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:
This is more or less the same as the modern formula given above, where a = 1.
1850 BC Eygpt
The Greatest Pyramid
A frustum of a pyramid is a pyramid with its top cut off. An ancient Egyptian manuscript gives a method for calculating the volume of this.
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Ancient Egypt is particularly famous for the construction of the Pyramids. The engineering skills that went into their construction have, unfortunately, been lost to us. Likewise we can now only guess at the mathematical skills the Egyptians possessed.
Take a solid like a cone or a pyramid, which slopes uniformly from its base to a point at the top. If we cut a slice off the top the result is a frustum. A yoghurt pot is an example of a frustum of a cone.
The Moscow papyrus, dating from about 1850 BC, contains a set of rules for finding the volume of a frustum of a pyramid. It goes:
Given a truncated pyramid of height six and square bases of side four on the base and two at the top. Square the four, result 16.
Multiply four and two, result eight. Square the two, result four.
Add the 16, the eight and the four, result 28.
Take a third of six, result two.
Multiply two and 28, result 56.
You will find it right.
Following these rules, this method gives a formula for the volume as:
1/3 x 6 (42 + 2 x 4 + 22) = 56.
This does give the correct volume.
Generalizing, if the frustum has height h, a square top of side r and a square base of side R, the method gives the following formula for its volume:
1/3h (R2 + Rr + r2)
The illustration shows truncated pyramids.
which is correct. No indication is given for how this method was reached. Was it by experiment, or from theory?
This mathematical result was described (by a mathematician, mind you) as the “Greatest Egyptian Pyramid.”
c. 3rd century BC Global
π
The ratio of the circumference of a circle to its diameter.
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The value of π has been found to higher and higher accuracy. It occurs in many places in mathematics besides the measurement of circles.
Circles come in different sizes of course. As the diameter (the length across) increases, so also does the circumference (the length around). The ratio between these two is the same for all circles and it is given the name π (Greek letter p, pronounced “pie”).
All civilizations have needed to find an approximation for π. An early Egyptian value was 4 x (8/9)2, which is 3.16, close to 3.14. In the Bible, I Kings 7, verse 23, the more approximate value of three is given.
The first-known