How to Convert Irrational Numbers

Instructor: Michelle Vannoy
What is an irrational number? How can you use it in a problem? Can you make it a rational number? This lesson will show you how to convert an irrational number to an approximate rational number, making it easier to use.

What Do I Do With This?

Irrational numbers can be intimidating to work with. How can you use a number that goes on and on forever without repeating? What place can you round it to, to make the number accurate enough? What does the square root of 2 mean? These are all questions that you may ask yourself when dealing with irrational numbers.

What is an Irrational Number?

An irrational number is any number that cannot be written as a ratio between two whole numbers. Irrational numbers include imperfect squares and non-terminating, non-repeating decimals like pi.

An imperfect square is a number that does not have a whole number multiplied by itself to equal that number. It is very difficult to understand its value, and it is confusing to work with. Converting an irrational number to an approximate decimal or fractional value makes the value of the number meaningful and easier to work with.

How Do We Convert an Irrational Number to a Fractional or Decimal Value?

To estimate an irrational, imperfect square to an approximate decimal or fractional value, we need to look at the rational, perfect squares around it. A perfect square is the product of a whole number that is multiplied by itself. Every imperfect square has a value between two perfect squares. Think of it as a square root number line.

Each imperfect square falls between an upper and lower perfect square
Square Root Number Line

For example, the perfect square root of 49 (7x7) and the perfect square root of 64 (8x8) have the imperfect squares of the square root of 50, the square root of 51, the square root of 52, the square root of 53, the square root of 54, and so on through the square root of 63 in between them.

Let's determine the approximate value of the square root of 56. We know that the lower perfect square that comes directly before square root of 56 is the square root of 49. The square root of 49 equals 7. This tells us the square root of 56 has an approximate value of slightly more than 7.

First 10 Perfect Squares

To determine the approximate fractional part or the decimal part, we have to count from the square root of 49 to the square root of 56. From the square root of 49 to the square root of 56 is 7. This 7 becomes the numerator in the fractional part.

To determine the denominator, we subtract the lower perfect square from the upper perfect square, the perfect square root that comes directly after the imperfect square root you are approximating. In this case, the upper perfect square would be the square root of 64. 64-49 is 15. This 15 becomes the denominator in the fractional part.

Put it all together, and you find that the square root of 56 is approximately 7 and 7/15. To make this a decimal, divide 7 by 15, and the square root of 56 is approximately 7.47.

The Steps

Let's go over the steps one more time:

Step 1 - Determine the lower and upper perfect square root your imperfect square falls between.

Step 2 - Find the difference between the imperfect square and the lower perfect square. This number becomes the numerator of your fractional part.

Step 3 - Find the difference between the upper perfect square and the lower perfect square. This number becomes the denominator in your fractional part.

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