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.
|
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
2
5
= 0.400000000
Notice that you can continue division to keep getting zeros for the decimal places after 4.
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
Test your understanding of this lesson with the rational numbers quiz below.
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