Decimals and Fractions

Learning Objectives

• Read and write numbers in decimal notation.
• Write decimals as fractions.
• Write fractions as decimals.

In addition to fraction notation, decimal notation is another way to write numbers between `0` and `1`. Decimals can also be used to write numbers between any two whole numbers. For example, you may have to write a check for `$2,003.38`. Or, in measuring the length of a room, you may find that the length is between two whole numbers, such as `35.24` feet. In this topic, you will focus on reading and writing decimal numbers, and rewriting them in fraction notation.

To read or write numbers written in decimal notation, you need to know the place valueThe value of a digit based on its position within a number. of each digit, that is, the value of a digit based on its position within a number. With decimal numbers, the position of a numeral in relation to the decimal point determines its place value. For example, the place value of the `4` in `45.6` is in the tens place, while the place value of `6` in `45.6` is in the tenths place.

Decimal numbersDecimal numbers are numbers whose place value is based on `10`s, including whole numbers and decimal fractions, which have decimal points and digits to the right of the decimal point. The numbers `18`, `4.12` and `0.008` are all decimal numbers. are numbers whose place values are based on `10`s. Whole numbers are actually decimal numbers that are greater than or equal to zero. The place-value chart can be extended to include numbers less than one, which are sometimes called decimal fractionsA fraction written as a decimal point and digits to the right of the decimal point.. A decimal point is used to separate the whole number part of the number and the fraction part of the number.

Let’s say you are measuring the length of a driveway and find that it is `745` feet. You would say this number as seven hundred forty-five. Then, a more accurate measurement shows that it is `745.36` feet. Let’s place this number in a place-value chart.

What you want to examine now are the place values of the decimal part, which are the numbers `3` and `6` in the chart below.

Notice how the place-value names start from the decimal point. To the left of the decimal point are the ones, tens, and hundreds places, where you put digits that represent whole numbers that are greater than or equal to zero. To the right of the decimal point are the tenths and hundredths, where you put digits that represent numbers that are fractional parts of one, numbers that are more than zero and less than one.

Again, the place value of a number depends on how far away it is from the decimal point. This is evident in the chart below, where each number has the digit “`4`” occupying a different place value.

Imagine that as a large balloon deflates, the volume of air inside it goes from `1,000` liters, to `100` liters, to `10` liters, to `1` liter. Notice that you’re dividing a place value by ten as you go to the right. You divide `100` by `10` to get to the tens place. This is because there are `10` tens in `100`. Then, you divide `10` by `10` to get to the ones place, because there are `10` ones in `10`.

Now, suppose the balloon continues to lose volume, going from `1` liter, to `0.1` liters, to `0.01` liters, and then to `0.001` liters. Notice that you continue to divide by `10` when moving to decimals. You divide `1` by `10` (`1/10`) to get to the tenths place, which is basically breaking one into `10` pieces. And to get to the hundreds place, you break the tenth into ten more pieces, which results in the fraction `1/100`. The relationship between decimal places and fractions is captured in the table below.

Consider a number with more digits. Suppose a fisherman has a net full of fish that weighs `1,357.924` kilograms. To write this number, you need to use the thousands place, which is made up of `10` hundreds. You also use the thousandths place, which is `1/10` of a hundredth. In other words, there are ten thousandths in one hundredth.

As you can see, moving from the decimal point to the left is ones, tens, hundreds, thousands, etc. This is the “no th side,” which are the numbers greater than or equal to one. Moving from the decimal point to the right is tenths, hundredths, thousandths. This is the “th side,” which are the numbers less than `1`.

The easiest way to read a decimal number is to read the decimal fraction part as a fraction. (Don’t simplify the fraction though.) Suppose you have `0.4` grams of yogurt in a cup. You would say, “`4` tenths of a gram of yogurt,” as the `4` is in the tenths place.

Note that the denominator of the fraction written in fraction form is always a power of ten, and the number of zeros in the denominator is the same as the number of decimal places to the right of the decimal point. See the examples in the table below for further guidance.

Notice that `0.5` has one decimal place. Its equivalent fraction, `5/10`, has a denominator of `10`—which is `1` followed by one zero. In general, when you are converting decimals to fractions, the denominator is always `1`, followed by the number of zeros that correspond to the number of decimal places in the original number.

Another way to determine which number to place in the denominator is to use the place value of the last digit without the “ths” part. For example, if the number is `1.458`, the `8` is in the thousandths place. Take away the “ths” and you have a thousand, so the number is written as `1 458/(1000)`.

Recall that a mixed numberAn expression in which a whole number is combined with a proper fraction. For example `5 2/3` is a mixed number. is a combination of a whole number and a fraction. In the case of a decimal, a mixed number is also a combination of a whole number and a fraction, where the fraction is written as a decimal fraction.

To read mixed numbers, say the whole number part, the word “and” (representing the decimal point), and the number to the right of the decimal point, followed by the name and the place value of the last digit. You can see this demonstrated in the diagram below, in which the last digit is in the ten thousandths place.

Another way to think about this is with money. Suppose you pay `$15,264.25` for a car. You would read this as fifteen thousand, two hundred sixty-four dollars and twenty-five cents. In this case, the “cents” means “hundredths of a dollar,” so this is the same as saying fifteen thousand, two hundred sixty-four and twenty-five hundredths. A few more examples are shown in the table below.

As you have seen above, every decimal can be written as a fraction. To convert a decimal to a fraction, place the number after the decimal point in the numerator of the fraction and place the number `10`, `100`, or `1,000`, or another power of `10` in the denominator. For example, `0.5` would be written as `5/10`. You’ll notice that this fraction can be further simplified, as `5/10` reduces to `1/2`, which is the final answer.

Let’s get more familiar with this relationship between decimal places and zeros in the denominator by looking at several examples. Notice that in each example, the number of decimal places is different.

Let’s look at an example in which a number with two decimal places is written as a fraction.

Now, examine how this is done in the example below using a decimal with digits in three decimal places.

You can write a fraction as a decimal even when there are zeros to the right of the decimal point. Here is an example in which the only digit greater than zero is in the thousandths place.

When writing decimals greater than `1`, you only need to change the decimal part to a fraction and keep the whole number part. For example, `6.35` can be written as `6 35/100`.

Just as you can write a decimal as a fraction, every fraction can be written as a decimal. To write a fraction as a decimal, divide the numerator (top) of the fraction by the denominator (bottom) of the fraction. Use long division, if necessary, and note where to place the decimal point in your answer. For example, to write `3/5` as a decimal, divide `3` by `5`, which will result in `0.6`.

Note that you could also have thought about the problem like this: `1/2=?/10`, and then solved for `?`. One way to think about this problem is that `10` is five times greater than `2`, so `?` will have to be five times greater than `1`. What number is five times greater than `1`? Five is, so the solution is `1/2=5/10`.

Now look at a more complex example, where the final digit of the answer is in the thousandths place.

Converting from fractions to decimals sometimes results in answers with decimal numbers that begin to repeat. For example, `2/3` converts to `0.666`, a repeating decimal, in which the `6` repeats infinitely. You would write this as `0.bar6,` with a bar over the first decimal digit to indicate that the `6` repeats. Look at this example of a problem in which two consecutive digits in the answer repeat.

With numbers greater than `1`, keep the whole number part of the mixed number as the whole number in the decimal. Then use long division to convert the fraction part to a decimal. For example, `2 3/20` can be written as `2.15`.

Decimal notation is another way to write numbers that are less than `1` or that combine whole numbers with decimal fractions, sometimes called mixed numbers. When you write numbers in decimal notation, you can use an extended place-value chart that includes positions for numbers less than one. You can write numbers written in fraction notation (fractions) in decimal notation (decimals), and you can write decimals as fractions. You can always convert between fractional notation and decimal notation.