Place value and expanded form

Every price tag, every ruler read­ing and every weight on a packet of dal is writ­ten with dig­its in places. Place value is the idea that where a digit sits decides what it is worth. Expanded form is a way of writ­ing a num­ber that shows those worths openly. Together they let you read any dec­i­mal with con­fi­dence and com­pare num­bers like $0.3$ and $0.03$ with­out guess­ing.

A price tag says Rs $24.50$. That means 24 whole rupees and 50 paise. The point shows where the rupees stop.

Num­bers like $457.368$ work the same way. The point splits the num­ber into two parts.

On the left of the point are whole things. On the right are parts of one thing.

A num­ber with a point in it is called a dec­i­mal. Rs $24.50$ is a dec­i­mal. So is $457.368$.

The sin­gle signs $4$, $5$, $7$, $3$, $6$ and $8$ are called dig­its. A digit is one num­ber sign on its own.

How to say it

We read $457.368$ as four hun­dred fifty seven and three tenths six hun­dredths and eight thou­sandths.

Say the dig­its after the point one at a time. So $12.35$ is twelve point three five. It is not twelve point thirty five.

Here is why. The $3$ means three tenths. The $5$ means five hun­dredths. They sit in two dif­fer­ent places.

Say­ing thirty five glues them into one num­ber. Then you can­not hear where each digit belongs.

Expanded form

Expanded form breaks a num­ber into the value of each digit. Then you add those val­ues back together.

Tak­ing a num­ber apart shows you what each digit is really worth. Once you can see that, you can tell $0.3$ from $0.03$ with­out guess­ing.

The read­ing above hands you the answer. Three tenths is $\displaystyle \frac{3}{10}$. Six hun­dredths is $\displaystyle \frac{6}{100}$. Eight thou­sandths is $\displaystyle \frac{8}{1000}$.

457.368 in expanded form

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

Each piece shows what one digit is worth. Add all six pieces and you get $457.368$ back.

Place value

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

The $4$ in $457.368$ stands in the hun­dreds place. So its place value is $400$.

The same digit is worth dif­fer­ent amounts in dif­fer­ent places. The $4$ in $457.368$ is worth $400$. The $4$ in $70.4$ is worth four tenths. Same digit, dif­fer­ent place, dif­fer­ent worth.

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

Read any row across and you have that dig­it's place value.

Look along the chart from right to left. Each place is worth ten times the place on its right. Ten thou­sandths make one hun­dredth. Ten hun­dredths make one tenth. Ten tenths make one unit. That is the same pat­tern as ten units mak­ing one ten, car­ried on past the point.

Zeros at the two far ends

A zero before the first whole digit adds noth­ing. A zero after the last dec­i­mal digit adds noth­ing.

So $0001035.5300120000$ can be writ­ten as $1035.530012$. It is the same num­ber.

More worked exam­ples

Exam­ple 1: a num­ber with no zeros

Write $23.45$ in expanded form

The $2$ is in the tens place, the $3$ in the units, the $4$ in the tenths and the $5$ in the hun­dredths.

$$\displaystyle \begin{aligned}23.45 &= 20 + 3 + \frac{4}{10} + \frac{5}{100}\end{aligned}$$

Exam­ple 2: 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 $\displaystyle \dfrac{6}{100}$.

$$\displaystyle \begin{aligned}305.06 &= 300 + 5 + \frac{6}{100}\end{aligned}$$

The two zeros do not appear in the sum, but they still mat­ter in the num­ber. With­out them the $3$ and the $6$ would slide into the wrong places.

Exam­ple 3: 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 $0$ (there are none), hun­dredths $7$, thou­sandths $2$. So the num­ber is $41.072$. Notice the zero you must write in the tenths place.

Where you have met the small places

The places after the point are not new to you. You meet all three most days.

• Look at a ruler. Between $1$ cm and $2$ cm there are ten small marks. One small mark is one tenth of a cen­time­tre. In dec­i­mals that is $0.1$ cm.
• One paise is $\displaystyle \frac{1}{100}$ of a rupee. In dec­i­mals that is Rs $0.01$.
• One gram is $\displaystyle \frac{1}{1000}$ of a kilo­gram. In dec­i­mals that is $0.001$ kg.

So tenths sit on your ruler. Hun­dredths sit in the paise on a price tag. Thou­sandths sit in the grams on a packet.

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

Try these

Write each of these num­bers in expanded form.

One thing first. If a place holds a zero, that piece is worth noth­ing, so leave it out.

In $8.07$ the tenths place holds $0$, so the answer starts at hun­dredths. In $70.4$ the units place holds $0$, so that piece goes too.

Now answer these three about $457.368$.

1. What is the place value of $7$?
2. What is the place value of $6$?
3. Which digit is in the tenths place?

Com­mon mis­takes

• Read­ing $12.35$ as "twelve point thirty five". Say each digit after the point on its own.
• Think­ing $0.35$ is big­ger than $0.4$ because $35$ is big­ger than $4$. Com­pare tenths first: $3$ tenths is less than $4$ tenths.
• Leav­ing out a zero that holds a place, as in writ­ing $8.7$ for $8.07$.
• Giv­ing a dig­it's face value instead of its place value. The $4$ in $457.368$ is worth $400$, not $4$.
• Writ­ing $\displaystyle \dfrac{6}{10}$ for a digit in the hun­dredths place. Count the places after the point: sec­ond place, hun­dredths.

Key terms

Dec­i­mal

A num­ber writ­ten with a point, such as $457.368$.

Digit

One num­ber sign on its own, from $0$ to $9$.

Dec­i­mal point

The dot that sep­a­rates whole things from parts of one thing.

Place

The posi­tion of a digit, such as hun­dreds, units or tenths.

Place value

What a digit is worth where it stands.

Expanded form

A num­ber writ­ten as the sum of the val­ues of its dig­its.

Units place

The ones place, just to the left of the point.

Answers

Expanded form

1. $\displaystyle 52.9 = 50 + 2 + \dfrac{9}{10}$
2. $\displaystyle 8.07 = 8 + \dfrac{7}{100}$
3. $\displaystyle 6.125 = 6 + \dfrac{1}{10} + \dfrac{2}{100} + \dfrac{5}{1000}$
4. $\displaystyle 70.4 = 70 + \dfrac{4}{10}$

About 457.368

1. $7$, because it is in the units place.
2. $\displaystyle \dfrac{6}{100}$, which is $0.06$, because it is in the hun­dredths place.
3. $3$ is in the tenths place.