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where | represents a unit,
The Chinese system writes numbers much as we say them. We say “three hundred and sixty-five:” in other words, so many hundreds, so many tens, and so many units. The number 365 is written as shown below.
It represents 3 x 100 + 6 x 10 + 5.
In both these systems there is no limit to the number of symbols required. We need a different symbol for millions, another symbol for 10 millions, and so on. The modern system uses precisely 10 symbols: the digits 0 to 9.
The value of each digit is shown by its place in the number. In 365, for example, the digit 5 on the right represents 5, the digit 6 represents 60, as it is one place to the left, and the 3 represents 300. This system came to the West from India via the Arab countries and is known as the Indo–Arabic system.
The ancient Babylonian place value system was even more economical. It used only two symbols:
represents 3 x 602 + 21 x 60 + 43 = 12 103.
3rd millennium BC Egypt & Babylonia
Fractions
There are different systems for writing fractions. This has always been the case, even in ancient times. For example, the Egyptian system was very limited, while the Babylonian system is still in use today.
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Any advanced civilization has a system of writing fractions. Despite their renowned technological prowess, the Ancient Egyptians had a system of fractions that was comparatively clumsy.
With the exception of 2/3, the only fractions recognized by the Ancient Egyptians were those with 1 on the top, called aliquot fractions, such as 1/2, 1/3, 1/4. Any other fraction had to be written in terms of these aliquot fractions. Furthermore, they were not allowed to repeat a fraction. If they wanted to write 2/5, for example, they could not write it as 1/5 + 1/5. For the second 1/5, they had to find aliquot fractions with sum 1/5, such as 1/6 + 1/30. So they wrote 2/5 as 2/5 = 1/5 + 1/6 + 1/30 (and there are other possibilities too).
Few examples of Ancient Egyptian mathematics survive, although one that does is a leather scroll, dated from about 1650 BC, which contains fractional calculations such as the one earlier.
The Babylonian system was more flexible, following their system of writing whole numbers. Each unit is divided into 60 smaller parts, called minute parts, then each minute is divided into 60 parts, called second minute parts, and this continues with third minute parts and fourth minute parts. This system is still used today for telling the time. We divide an hour into 60 minutes and a minute into 60 seconds. (Seconds are divided into decimal fractions rather than thirds and fourths, however.)
Why was 60 chosen both for whole numbers and for fractions? Most probably because it has so many divisors and, consequently, many fractions terminate.
Consider the fractions 1/2, 1/3, 1/4 up to 1/9. Using ordinary decimals, four of them, 1/2, 1/4, 1/5, and 1/8, have a terminating representation. The other four, 1/3, 1/6, 1/7, and 1/9, have a recurring representation, such as 1/3 = 0.3333… (the threes go on ad infinitum). Using Babylonian fractions, only 1/7 does not have a terminating representation.
Nowadays, we have two ways of writing fractions. When 5 is divided by 8, the result can be written either as 5/8 or as 0.625.
See: Writing Numbers, page 8
2000 BC Babylonia
Quadratic Equations
A quadratic equation includes the square of the unknown. Thousands of years ago mathematicians in Babylonia knew how to solve quadratic equations.
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The measurement of land has always been important to any civilization. To find the area of a square piece of land you multiply the side by itself, which is called the square of the side. The Latin for square is quadratus, and this is where the word quadratic comes from. There is always a square term.
Algebraically, a quadratic equation is of the form:
ax2 + bx + c = 0
where a, b and c are numbers.
The solution (in other words the formula for x) is very well known in school mathematics all over the world.
This, of course, uses modern algebraic notation. However, a method for solving quadratic equations has been known for thousands of years.
A Babylonian clay tablet in the British Museum in London contains the solution to the following problem:
The area of a square added to the side of the square comes to 0.75. What is the side of the square?
The working shown on the tablet is illustrated on the left of the table overleaf (see page 12). The modern algebraic equivalent is shown on the right.
| Babylonian tablet | Modern notation |
| I have added the area and the side of my square. 0.75You write down 1, the coefficientYou break half of 1. 0.5You multiply 0.5 and 0.5. 0.25You add 0.25 and 0.75. 1This is the square of 1Subtract 0.5, which you multiplied0.5 is the side of the square | x2 + x = 0.75Coefficient of x is 1Half of 1 is 0.5(0.5)2 = 0.250.25 + 0.75 = 1√1 = 11 – 0.5 = 0.5x = 0.5 |
In general, the method gives the following formula to solve the equation x2 + bx = c:
