What decimals are

Dec­i­mals are every­where in daily life: on price tags, on the petrol pump, on a doc­tor's ther­mome­ter and on the weigh­ing scale at the kirana shop. They let us write amounts that fall between whole num­bers. This les­son explains what a dec­i­mal really is, how the places after the point work, how to read a dec­i­mal aloud, and which zeros you may leave out.

A rupee can be split into 100 paise. So a price tag may say Rs $12.35$. That is more than 12 rupees. It is less than 13 rupees.

Look at the small dot in Rs $12.35$. That dot is called the point. The part before the point is the whole rupees. The part after it is the paise.

A ruler works the same way. One cen­time­tre is marked off into 10 small parts. A pen­cil line can stop between two whole marks. There is a way to write that. It is called a dec­i­mal.

A dec­i­mal is a frac­tion

A dec­i­mal is another way to write a frac­tion. It uses base ten. That means the whole is cut into 10 parts, or 100, or 1000.

Frac­tions can be turned into dec­i­mals. Dec­i­mals can be turned back into frac­tions. It is the same num­ber either way. Only the writ­ing changes.

This les­son uses only tenths, hun­dredths and thou­sandths. So every frac­tion here is cut into 10 parts, or 100, or 1000.

Take $\displaystyle \frac{5}{10}$. It is a frac­tion. As a dec­i­mal you write it $0.5$.

From frac­tion to dec­i­mal

$$\displaystyle \begin{aligned}&\frac{5}{10} \\ &= 0.5\end{aligned}$$

Remem­ber. A dec­i­mal that stops is a frac­tion. It is writ­ten with a point instead of the line in a frac­tion.

Tenths, from base 10

Go back to the ruler. One cen­time­tre holds 10 small parts. One small part is one tenth of a cen­time­tre.

As a frac­tion that is $\displaystyle \frac{1}{10}$. As a dec­i­mal it is $0.1$. You read it as zero point one.

A digit is one of the num­ber sym­bols. They run $0$, $1$, $2$ and so on up to $9$. Where a digit sits tells you how big it is. That spot is called its place.

The same digit means some­thing dif­fer­ent in each place. Only one digit fits in a place.

The first place after the point counts tenths. One cen­time­tre holds exactly 10 small parts. Once you have 10 small parts you have a whole cen­time­tre again. So that place never needs to go past 9.

Now take one whole cen­time­tre and 2 small parts more. As a dec­i­mal that is $1.2$. You read it as one point two.

Hun­dredths, from base 100

Money cuts a rupee into 100 paise. One paisa is one hun­dredth of a rupee.

As a frac­tion that is $\displaystyle \frac{1}{100}$. As a dec­i­mal it is $0.01$. You read it as zero point zero one.

Look at the zero just after the point. It holds the tenths place. It says there are no tenths. With­out it you would have $0.1$, which is ten times big­ger.

A longer one is $12.35$. You read it as twelve point three five.

Say the dig­its one at a time

Do not read $12.35$ as twelve point thirty five. There is a good rea­son for this.

Each digit after the point sits in its own place. The $3$ means 3 tenths. The $5$ means 5 hun­dredths.

In Rs $12.35$ the $3$ is three ten-paise coins. The $5$ is five sin­gle paise. Those are dif­fer­ent coins.

Say thirty five and you have tipped the two piles together. So read the dig­its one by one, from left to right.

Remem­ber. To read a dec­i­mal aloud, say the whole part as a num­ber. Then say each digit after the point on its own.

There is a sec­ond job you can do with a dec­i­mal. You can say what each digit is worth. $12.35$ is 12 wholes, 3 tenths and 5 hun­dredths. That is nam­ing the parts, not read­ing it aloud.

The grid shows why $0.35$ can be read two ways and still be the same amount. As one frac­tion it is $\displaystyle \dfrac{35}{100}$. Split into places it is $\displaystyle \dfrac{3}{10} + \dfrac{5}{100}$, because each col­umn of ten squares is one tenth.

Thou­sandths, from base 1000

A litre jug holds 1000 mil­li­l­itres. One mil­li­l­itre is one thou­sandth of a litre.

As a frac­tion that is $\displaystyle \frac{1}{1000}$. As a dec­i­mal it is $0.001$. You read it as zero point zero zero one.

The two zeros hold the tenths place and the hun­dredths place. The $1$ then lands in the thou­sandths place.

Another one is $1.235$. You read it as one point two three five. It is one whole, 2 tenths, 3 hun­dredths and 5 thou­sandths.

These three are read the same way, digit by digit.

Zeros that do a job, and zeros that do not

Some zeros hold a place open. Oth­ers just sit there doing noth­ing. The lazy ones can go.

The places in front of the point count ones, then tens, then hun­dreds. In $007$ there are no hun­dreds and no tens. So those two zeros count noth­ing at all. You can leave them out, and $007$ is just $7$.

Extra zeros at the very end after the point do noth­ing either. $0.50$ and $0.5$ are the same amount. Fifty paise is half a rupee, writ­ten either way.

So a price can be tidied. Rs $012.350$ is just Rs $12.35$. One zero went from the front. One zero went from the tail.

Now a long num­ber to tidy. The rule is the same, but there are more zeros to count.

A big­ger exam­ple

$$\begin{aligned}&0001035.5300120000 \\ &= 1035.5300120000 \\ &= 1035.530012\end{aligned}$$

Three zeros went from the front. Four zeros went from the tail. The num­ber did not change.

Two zeros to leave alone

The first kind sits between other dig­its. Rs $105$ and Rs $15$ are not the same money.

The zero in $105$ holds the tens place empty. Take it out and the $1$ slides down from hun­dreds to tens. The num­ber gets smaller.

Zeros inside a num­ber work the same way after the point. In $1035.530012$ there are two zeros after the $53$.

Drop one of them and every digit behind it slides one place to the left. Each of those dig­its becomes ten times big­ger.

You would get $1035.53012$. That is a dif­fer­ent num­ber.

The sec­ond kind is the zero in $0.5$. It does not change the amount. $.5$ and $0.5$ are the same num­ber.

You write it so the point can­not be missed. It also stops the whole part being left blank. That is a rule of writ­ing, not a rule of value.

Remem­ber. A zero at the front can go only while another digit of the whole part is left stand­ing. $007$ is $7$, but $0.5$ keeps its zero. A zero at the tail can go only after the point. $0.50$ is $0.5$, but $50$ is not $5$. A zero doing a job stays.

More worked exam­ples

Exam­ple 1: tenths

Write $\displaystyle \dfrac{7}{10}$ as a dec­i­mal.

Seven tenths means the digit $7$ goes in the first place after the point. There are no whole ones, so write $0$ before the point: $\displaystyle \dfrac{7}{10} = 0.7$. Read it as zero point seven.

Exam­ple 2: hun­dredths

Write $\displaystyle \dfrac{43}{100}$ as a dec­i­mal.

Forty-three hun­dredths is $4$ tenths and $3$ hun­dredths, because $40$ hun­dredths make $4$ tenths.

$$\displaystyle \begin{aligned}\frac{43}{100} &= \frac{4}{10} + \frac{3}{100} \\ &= 0.43\end{aligned}$$

Exam­ple 3: a zero that holds a place

A par­cel weighs $2$ kg and $70$ g. Write this in kilo­grams.

One gram is one thou­sandth of a kilo­gram, so $70$ g is $\displaystyle \dfrac{70}{1000}$ kg. That is $0$ tenths, $7$ hun­dredths and $0$ thou­sandths.

$$\displaystyle \begin{aligned}2 + \frac{70}{1000} &= 2 + \frac{7}{100} \\ &= 2.07\end{aligned}$$

The par­cel weighs $2.07$ kg. The zero after the point must stay. With­out it you would write $2.7$ kg, which is $2$ kg and $700$ g.

Prac­tice

1. Write $\displaystyle \frac{3}{10}$ as a dec­i­mal.
2. Write $\displaystyle \frac{7}{100}$ as a dec­i­mal.
3. Write $\displaystyle \frac{9}{1000}$ as a dec­i­mal.
4. Read $12.35$ aloud. Write down the words you said.
5. Tidy $00045.6700$.
6. Why do you write the zero in $0.5$?

Com­mon mis­takes

• Writ­ing $\displaystyle \dfrac{7}{100}$ as $0.7$. Hun­dredths need two places after the point: $0.07$.
• Read­ing $12.35$ as "twelve point thirty five". Say the dig­its one at a time.
• Remov­ing a zero from between dig­its, such as writ­ing $1035.53012$ for $1035.530012$.
• Remov­ing a tail zero from a whole num­ber. $0.50$ is $0.5$, but $50$ is not $5$.
• Putting a digit big­ger than $9$ in one place. Ten tenths make a whole, so carry it to the units place.

Key terms

Dec­i­mal

A way of writ­ing a frac­tion in base ten, using a point.

Point

The dot that sep­a­rates the whole part from the parts of one.

Base ten

Count­ing in tens, so each place is worth ten times the place to its right.

Tenth

One of ten equal parts of a whole: $\displaystyle \dfrac{1}{10} = 0.1$.

Hun­dredth

One of a hun­dred equal parts: $\displaystyle \dfrac{1}{100} = 0.01$.

Thou­sandth

One of a thou­sand equal parts: $\displaystyle \dfrac{1}{1000} = 0.001$.

Place

The posi­tion of a digit, which decides what it is worth.

Answers

1. $\displaystyle \dfrac{3}{10} = 0.3$
2. $\displaystyle \dfrac{7}{100} = 0.07$
3. $\displaystyle \dfrac{9}{1000} = 0.009$
4. Twelve point three five.
5. $00045.6700 = 45.67$. Three zeros go from the front and two from the tail.
6. Model answer: the zero in $0.5$ does not change its value, since $.5$ is the same num­ber. It is writ­ten so that the point can­not be missed and the whole part is not left blank.