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OSAT Advanced Mathematics (CEOE) (111): Practice & Study Guide59 chapters | 520 lessons | 23 flashcard sets

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.

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.

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 | ||||

32 | |||||

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.

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.

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

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.

- Guess: 50.
- Divide: 25,550 / 50 = 511.
- Divide again: 511 / 50 = 10.22.
- Average: (50 + 50 + 10.22) / 3 = 36.75.
- Guess: 35.
- Divide: 25,550 / 35 = 730.
- Divide again: 730 / 35 = (roughly) 21.
- Average: (35 + 35 + 21) / 3 = (roughly) 30.
- Guess: We'll try 30.
- Divide: 25,550 / 30 = 852.
- Divide again: 852 / 30 = 28.4.
- 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?

- Guess: Well, 10 x 10 x 10 x 10 = 10,000, so let's try 20.
- Divide: 123,456 / 20 = about 6173.
- Divide again: 6173 / 20 = about 309.
- Divide again (since it's a 4th root): 309 / 20 = about 15. Wow, we're already close!
- Average: (20 + 20 + 20 + 15) / 4 = 18.75. Our root will be somewhere around 18.75.

An **algorithm** is a set of steps that will produce precise results, common in mathematics. The people of Mesopotamia are known for their remarkable floating point, base 60 math system (we use base 10). Their method for extracting roots is significant in that it is a precise, repeatable process, one that will produce as many levels of precision in an irrational root as a mathematician desires. It is considered one of the earliest examples of a reliable mathematical algorithm. During a time when people were using clay tablets to perform mathematical operations, this computer-friendly system and method was a reality.

A **root** of any number is a piece of that number that can be multiplied by itself to get back to the original number. A square root is multiplied by itself once; a cube (3rd) root must be multiplied by itself twice.

Finding roots, especially beyond the square root, can be a very difficult and time-consuming process. Many roots are **irrational numbers**, which cannot be expressed accurately in decimal or fraction form. This makes them harder to work with and difficult to extract from the original number. The **Babylonian-Sumerian** or Heron's method uses a guess-divide-average process to rapidly close in on any root that is difficult to find and is an early example of a reliable mathematical **algorithm**.

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OSAT Advanced Mathematics (CEOE) (111): Practice & Study Guide59 chapters | 520 lessons | 23 flashcard sets

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