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 numbers that are rational are:

 
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 examples of such numbers?

It is because 2 and 0 can be written as

 
2 / 1
  and  
0 / x
 
2 / 1
  and  
0 / x

Decimal expansion of rational numbers


x can be any number since 0 divided by any number is zero.

We can also write rational numbers as decimals.

We do this by performing a quick division that is dividing the numerator by the denominator.

For instance,

2 / 5
= 0.4
or

2 / 5
= 0.400000000

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

Terminating decimal

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


Diving 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.

Figure summarizing rational numbers

Rational numbers

Test your understanding of this lesson with the rational numbers quiz below.

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