Babylonian-Sumer Method of Extracting a Root

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Manually finding the 2nd, 3rd, 4th, etc. root of many numbers can be a difficult task for almost anyone, yet the people of Mesopotamia developed a simple way to quickly estimate the root of a number. In this lesson, we'll see how that works.

Defining the Root of a Number

Ever been driven crazy trying to solve a Rubik's Cube? With trillions of possible combinations, it's not easy, but it can be done if you apply the right steps.

Estimating roots is similar to finding the right Rubik combination. Taking an educated guess and checking your math will eventually get you to a reasonable answer, but you can spend a lot of time doing it, especially if you're talking about a 3rd, 4th, 5th, 6th, or higher root of a number. In this lesson, we'll use a 4,000-year-old trick to make it a lot simpler.

A root of a number is that part of the number that will, when multiplied by itself, end up reaching the original number. Lots of roots are quite familiar to us. For example, 2 is the square (2nd) root of 4 because if you multiply two 2's together, you get 4. Two is also the cube (3rd) root of 8 because if you multiply 2 x 2 x 2 you'll get 8, as well as the 4th root of 16 because 2 x 2 x 2 x 2 = 16.

Popular Roots & Irrational Numbers

Now, it's not too hard to get the root of a number if the root is nice and even. In fact, it's useful to memorize a bunch of the square roots and a few of the larger roots for use in your various math adventures. Here are some of the more popular ones:

Number Square Root Cube Root 4th Root 5th Root 6th Root
1 1 1 1 1 1
4 2
8 2
9 3
16 4 2
25 5
27 3
36 6
49 7
64 8 4 2
81 9 3
100 10
121 11
144 12

Did you notice that some of the columns in the table are empty? That's because those roots aren't so neat. They are strange numbers called irrational numbers because there is no rational way to accurately show their value using decimals or fractions. The only way to accurately express them is with a radical (√) sign, showing the level of root they represent. For example, the square root of 8 may be written as √8.

But what if we need to use irrational numbers in our work? We need to be able to get an idea of how big they are, otherwise we can't graph points that include them, get a feel for their quantity, or use them in decimal work. So, what do we do? Well, usually we pick up a calculator, but let's look at a system developed by the Babylonian-Sumerian people of Mesopotamia.

Estimating Square Roots

The Babylonian-Sumerian method of extracting a root, also called Heron's Method, uses a guess-divide-average method to extract irrational roots. You start with some reasonable number as your first guess, divide your original number by that root, and then take the average of your guess and your division result. The average will be closer to the root than either one, and each time you go through this process, your answer will become more accurate. Let's try the method on the square root of 25,550. We'll start with 100 for our guess.

  1. Guess: 100.
  2. Divide: 25,550 / 100 = 255.5.
  3. Average: (255.5 + 100) / 2 is about 178.
  4. Guess: Let's try 175.
  5. Divide: 25,550 / 175 = 146.
  6. Average: 146 + 175 is 160.5.
  7. Guess: Let's try 160.
  8. Divide: 25,550 / 160 = about 159.7! The square root of 25,550 is pretty close to 160.

Extracting Higher Roots

With higher roots, the level of the root determines how many times you divide by your guess. For example, let's find the cube (3rd) root of 25,550.

  1. Guess: 50.
  2. Divide: 25,550 / 50 = 511.
  3. Divide again: 511 / 50 = 10.22.
  4. Average: (50 + 50 + 10.22) / 3 = 36.75.
  5. Guess: 35.
  6. Divide: 25,550 / 35 = 730.
  7. Divide again: 730 / 35 = (roughly) 21.
  8. Average: (35 + 35 + 21) / 3 = (roughly) 30.
  9. Guess: We'll try 30.
  10. Divide: 25,550 / 30 = 852.
  11. Divide again: 852 / 30 = 28.4.
  12. Average: (30 + 30 + 28.4) / 3 = about 29.5. Our cube root will be around 29.5.

Let's extract a 4th root. What's the 4th root of 123,456?

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