计值法

(Video) Writing Decimals in Expanded Notation

by larryschmidt

This video by larryschmidt shows you how to write a decimal in expanded notation.

$\large{\text{Summary}}$

We already know how to do expanded notation with integers, so now it's time to learn decimals.

With decimals it's not much different. We take each decimal place and multiply it by a fraction where the denominator is a multiple of $10$. The further right we go, the larger the multiple of $10$ in the denominator.

For example, suppose we want to expand $1.234$. First, we have the ones place. $$1$$ Next we have the first decimal place, the tenths. We take $2$ and multiply it by the fraction $\frac1{10}$. $$1+\left(2\times\frac1{10}\right)$$ Then, we have the hundredths place. We move to higher multiple of $10$ for the denominator. In other words, we add another $0$. $$1+\left(2\times\frac1{10}\right)+\left(3\times\frac{1}{100}\right)$$ Finally, we have the thousandths place. Add another $0$ in the denominator. $$1+\left(2\times\frac1{10}\right)+\left(3\times\frac{1}{100}\right)+\left(4\times\frac{1}{1000}\right)$$

Add all the digits together

"Expanded form" basically just means writing a number out as a sum of all its digits in their place values. For $1.582$, this would mean adding the digit in the ones place to the digit in the tenths place, to the digit in the hundredths place, to the digit in the thousandths place.

For example: $$1.582 = 1 + 0.5 + 0.08 + 0.002$$

If that seems tricky, there's a quick trick you can use to make sure your digits have the right number of $0$'s in front of them.

First, write down the number. Let's just use $1.582$ again.

$$1.582$$

Next, move each digit straight down to its own line. Pull the decimal point down to each line, too. $$\begin{array}{cccc} 1.& & & \\ .&5& & \\ .& &8& \\ .& & &2 \end{array}$$

Fill in the bottom corner (or the corner directly beneath the decimal point) with zeros. $$\begin{array}{cccc} 1.& & &\\ \color{#29A603}{0}.&5&\\ \color{#29A603}{0}.&\color{#29A603}{0}&8&\\ \color{#29A603}{0}.& \color{#29A603}{0}&\color{#29A603}{0}&2 \end{array}$$

Finally, put $+$'s at the beginning of each line except the first. $$\begin{array}{ccccc} &1.& & &\\ +&0.&5&\\ +&0.&0&8&\\ +&0.& 0&0&2 \end{array}$$ Writing this on a single line is your expanded notation! $$1+0.5+0.08+0.002$$

Expanded Notation for Decimals

The expanded notation for decimals is writing out the decimal as a sum of each of the digits’ place values.

Example 1

Write $4.89$ in expanded notation.

Since the $4$ is in the ones place, $4$ would equal $4 \times 1$. Then, we add $(4 \times 1)$ to the tenth place, which is $8 \times 0.1$. Next, we add $(4 \times 1)+(8 \times 0.1)$ to the hundredth place, which is $9 \times 0.01$. Lastly, we add all theses together to get the expanded notation: $$(4 \times 1)+(8 \times 0.1)+(9 \times 0.01)$$ This is the expanded notation for $4.89$.

You can also write the expanded notation for decimals in fraction form. $\\$

Example 2

Write $5.16$ in expanded notation.

Let's start with the ones value. We can write the ones value as $(5 \times 1)$. Next, we can write the tenth place in fraction form: $\left(1 \times \frac{1}{10}\right)$. Then, we can write the hundredth place in fraction form: $\left(6 \times \frac{1}{100}\right)$. Lastly, we need to add these values together to get the expanded notation: $$(5 \times 1)+ \left(1 \times \frac{1}{10}\right)+ \left(6 \times \frac{1}{100}\right)$$

We are now going to take a look at some of the types of problems you might encounter about the expanded notation for decimals.

Before you start these problems, make sure to go over the basics of the expanded notation for decimals.

$$\begin{align} &5 \cdot 10^0 + 1\cdot 10^{-1} + 4\cdot 10^{-2}\\ &= 5 \cdot 1 + 1 \cdot \frac{1}{10} + 4 \cdot \frac{1}{100} \\ &= 5 + \frac{1}{10} + \frac{4}{100} \end{align}$$

$$\begin{align} &3 \cdot 10^0 + 8\cdot 10^{-1} + 9\cdot 10^{-2} + 2\cdot 10^{-3}\\ &= 3 \cdot 1 + 8 \cdot \frac{1}{10} + 9 \cdot \frac{1}{100} + 2 \cdot \frac{1}{1000}\\ &= 3 + \frac{8}{10} + \frac{9}{100} + \frac{2}{1000} \end{align}$$

Let’s write these our from the smallest denominator to the largest. $$\frac{2}{10}+\frac{7}{100}+\frac{4}{1000}$$ Our decimal is: $$0.274$$