Tom has taught math / science at secondary & post-secondary, and a K-12 school administrator. He has a B.S. in Biology and a PhD in Curriculum & Instruction.
Numbers Without End
Sometimes in math it can feel like the numbers we see go on and on without any real pattern. And sometimes, that's indeed the case. However, more often, those numbers do end up either terminating or forever repeating.
When we express a number as a decimal, it is called decimal expansion. Decimal expansion is the form of a number that has a decimal point, either actual or implied. Examples of numbers with actual decimal points are 10.2 and 0.0084. An example of a number with implied decimal points is the whole number 17, which could actually be written as 17.000000000000. We often leave off the repeating zeros for ease of reading and figuring.
Decimals either terminate or repeat, which is a characteristic of rational numbers. When we use the term rational in math, we're not talking about a number that makes logical sense. Rational numbers can either be written as a fraction of two whole numbers or a ratio.
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Finite and Repeating Decimals
Rational decimals that end with repeating zeroes are known as finite decimals, like 6/3 = 2 or 10/2.5 = 4. They're the opposite of infinite or forever. Rational decimals that are finite are those that originate from a fraction with a denominator that is a product of 2, 5, or both. For example, 3/8, 13/25, and 7/50 are all finite decimals because:
8 = 2^3
25 = 5^2
50= 5^2 * 2
Rational decimals that forever repeat are known as repeating decimals. Repeating decimals originate from fractions whose denominators are not entirely products of 2 and 5 only. For example, 3/14 and 7/22 are repeating decimals because 14 = 2 * 7 and 22 = 2 * 11. The number of digits that repeat is referred to as the period of the repeating decimal.
- Thirds have a period of 1 (1/3 = 0.3333333...)
- Elevenths have a period of 2 (2/11 = 0.1818181818….)
- Sevenths have a period of 6 (2/7 = 0.285714285714285714…..)
Converting Repeating Decimals to Rational Numbers
If we want to show that a repeating decimal is rational, we can use the following simple steps:
- Identify the repeating decimal's period and raise 10 to that power
- Set the original repeating decimal equal to x, which is equation A
- Multiply both sides of equation A by 10 raised to the period's power for equation B
- Subtract equation A from equation B and solve
The result will be your repeating decimal in rational form.
For example, convert 0.2727272727272727... into a rational number
1. The period is 2, so 10^2 = 100
2. x = 0.272727272727….
3. 100x = 27.272727272
4. (100x = 27.272727272) - (x = 0.272727272727….)
5. 99x = 27
6. x = 27 / 99 = 3 / 11
The decimal 0.2727272727.... is the rational number 3/11. This method works for any repeating decimal of any repeating period.
Numbers can be represented in many different forms, one of which is decimal expansion. Numbers expressed through decimal expansion contain a decimal point, which can be either actual or implied; for example: 7 = 7.0. In this lesson, we focused on rational numbers and rational decimals.
Rational decimals can be either finite or repeating. Finite decimals end, like 0.5 and 0.375. Decimals are terminal only when their representative fractions have denominators that are entirely multiples of 2 or 5. All others are repeating decimals. For instance: 1/3 = 0.333333…., where the number of digits that repeat are called the period.
Repeating decimals can be converted to rational numbers by:
- Identifying the decimal's repeating period and raising 10 to that power
- Writing an equation where x equals a repeating decimal, or equation A
- Multiplying both sides of equation A by the result obtained in the first step for equation B
- Subtracting equation A from equation B and solving for x
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Decimal Expansion of Rational Numbers
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