We use decimals every day without even realizing it. We use them in everything from money to measuring distances to baking cakes. Essentially another way to write fractions, we cannot do anything without these fantastic numbers. They let us show values between whole numbers. They even make doing math with fractions a breeze and easy to write. With so much riding on our understanding of decimals, we have to do them correctly, but decimals can difficult to do even by experts. The trick is to remember that they are just fractions, and with this relationship, you will know how to do decimals.

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## What are Fractions and Decimals?

Decimals are fractions, and we use them as such. Therefore, if we know how to use fractions we can use decimals.

Fractions are ratios between two whole numbers separated by a line. The first number is called the numerator and it represents how much stuff we have. The second number is called the denominator and it represents how much we need to complete the whole set. When we write fractions by hand, we usually write the line horizontally with the numerator on top and the denominator below the line. We use a backslash for the line if we have to type out the fractions on a computer, tablet, or smartphone.

Fractions let is write values that we cannot express as whole numbers such as half (1/2) and quarter (1/4). We can even combine them with whole numbers to create mixed numbers if we need to express values such as 346¾. We can turn any whole number into a mixed number just by adding a fraction with a numerator of 0 to it.

Decimals are fractions where the denominator is a power of ten. We use them to streamline who we write and use fractions. Instead of writing them as two numbers with a line between them, we write decimals as a single number with a period, called the decimal point that indicates where the fractional component begins. For example, 0.3456 is a decimal, and so is 64.346.

As you can see, we typically include whole numbers as decimals. This is the power of decimals which we will show later. For now, you can simply understand it as the way save space when writing fractions. As you might have guessed we don’t have mixed numbers when dealing with decimals. Instead, we have numbers called rational and irrational. Rational numbers are numbers we can express as fractions. Irrational numbers are numbers we cannot.

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## Reading Decimals

We write decimals with the decimal number system which expresses values as sequences of symbols called digits. These digits are the familiar 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Each digit in the sequence represents the power of ten we have to multiply that digit to put it in place. We use positive powers for whole numbers, and negative powers (or fractional powers of ten) for the fraction part of the number.

For example:

3598.726

3 is in the 10^{3} or thousands position

5 is 10^{2} or hundreds position.

9 is in the 10 or tens position.

8 is in the 1 or ones position.

7 is in the 1/10, 10^{-1} or tenths position.

2 is in the 1/100, 10^{-2}, or hundredths position.

6 is in the 1/1000, 10^{-3}, or thousandths.

These powers of 10 are how we read decimals. We read decimals by reading their whole number parts as is. We then read the fractional parts as they were whole numbers by place the name of the position of the rightmost digit. For instance we read 3598.726 as three thousand five hundred ninety-eight and seven hundred twenty-six thousandths. The word “add” indicates where the decimal point is.

## Decimal Math

### Adding and subtracting decimals

Now that you know how to read and write decimals, it is time to learn how to use them in math problems. We can use decimals anywhere that needs a fraction to reduce the difficulty of the problem. Their streamlined nature lets us use the same math rules we have for whole numbers on them provided we keep track of the decimal point in our answers.

For addition and subtractions, we can keep track of the decimal point by aligning up the decimal points of all numbers together on a single vertical line. If you get good enough at it, you can do this in your mind saving space on the paper. Once properly aligned, we continue our addition or subtraction problem as if the decimal points didn’t exist. We then insert a decimal point in our answer when we come to the points in the numbers we are adding or subtracting. For instance

4.5

+7.45

11.95

### Multiplying decimals

To multiply decimals, we first count the number of fractional positions in the numbers we a multiplying together. We then continue the problem as if the decimal points didn’t exist. Once we have our product we count the number of positions from the right side of the number. Once we reach the number we counted earlier, we add a decimal point at that spot. The resulting number is the produce of out multiplication problem. For instance, If we multiply 4.5 and 7.45 together, we can see that 4.5 has one fractional position while 7,45 has two, giving us three fraction positions. We then continue the problem as if the decimal points didn’t exist. That is we multiply 45 and 745. We then add a decimal point three positions from the right side of the product. The resulting number is the product of our original problem

4.5 x 7.45 à 45 x 745 = 33525 with 3 decimal places

We add a decimal point three positions from the right of 33525 to get 33.525.

4.5 x 7.45 = 33.525

### Dividing Decimals

We divide decimals by remembering that decimals are fractions and that fractions themselves are ways we can write the division between two numbers. We multiple both the dividend and divisor by the power of ten we need to remove the decimal point from the divisor. We then continue with the division as if the decimal point in the dividend didn’t exist. We then write a decimal point in our quotient when we reach the decimal point in the dividend.

10.08/1.2 = 100.8/12 = 8.4

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