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.
Rational numbers FAQs
No. 0.6666666 is a rational number. However, it is not an integer. Some rational numbers are integers. For example, -8/1, 5/1 are rational numbers and integers as well since -8/1 = -8 and 5/1 = 5. Nonetheless, every integer is a rational number!
Yes, rational numbers are real numbers since real numbers include all sets of numbers except complex numbers.
No, not necessarily!There are negative rational numbers and positive rational numbers such as -1/2, 15/6, -4/3, 6/20.
3.14 is a famous irrational number known as pi. 3.14 has an infinite number of digits after the decimal point that do not have repeating patterns. Therefore, pi cannot be written as a ratio or be a rational number.
As a rule of thumb, if the number has a bunch of different digits after the decimal point with no specific pattern, the number is not rational.
Test your understanding of this lesson with the rational numbers quiz below.