Algebra: An exploration into Ancient algebra before Al-Kawarimi

From the first crude tablets in the murky swamps of Mesopotamia with instruction for tax collecting to the large supercomputers of today trying to calculate the most efficient way to get to the stars. They are all based on the abstracting real life problems into a mathematical equation and deriving information from them something which algebra helps greatly with and lends algebra it’s rich history

Algebra is the very essence of problem-solving and maths formalised without which we struggle to solve anything more than the most basic of maths problems. It can create elegant theorems proving the irrationality of the square root of two beautifully as well as deciding how much you will have to pay in taxes.Its utility extends into almost all fields responsible for our modern society from rocket science to agriculture and is simple enough to be taught to most of the world’s middle schoolers. The simple fundamental rules of algebra can allow for some of the most complicated problems to be modelled and solved. From deciding how many apples each child in a class has to simulate the 12 dimension of string theory, the number of problems that can be modelled by Algebra is almost limitless.

Algeria also has a rich history that traces its origin in the dawn of agriculture to its current status as the bedrock for the teaching of modern mathematics. From a base 60 number system and estimating the tax burden on the farmer in the swampy land of Mesopotamia to trying to prove the existence of god in the court of the Russian tsar, Algebra has left it’s marked on history just as history as left its mark on Algebra.  It follow the very history of human civilisation as it snakes from Babylon

 

Many mathematical stories begin in Ancient Greece, Algebra, however, is an exception. It traces it’s historical roots back to the very dawn of humans civilisation in the fertile crescent and snakes from there to Egypt and only much later goes to the Greeks.The world’s first cities were constructed here and the first words were written. The wrote the world’s first Epics “The epic of Gilgamesh”. In maths, they created a base 60 number system that bears more resemblance to our modern number system than the roman number system. It was similar to our modern system where each place value increases the value by a magnitude of 10 due to being base 10. In their system, each place value increases the value of a number by the magnitude of  60.

Image result for babylonian mathematical tablets
One of the world’s first example of What is today called Pythagoras theorem. The number in the middle of the square is an approximation of the square root of 2 to six significant digits. This was probably a student exercise to teach scribes and other how to do maths. This predates Pythagoras by a thousand years Link:^2

 

Image result for square root of 2 proof
This is a proof by contradiction that uses algebra to prove that the square root of 2 is irrational and is used today as a study aid for students Link:^1

 

 

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Babylonian Numbers explained

Like the Egyptians, the Babylonians used two ones to represent two, three ones for three, and so on, up to nine. However, they tended to arrange the symbols into neat piles. Once they got to ten, there were too many symbols, so they turned the stylus on its side to make a different symbol. Eleven was ten and one, twelve was ten and one and one, twenty was ten and ten, just like the Egyptians. This is a unary system. However, something strange happened at sixty (see below). The symbol for sixty seems to be exactly the same as that for one. Sixty one is sixty and one, which therefore looks like one and one, and so on. Surely this is very confusing! However, the Babylonians were working their way towards a positional system (see below).

1

2

3

4

5

6

7

8

9

10

1-10

11-20

21-30

31-40

41-50

51-60

61-70


Positional number system

A positional number system is one where the numbers are arranged in columns. We use a positional system, and our columns represent powers of ten. So the right hand column is units, the next is tens, the next is hundreds, and so on. If you want to add large numbers (and you’ve lost your calculator!) you line the numbers up so their units are in the same column. Then you can add each column, carrying forward to the next, if necessary. The Babylonians had the same system, but they used powers of sixty rather than ten. So the left-hand column were units, the second, multiples of 60, the third, multiplies of 3,600, and so on.

x 3600

x 60

Units

Value

1

1 + 1 = 2

10

10 + 1 = 11

10 + 10 = 20

60

60 + 1 = 61

60 + 1 + 1 = 62

60 + 10 = 70

60 + 10 + 1 = 71

2 x 60 = 120

2 x 60 + 1 = 121

10 x 60 = 600

10 x 60 + 1 = 601

10 x 60 + 10 = 610

3600 (60 x 60)

2 x 3600 = 7200

You can now see why they piled the units up into neat piles! They needed to distinguish one plus one or two, from one times sixty plus one meaning sixty one. Both these have two symbols for one. But the representation of two has the two ones touching, while the representation for sixty one has a gap between them. A careless clerk might make mistakes that way, but if you were careful, it should be all right.

source:http://gwydir.demon.co.uk/jo/numbers/babylon/

They managed to create the quadratic equation and even solve some cubic equation without the concept of variables. You might be wondering why the Babylonian needed such an advanced maths system given their rudimentary tech level compared to ours. The reason was taxes, the Babylonians needed a way to tax farmers and quadratics was required to estimate how many crops they would be able to grow and how much tax they would pay.

 

Image result for babylonian numerals
The 59 different values that made up the Babylonian number system^3

 

Ancient question 1:

A babylonian farmer owned a square bit of land with the side lenght  yards  and produced  bushels of wheat per square yard. He had to give    (two spaces) bushels of wheat to the the goverment in tax. what percentage of his land does he have to cultivate to still pay his taxes ? (give your answer in babylonian numerals)

Babylonian quadratic

The Babylonian version of the quadratic formula is different from ours but still is early similar to the version we use today. The only reason for the d difference is due to their lack of zero. So instead of solving a quadratic equation in this form,ax^{2}+bx+c=0. They solved the question in in the format “x2 +/- bx = c” using the formula shown below. T

x=-{\frac  {b}{2}}+{\sqrt  {\left({\frac  {b}{2}}\right)^{2}+c}}.

They also managed to solve some cubic equation by putting them into the form of n3 + nusing a table of values to figure out what n was equal to. This method is quite versatile and didn’t require the usage of the huge cubic formula which would only be discovered 2 millennium later.

A pottery maker is taxed on both the number of potter wheels he owns and the number of pots he produces.

His production of pots is given by this formula  . x^

where x is equal to the number of potter wheels he owns and the the output is the amount of pots he produced ?

He has to pay gold coins for every pottery wheal he owns and gold coin for every pot he produces. Given that he he payed in tax, how many potery wheels does he own  ?

Egyptian mathematics

Ancient Egypt is such a tragically oversimplified part of human history. Glossed over with a facade of pyramids and mummies which hide the fact, that the fall of Egypt to the Romans was twice as close to today then it was to the dawn of the old kingdom in Egypt. Egyptian mathematic was equally diverse, ranging from the crude papyrus conclusion of the old kingdom to the complicated astronomical observation of the greek era. Egyptian numerals resemble are own far more than they did roman or Babylonian numerals given their use of a base 10 system

Eygpitan Numerals explained

The Egyptians had a bases 10 system of hieroglyphs for numerals. By this we mean that they has separate symbols for one unit, one ten, one hundred, one thousand, one ten thousand, one hundred thousand, and one million.

Here are the numeral hieroglyphs.

To make up the number 276, for example, fifteen symbols were required: two “hundred” symbols, seven “ten” symbols, and six “unit” symbols. The numbers appeared thus:

276 in hieroglyphs.

Here is another example:

4622 in hieroglyphs.

Note that the examples of 276 and 4622 in hieroglyphs are seen on a stone carving from Kranak, dating from around 1500 BC, and now displayed in the Louvre in Paris.

source:http://www-history.mcs.st-and.ac.uk/HistTopics/Egyptian_numerals.html

 

They also made great advances in the field of algebra as they were the first to use a character to represent an unknown quantity when solving a problem creating the first variable which still forms the foundation of how we use algebra today. Unlike the Babylonian, they focused more on the basics such as linear equations rather than the more advanced quadratics and cubics favoured by the Babylonian but their system resembled the ones used today unlike the ancient Babylonian whose number system bears little relation to the one we use today

Image result for egyptian algebra
The rind papyrus is a famous document where much of our knowledge of Egyptian knowledge comes from, It contains numerous problem that requires variables to solve Link:^3

Historical Question 3

If we define the time of 1 historical hour as equal to the the time period between the first dynasty in Eygpts and Eygpts annexation by the roman empire. How many historical minutes would seperate us from the end of eygpt ?

 

 

The future

Until Humanity goes extinct Algebra will continue to form a key backbone of our society as our entire modern maths system is based on its principles and the increasing computerization of our society is only spreading it further as they often operate on algebraic principles such as variables. Algebra is only going to spread it too fundamental to the way our maths work to be replaced. It is useful to look back at it’s history to see how and why it became the building block of modern maths. Looking back at its pre history can give insight towards it and why we use it.

 

Footnotes (There is an elephant in the room that has been ignored by my exploration, Al-Kawarimi and his book “The Compendious Book on Calculation by Completion and Balancing” which many believe is resolve for creating algebra. This is, however, a misconception as the book was a formalisation of the previous knowledge that was compiled in one place. Al-Kawarimi is already very wll known and an exploration into his accomplishments and style of algebra wouldn’t be very interesting as it has been done many times before. His style of algebra is the grand-father of what we do today and wouldn’t be novel to anyone reading it who was familiar with algebra. I instead focused on the less well known Babylonian and Egyptian Algebra which isn’t as well known.)

Bibliography

1:https://blogs.adelaide.edu.au/maths-learning/2015/07/12/the-square-root-of-two/

2:https://www.pinterest.com/pin/286682332500110782/

3:http://mathsforeurope.digibel.be/pyramids.htm

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