get to 1, but you will never actually reach 1 unless you carry on to infinity 1 2 - 64 4 8 16 32 + 1 128 - 256 + 1024 512 1 4096 - 2048 8192 +

Units of Measurement 4 Digits = 1 Palm 7 Palms = 1 Cubit The Egyptians used their own body to measure things around, which often meant that measurements were not very accurate or consistent 1 Digit 4 Digits = 1 Palm 7 Palms = 1 Cubit 1 Cubit

Thank you for watching my presentation By Edwin Barnes 7 Kolbe St Joseph’s Catholic and Anglican School