# 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

Perform each multiplication in the table below.

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 and 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, $0.\ov{3}$ means the same thing as $0.33333...$, so $$1/3 = 0.\ov{3}$$.

Perform each multiplication in the table below.

 Using scratch paper and long division, compute $$2/9$$, by dividing $2.00000...$ by 9.

To the left you can see the result of dividing $4.00000...$ by 33. Again, use the and buttons to step through the division. In this example the remainder isn’t always the same, but it alternates between two numbers (7 and 4), which means that the division repeats after every other step. We’ll use an overline that goes over two digits to represent this. For example, $0.\ov{12}$ means $0.1212121212...$, so $$4/33=0.\ov{12}$$.

 What is $33(0.1212)$?
 What is $33(0.1213)$?

Click to see the result of dividing $11.00000...$ by 37. Notice that this time the remainder rotates between three different numbers.

 What is $$11/37$$ as a repeating decimal?
 Using scratch paper and long division, compute $$5/11$$ as a repeating decimal.

To the left we compute $$5/6$$, by dividing $5.00000...$ by 6. Notice that the first subtraction is different from all the rest: the result is $0.83333...$, which we write as $0.8\ov{3}$.

 What is $6(0.833333)$?
 What is $6(0.833334)$?

Click to see the result of dividing $59.00000...$ by 54.

 What is $$59/54$$ as a repeating decimal?
 Using scratch paper and long division, compute $$49/18$$ as a repeating decimal.

You can use long division to write any fraction (or rational number) as a decimal number in this way. Eventually the remainders must always start to repeat, which means the decimal number will be a repeating decimal. Some rational numbers, like $$1/5=0.2$$, don’t go on forever when you write them as decimals. You can always think of them as repeating decimals where the repeating part is $0$ ($0.2=0.20000...$).

## Turning repeating decimals into fractions

If you subtract two repeating decimals which repeat in the same way, the repeating parts will cancel out. For example, we can subtract the following two repeating decimals:

 1.0121212... $-$ 0.2121212... 0.8000000...

and so $1.0\ov{12}-0.2\ov{12}=0.8$.

What is $1.\ov{3}-0.1\ov{3}$? (Use six decimal places in each intermediate answer blank.)

If you multiply a decimal number by a power of 10, you get another decimal number with the same digits, but with the decimal point shifted to a new place. If you multiply a repeating decimal by the right power of 10, you will get another decimal which repeats in the same way (with the same repeated digits in the same places). For example, $1000(0.\ov{459})=1000(0.459\ov{459})=459.\ov{459}$.

 What is $459.\ov{459}-0.\ov{459}$?

Notice that, in order to get this to happen, we multiplied by $10^3$ to shift the decimal point by 3 places, because $0.\ov{459}$ has 3 repeated digits.

By using some algebra and the observations from manipulate-decimals-qn, you can convert any repeating decimal back into a fraction. For example, you can convert $1.\ov{27}$ to a fraction as follows:

$$\cl"tight"{\table x, , , =, 1.\ov{27}; x, , , =, 1.27\ov{27}, , , \; \cl"hint"\text"pulling out the first repetition"; 100x, , , =, 127.\ov{27}, , , \; \cl"hint"\text"multiplying both sides by 100"; 100x, -, x, =, 127.\ov{27}, -, x, \; \cl"hint"{\text"subtracting"\,x\,\text"from both sides..."}; \colspan 3 99x, =, 127.\ov{27}, -, 1.\ov{27}, \; \cl"hint"{\text"...and using the fact that"\,x= 1.\ov{27}\,\text"..."}; \colspan 3 99x, =, \colspan 3 126, \; \cl"hint"\text"...to make the repeated part cancel out"; \colspan 3 {99 x} / 99, =, \colspan 3 126/99; \colspan 3 x, =, \colspan 3 14/11 }$$

We multiplied by $100=10^2$ in order to shift the repeating decimal over by 2 decimal places, because there were 2 repeated digits. In general, if there are $n$ repeated digits, then we multiply by $10^n$ to shift the repeating decimal over by $n$ decimal places.

Convert the following repeating decimals into fractions.

## Irrational numbers

If you write some numbers as decimals, they go on forever without ever repeating exactly. For example,

$$√2=1.4142135623730950488016887242096980785696...$$
 As you saw earlier in this lesson, every rational number (ratio of two integers) is a repeating decimal. Can a decimal which doesn’t repeat be a rational number?

A real number which is not rational is called an irrational number. Another irrational number is $π$, the circumference of a circle with diameter 1. It has a value of

$$3.141592653589...$$
 What is $7(√2)$? Round your answer to four decimal places.
 What is $π^2$? Round your answer to four decimal places.
 Is $π^2$ larger than $7(√2)$, smaller than $7(√2)$, or equal to $7(√2)$?