What are decimals?

Decimals are a way of writing numbers that don't only consist of whole numbers. A decimal contains a decimal point, and the digits that come after the point represent a part of a whole.

That is really all a decimal is: a whole-number part, a point, and then some parts of a whole. Once you know what each part is worth, a decimal is just a fraction written in a different way.

The parts of a whole

Let's look at the number $9.123456$. Before the point we have 9 wholes. After the point, every digit has its own place, and each place is worth a certain part of a whole.

The first decimal, so the $1$, is on the tenths place. This means that we need 10 of them before we get a whole:

$$0.1 = \frac{1}{10}$$

The $2$ is on the hundredths place, which means that we need 100 of them to get a whole. It continues like that, so next come thousandths, then ten-thousandths, and so on. The further a digit is to the right of the point, the smaller the part it stands for. The whole walk through the number, digit by digit, is in the guide on decimal place value.

Keep the parts separate

When we work with decimals, it's important that we don't mix the parts up with each other. That means we mustn't mix tenths with hundredths, and so on, because a tenth and a hundredth don't mean the same thing. So when we add decimals or subtract them, we keep the different parts separate. For example:

$$1.234 + 2.542 = 3.776$$

In the first number we have 1 whole, 2 tenths, 3 hundredths and 4 thousandths, and in the second we have 2 wholes, 5 tenths, 4 hundredths and 2 thousandths. We add wholes to wholes, tenths to tenths, and so on down the line. The full walkthrough, including what happens when a place overflows, is in how to add decimals.

The parts are connected, though. Ten hundredths are the same as one tenth:

$$\frac{10}{100} = \frac{1}{10}$$

That's exactly the same idea as equivalent fractions, and it's the reason we can carry from one place to the next.

Decimals and fractions

As we said, decimals are a way of writing numbers that aren't whole. That could be, for example, $1.5$. Here we see $1$, which is a whole, and $0.5$, which is half of $1$, so a half. We can also write this decimal as a fraction:

$$1.5 = 1 + \frac{1}{2}$$

Or, if we only want one fraction, $\tfrac{3}{2}$. Why $\tfrac{3}{2}$? Because $\tfrac{2}{2}$ is, after all, a whole, and from there we have $\tfrac{1}{2}$ left.

So a decimal and a fraction are two ways of writing the same number. Some of these pairs are worth having at your fingertips, like $0.5 = \tfrac{1}{2}$ and $0.25 = \tfrac{1}{4}$, and the whole list is in fractions, decimals and percentages. Percentages fit in there too, because $10\% = \tfrac{10}{100} = 0.1$.

Common misunderstandings

• "I can add $1.234 + 2.542$ by just adding the digits and not worrying about where the point is." The point tells you which digit is which part. We keep the parts separate, wholes with wholes and tenths with tenths, and that is what the point is for.
• "$0.09 + 0.02$ gives $0.011$." No. Nine hundredths plus two hundredths is eleven hundredths, and ten of those are one tenth. When there are more parts than there is room for on a place, we move over to the next part, so the answer is $0.11$.
• "$0.5$ and $\tfrac{1}{2}$ are different numbers, because they look different." They're the same number. $0.5$ is half of $1$, and that is what $\tfrac{1}{2}$ means.

Related guides

• Decimal place value: tenths, hundredths and thousandths
• How to add decimals
• Fractions, decimals and percentages: the same number in three forms
• What is a fraction?