Is it possible to express every fraction as a decimal

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The 0 means we're done dividing. We're going to rewrite 0. To convert a decimal into a fraction, we'll use place values. In decimals, the number immediately to the right of the decimal point is in the tenths place. This means our decimal is equal to 85 hundredths. Now we have our fraction. But it's always a good idea to reduce fractions when we can—it makes them easier to read. To reduce, we need to find the largest number that will go evenly into both 85 and So we'll divide both parts of our fraction by 5.

First we'll divide the numerator. Now we'll divide the denominator. Reducing a fraction may seem unnecessary when you're converting a decimal. But it's important if you're going to use the fraction in a math problem. If you're adding two fractions, you may even need to reduce or change both fractions so they have a common denominator.

Convert these decimals into fractions. Be sure to reduce each fraction to its simplest form! Knowing how to convert percents and decimals will help you calculate things like sales tax and discounts. To learn how, check out our Percentages in Real Life lesson. Next, we'll move the decimal point two spaces to the left.

We've converted our percent to a decimal. Let's look at another example. First, we'll replace the percent sign with a decimal point. Then, we'll move the decimal point two spaces to the left. Notice there is an extra space next to the 8. We can't just leave an open space with nothing in it. Since zero equals nothing, we'll replace the space with zero.

Converting percents into decimals is so easy that you may feel like you've missed something. But don't worry—it really is that simple! Here's why the method we showed you works. Does this happen for all fractions? Let us look at the decimal for 3 Following the same division process, we get a 1 on top with a remainder of 8, a 3 on top with a remainder of 14, a 6 on top with a remainder of 8, a 3 on top with a remainder of 14 … but wait!

We have already seen these remainders, and we know that the next number on top is a 6 with a remainder of 14 again. This means that if you try to guess the number 3 22 one decimal place at a time, you will be guessing forever!

All of the numbers we have considered so far are called rational numbers. A rational number is any number that we can write as a fraction a b of two integers whole numbers or their negatives , a and b.

This means that 2 5 is a rational number since 2 and 5 are integers. Even if we do not write 3 and 4. We have seen that some rational numbers, such as 7 16 , have decimal expansions that end. We call these numbers terminating decimals.

Other rational numbers, such as 3 22 , have decimal expansions that keep going forever. But we do know that even the decimal expansions that do not terminate repeat, so we call them repeating decimals. For example, when we were changing 3 22 into a decimal, the only options we had for remainders were 0, 1, 2, 3, …, 20, Because there are only a finite number of remainders, the remainders must start to repeat eventually. This is true for all fractions whose decimals do not terminate.

Even though there is a repeating pattern to the decimals for these fractions, we will never guess the exact number in the guessing game if we are guessing one decimal place at a time because the decimal goes on forever. We cannot say infinitely many digits! We can go in the reverse direction and change decimals to fractions, too! When we have a terminating decimal expansion, such as 4. The 2 of 4.

If we are starting with a repeating decimal, we have to do a bit more work to find its corresponding fraction. For example, consider 0. Call this number A. The repeating portion 35 has two digits, so we multiply A by to move the decimal over two places.

Notice that all the decimal places in A and A match up. We subtract A from A to get 99 A. When we subtract the decimals, the 0. Therefore, we are left with only whole numbers!

For any repeating decimal, we can use the same process to find the corresponding fraction. We multiply by 10, , , or whatever is necessary to move the decimal point over far enough so that the decimal digits line up. Then we subtract and use the result to find the corresponding fraction. This means that every repeating decimal is a rational number! What if we have a decimal expansion that does not end, but the digits do not repeat?

For example, look at 0. In this number, we increase the number of 0s between each pair of 1s, first having one 0 between, then two 0s, then three 0s, etc. This cannot be a rational number since we know the decimals for rational numbers either terminate or repeat.