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This lesson describes a method for finding square roots used by the Babylonian people of Mesopotamia. The method involves dividing and averaging, over and over, to find a more accurate solution with each repeat of the process.
To introduce students to the concept of square root.
To reinforce the ideas of estimation and accuracy.
To expose students to a mathematical method from a non-European culture.
Babylonian Square Roots could be used in a basic skills mathematics, prealgebra or algebra course when reviewing or teaching square roots and the methods for finding or estimating square roots.
BABYLONIAN SQUARE ROOTS
Ancient Mesopotamia was a civilization that existed in the area of modern Turkey, Syria, Iraq and Iran, between the Mediterranean Sea and the Persian Gulf. In the period 1900-1600 BC, Babylon was the capital city of Mesopotamia and the mathematics recorded at this time came to be known as Babylonian Mathematics.
Babylonian scribes used wet clay tablets written on with reeds the size of pencils. Corrections were difficult to make, since once the clay dried it was no longer possible to write on or change the tablet. Tablets from the size of postage stamps to pillows have been found in the area of the Babylonian civilization and now can be found in museums around the world.
The Babylonians had an accurate and simple method for finding the square roots of numbers. This method is also known as Heron’s method, after the Greek mathematician who lived in the first century AD. Indian mathematicians also used a similar method as early as 800 BC. The Babylonians are credited with having first invented this square root method, possibly as early as 1900 BC.
The Babylonian method for finding square roots involves dividing and averaging, over and over, to obtain a more accurate solution with each repeat of the process.
Babylonian Square Roots
Step 1: Make a guess.
Step 2: Divide your original number by your guess.
Step 3: Find the average of these numbers.
Step 4: Use this average as your next guess.
REPEAT THE PROCESS THREE TIMES.
For example, find sqrt 5
Step 1: Guess 2 (because 2*2=4, close to 5)
Step 2: Divide 5 by 2 = 2.5
Step 3: Find average of 2 and 2.5 = 2.25 (because (2+2.5)/2 = 2.25)
Step 4: Next guess is 2.25
Step 1: Guess 2.25
Step 2: Divide 5 by 2.25 = 2.22222222 (go 8 decimal places for accuracy)
Step 3: Find average of 2.25 and 2.22222222 = 2.23611111
Step 4: Next guess is 2.23611111
Step 1: Guess 2.236111111
Step 2: Divide 5 by 2.23611111 = 2.2360248
Step 3: Find average of 2.23611111 and 2.2360248 = 2.2360679
Step 4: FINAL guess is 2.2360679
Now CHECK your final guess with a calculator: sqrt 5 = 2.2360679
1. Find sqrt 1000 using the Babylonian square root method.
2. Write out each of your steps.
3. Check your result using a calculator.
References: Babylonian Square Roots
Gullberg, Jan. (1997). Mathematics: From the Birth of Numbers. New York: W.W. Norton & Company.
Joseph, George Gheverghese. (1991). The Crest of the Peacock: Non-European Roots of Mathematics. London: Penguin Books.
Nelson, D., Joseph, G. and Williams, J. (1993). Multicultural Mathematics: Teaching Mathematics from a Global Perspective. New York: Oxford University Press.
San Joaquin Delta College
Basic Mathematics Program
Communications Skills Division
5151 Pacific Avenue
Stockton, CA 95207
Tel. (209) 954-5252
Division Chairperson: Mary Ann Cox
Division Secretary: Joann Hymes
Designed by Patricia Donovan
San Joaquin Delta College