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. So what exactly is a dec­i­mal? How do the places after the point work, how do you read one aloud, and which zeros can you safely leave out? Let's find out.

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

See the small dot in Rs $12.35$? That dot is called the point. The part before it is the whole rupees, and the part after it is the paise.

A ruler works the same way. One cen­time­tre is marked off into 10 small parts, and a pen­cil line can eas­ily stop between two whole marks. The way we write such an amount is called a dec­i­mal.

A dec­i­mal is a frac­tion#

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

You can turn frac­tions into dec­i­mals and dec­i­mals back into frac­tions. It is the same num­ber either way; only the writ­ing changes.

Here we'll stick to tenths, hun­dredths and thou­sandths, so every frac­tion you see is cut into 10 parts, or 100, or 1000.

Take $\displaystyle \frac{5}{10}$. 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, so 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 $0$, $1$, $2$ and so on up to $9$. Where a digit sits tells you how big it is, and that spot is called its place.

The same digit means some­thing dif­fer­ent in each place, and only one digit fits in a place.

The first place after the point counts tenths. One cen­time­tre holds exactly 10 small parts, and once you have 10 of them you have a whole cen­time­tre again. That is why this 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, so 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 is hold­ing the tenths place, telling you there are no tenths. Leave it out and you would have $0.1$, which is ten times big­ger. Small zero, big job!

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

Say the dig­its one at a time#

Don't read $12.35$ as twelve point thirty five. There's a good rea­son for this.

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

In Rs $12.35$, the $3$ is three ten-paise coins and 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.

You can also do a sec­ond job with a dec­i­mal: say what each digit is worth. $12.35$ is 12 wholes, 3 tenths and 5 hun­dredths. That is nam­ing the parts, which is dif­fer­ent from 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 $\displaystyle \frac{35}{100}$. Split into places it is $\displaystyle \frac{3}{10} + \frac{5}{100}$, because each col­umn of ten squares is one tenth.

Thou­sandths, from base 1000#

A litre jug of milk holds 1000 mil­li­l­itres, so 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 and hun­dredths places open, so the $1$ 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, and those 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 and one from the tail.

Now try a long num­ber. The rule is the same; there are just more zeros to count.

A big­ger exam­ple

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

Three zeros went from the front and four from the tail, yet the num­ber did not change one bit.

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, so the num­ber shrinks.

Zeros inside a num­ber behave 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, becom­ing ten times big­ger.

You would get $1035.53012$, which is a dif­fer­ent num­ber alto­gether.

The sec­ond kind is the zero in $0.5$. It does­n't change the amount at all; $.5$ and $0.5$ are the same num­ber.

We write it so the point can't be missed, and so the whole part isn't left blank. That is a rule of neat 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 $\displaystyle \frac{7}{10}$ 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: $\displaystyle \frac{7}{10} = 0.7$. Read it as zero point seven.

Exam­ple 2: hun­dredths#

Write $\displaystyle \frac{43}{100}$ 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 $\displaystyle \frac{70}{1000}$ 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 $\displaystyle \frac{7}{100}$ 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: $\displaystyle \frac{1}{10} = 0.1$.

Hun­dredth

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

Thou­sandth

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

Place

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

Answers#

Show answers

1. $\displaystyle \frac{3}{10} = 0.3$
2. $\displaystyle \frac{7}{100} = 0.07$
3. $\displaystyle \frac{9}{1000} = 0.009$
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.