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When you write some numbers as decimals, they go on forever. For example,
In this lesson, we will study what happens when you write numbers this way.
Question 1 of 6. 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 13, 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 13=0.3.
Perform each multiplication in the table below.
Question 2. To the left you can see the result of dividing 4.00000... by 33. Again, use the Previous and Next 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, 0.12 means 0.1212121212..., so 433=0.12.
Click to see the result of dividing 11.00000... by 37. Notice that this time the remainder rotates between three different numbers.
Question 3. To the left we compute 56, 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 0.83.
Click to see the result of dividing 59.00000... by 54.
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 15=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...).
Question 4. 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:
and so 1.012−0.212=0.8.
What is 1.3−0.13? (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(0.459)=1000(0.459459)=459.459.
Notice that, in order to get this to happen, we multiplied by 103 to shift the decimal point by 3 places, because 0.459 has 3 repeated digits.
Question 5. By using some algebra and the observations from Question 4, you can convert any repeating decimal back into a fraction. For example, you can convert 1.27 to a fraction as follows:
We multiplied by 100=102 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 10n to shift the repeating decimal over by n decimal places.
Convert the following repeating decimals into fractions.
Question 6. If you write some numbers as decimals, they go on forever without ever repeating exactly. For example,
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