Rational Numbers

Rational numbers are any numbers that can be written as a fraction. In other words, you can rewrite the number so it will have a numerator and a denominator.

They have the form

a / b

in which a and b are integers and b not equal to zero.

Notice that we said b cannot be zero. It is because any number divided by 0 has no answer.

Examples of rational numbers

2 / 3

5 / 2

1 / 4

, 2 ,

-8 / 2

, and 0

2 / 3

5 / 2

1 / 4

, 2 ,

-8 / 2

, and 0

Now, why are 2 and 0 rational numbers as well?

It is because 2 and 0 can be written as

2 / 1

and

0 / x

2 / 1

and

0 / x

x can be any number since 0 divided by any number is zero. Notice though that according to the definition x cannot be zero!

Decimal expansion of rational numbers

We can also write rational numbers as decimals.

We do this by performing a quick division. We divide the numerator by the denominator.

For instance,

2 / 5

= 0.4

2 / 5

= 0.400000000

Notice that you can continue division to keep getting zeros for the decimal places after 4.

You can also write 0.400000000000 as 0.40̄

The bar on top of 0 means that if we continue to perform long division, we will keep getting an infinite number of zeros.

Another way to convert

2 / 5

into a decimal is to notice that we can multiply 5 by 20 and 2 by 20

We will get

40 / 100

And

40 / 100

= 0.40

Dividing by 100 or any other power of 10 is a straightforward process.

If you are dividing by 10, just move the decimal point one place to the left.

If you are dividing by 100, just move the decimal point 2 places to the left.

and so forth...

For

40 / 100

, the decimal point is after 0 for 40.

Moving that two places to the left bring the decimal point right before the 4.

A rational number can have either repeating decimal expansion or terminating decimal expansion.

Repeating decimal expansion:

A decimal expansion in which the numbers repeat in exactly the same order.

For example, 0.251251251251 is a repeating decimal expansion because 251 keeps repeating in the same order.

Terminating decimal expansion:

A decimal expansion that ends in all zeros.

For example, 0.150 is a terminating decimal expansion.

Here is a figure summarizing rational numbers or the "different types of rational numbers"

Natural numbers are rational numbers since any natural number can be written in the form a/b. For example, 2 can be written as 2/1.

Whole numbers are rational numbers since any whole number can be written in the form a/b. For example, 9 can be written as 9/1.

Integers are rational numbers since any integer can be written in the form a/b. For example, -8 can be written as -8/1.

Non-terminating decimal numbers with repeating patterns are rational numbers. For example, 0.2121212121 is a rational number since it can be written as 7/33.

Terminating decimal numbers are rational numbers. For example, 0.75 is a rational number since it can be written as 3/4.

Simple fractions whose numerators and denominators are whole numbers are rational numbers. For example, the fraction 2/5 is a rational number. However, the fraction 1.3/5 is not a rational number since 1.3 is not an integer according to the definition of a rational number.