0.
31.00000
9
10
9
10
9
10
9
10
9
1
   

Rational Numbers and Repeating Decimals

When you write some numbers as decimals, they go on forever. For example,

1
3
=0.3333333...

In this lesson, we will study what happens when you write numbers this way.


Long division and repeating decimals

Question 1 of 6. Perform each multiplication in the table below.

3(0.3)0.9
3(0.4)
3(0.33)
3(0.34)
3(0.33333)
3(0.33334)

Notice that these products are very close to 1, and getting closer. That means that the numbers 0.3,0.33,... which you are multiplying by 3 are very close to

1
3
, and getting closer.

By using long division, you can divide 1.00000... by 3 to get 0.33333..., as shown to the left. Use the Previous and Next buttons to step through the division. In this example, the remainder is the same every time (it is always 1), which means that each step of the division is identical. So the digits in the quotient repeat. We use an overline to indicate decimals that repeat. For example, 0.3 means the same thing as 0.33333..., so

1
3
=0.3
.

Perform each multiplication in the table below.

9(0.2)1.8
9(0.3)
9(0.22222)
9(0.22223)
Using scratch paper and long division, compute
2
9
, by dividing 2.00000... by 9.
2
9
=0.
If you’re stuck, and can’t ask another student or your teacher for help, you can ask us a question via e-mail.