Place value and expanded form

Every digit in a dec­i­mal has a job. Where it stands decides what it is worth. In this les­son you will take a dec­i­mal apart, see the worth of each digit, and put it back together again. This is called the expanded form, and it is the surest way to tell $0.3$ from $0.03$ with­out guess­ing.

Dig­its and places#

A digit is one num­ber sign on its own: $0$, $1$, $2$ and so on up to $9$. The num­ber $457.368$ is made of six dig­its.

Each digit stands in a place. To the left of the point the places are units, tens, hun­dreds and so on. To the right of the point they are tenths, hun­dredths, thou­sandths and so on.

The units place is the plain ones place. It sits just to the left of the point.

Place value#

The place value of a digit is what that digit is worth where it stands. Take $457.368$.

So the place value of $4$ is $400$, and the place value of $8$ is $\displaystyle \frac{8}{1000}$.

Read the chart from right to left. Ten thou­sandths make one hun­dredth. Ten hun­dredths make one tenth. Ten tenths make one unit. It is the same pat­tern as ten units mak­ing one ten, car­ried on past the point.

The same digit can be worth very dif­fer­ent amounts. The $4$ in $457.368$ is worth $400$. The $4$ in $70.4$ is worth only $\displaystyle \frac{4}{10}$. Same digit, dif­fer­ent place, dif­fer­ent worth.

Expanded form#

The expanded form of a num­ber writes the place value of every digit and adds them up.

$$\displaystyle 457.368 = 400 + 50 + 7 + \frac{3}{10} + \frac{6}{100} + \frac{8}{1000}$$

You read $457.368$ as four hun­dred fifty seven and three tenths, six hun­dredths and eight thou­sandths. Notice that the words already give you the expanded form.

You can also write the parts after the point as dec­i­mals.

$$457.368 = 400 + 50 + 7 + 0.3 + 0.06 + 0.008$$

Add the six pieces and you get $457.368$ back.

Exam­ple 1#

Write $1257.349$ in expanded form.

The $1$ is in the thou­sands place, the $2$ in the hun­dreds, the $5$ in the tens and the $7$ in the units. After the point, $3$ is tenths, $4$ is hun­dredths and $9$ is thou­sandths.

$$\displaystyle 1257.349 = 1000 + 200 + 50 + 7 + \frac{3}{10} + \frac{4}{100} + \frac{9}{1000}$$

Exam­ple 2#

Write $23.45$ in expanded form.

$$\displaystyle 23.45 = 20 + 3 + \frac{4}{10} + \frac{5}{100}$$

Exam­ple 3: zeros that hold places#

Write $305.06$ in expanded form.

Go place by place. Hun­dreds: $3$, worth $300$. Tens: $0$, worth noth­ing. Units: $5$. Tenths: $0$, worth noth­ing. Hun­dredths: $6$, worth $\displaystyle \frac{6}{100}$.

$$\displaystyle 305.06 = 300 + 5 + \frac{6}{100}$$

The two zeros do not show in the sum, because they are worth noth­ing. But they mat­ter in the num­ber. With­out them the $3$ and the $6$ would slide into the wrong places.

Exam­ple 4: going back­wards#

Which num­ber is $\displaystyle 40 + 1 + \frac{7}{100} + \frac{2}{1000}$?

Put each worth in its place. Tens: $4$. Units: $1$. Tenths: there are none, so write $0$. Hun­dredths: $7$. Thou­sandths: $2$.

The num­ber is $41.072$. The zero in the tenths place is a must.

Exam­ple 5: place value in a long num­ber#

In $3046.271$, find the place value of $6$, $7$ and $1$.

The $6$ is in the units place, so its place value is $6$. The $7$ is in the hun­dredths place, so it is worth $\displaystyle \frac{7}{100} = 0.07$. The $1$ is in the thou­sandths place, so it is worth $\displaystyle \frac{1}{1000} = 0.001$.

Where you meet the small places#

You already meet these places every day.

• On a ruler, each cen­time­tre is cut into ten small marks. One mark is $\displaystyle \frac{1}{10}$ cm, which is $0.1$ cm.
• One paisa is $\displaystyle \frac{1}{100}$ of a rupee, which is ₹0.01.
• One gram is $\displaystyle \frac{1}{1000}$ of a kilo­gram, which is $0.001$ kg.

The pen­cil line runs 1 whole cen­time­tre and 3 small marks more. That is $\displaystyle 1 + \frac{3}{10} = 1.3$ cm.

Remem­ber. Every digit has a place. The place tells you what the digit is worth. The expanded form writes down all those worths and adds them up.

Com­mon mis­takes#

• Giv­ing the digit itself instead of its place value. The $4$ in $457.368$ is worth $400$, not $4$.
• Writ­ing $\displaystyle \frac{6}{10}$ for a digit in the hun­dredths place. Count the places after the point: the sec­ond place is hun­dredths.
• Leav­ing out a zero that holds a place, as in writ­ing $8.7$ for $8.07$.
• Think­ing $0.35$ is big­ger than $0.4$ because $35$ is big­ger than $4$. Com­pare the tenths first: 3 tenths is less than 4 tenths.

Prac­tice#

1. Write $52.9$ in expanded form.
2. Write $8.07$ in expanded form.
3. Write $6.125$ in expanded form.
4. Write $70.4$ in expanded form.
5. In $457.368$, what is the place value of $7$?
6. In $457.368$, what is the place value of $6$?
7. In $1257.349$, which digit is in the hun­dredths place?
8. Which num­ber is $\displaystyle 500 + 3 + \frac{9}{10} + \frac{1}{1000}$?
9. Which is big­ger, $0.3$ or $0.03$? Use place value to explain.
10. Write $2008.502$ in expanded form.

Answers#