Math Help on Rational Numbers

What's a Rational Number?

A rational number is any number that you can write as a fraction. Many of the numbers that you use on a daily basis are rational numbers. All integers are rational numbers, and so are all fractions. Most decimals are rational numbers, but a few aren't. Some square roots are also rational numbers. Numbers that are not rational are called irrational numbers.

Integers

Integers include all of the whole numbers, which are the numbers from zero to infinity. They also include all of the negative numbers from negative one to negative infinity. Examples of integers include 25, three and -48. Decimals are not integers. Integers are rational numbers because any integer can be written as a fraction by putting it over one. For example, 5/1 = 5 and -9/1 = -9. This even works with zero, since 0/1 = 0.

Fractions

Since rational numbers are numbers that can be written as fractions, it's no surprise that fractions are rational numbers. Proper fractions like 3/5, improper fractions like 16/7 and even mixed fractions like 5 2/3 are all rational numbers. Mixed fractions are rational because they can be converted into improper fractions.

Decimals

Decimals that have a definite ending point are rational numbers, because they have an equivalent fraction. The only decimals that are not rational are decimals that continue infinitely and follow no particular pattern. Because some decimals continue forever in a random manner, mathematicians have not been able to pinpoint an equivalent fraction for them. Examples of decimals that fit this description are pi (3.14159265358979323...) and e (2.71828182845904523...). They are considered irrational numbers.

Repeating Decimals

Some decimals that continue forever, like 0.3 repeating, are rational numbers. The difference between these decimals and the irrational decimals, like pi, is that they go on forever in a predictable pattern. Because of this, it is possible to write them as fractions. For instance, 0.3 repeating can be written as 1/3. If you divide one by three, you'll see that the division problem never ends!

Converting Decimals to Fractions

Although it's not necessary to actually convert a decimal to a fraction in order to determine if it's rational, it's helpful to know how to do this. For simple decimals that just go to the tenths place, you can determine what their equivalent fraction is by putting the number in the tenths place over ten, and then simplifying it if necessary. For instance, 0.2 is represented by the fraction 2/10, which can be simplified to 1/5. This means that 0.2 is rational. You can extend this process to any decimal that has a definite ending point. For example, you could write 0.8765 as the equivalent fraction 8765/10,000, and 0.000057 as 57/1,000,000.

Square Roots

Some square roots that are perfect squares, like the square root of 25, are rational numbers because they are equal to an integer that can be written as a fraction. For instance, the square root of 25 is five, and five is equal to the fraction 5/1. Mathematicians have proven that some square roots, like the square root of two and the square root of three, are not rational numbers. They're irrational because they're equal to decimals that continue forever and don't follow a pattern.

Irrational Numbers

Numbers that are not rational are irrational. Most of the numbers you'll encounter in daily life are rational, but here's a quick reference of commonly used irrational numbers:

• Pi
• Euler's number
• The golden ratio
• The square roots of two and three