What decimals are

Dec­i­mals are every­where. A price tag says ₹12.35. A ruler shows 1.3 cm. A ther­mome­ter reads 98.6. Each of these num­bers has a small dot in it, and each one tells you about an amount that falls between two whole num­bers. In this les­son you will see what a dec­i­mal really is: a frac­tion whose bot­tom is 10, 100, 1000 and so on, writ­ten in a shorter way.

A dec­i­mal is a frac­tion writ­ten with a point#

Take a strip of paper and cut it into 10 equal parts. Shade one part. The shaded part is one tenth of the strip.

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

The dot is called the dec­i­mal point. Whole things go on its left. Parts of one thing go on its right.

So a dec­i­mal is just another way to write a frac­tion whose bot­tom is $10$, $100$, $1000$ and so on. The amount is the same. Only the writ­ing changes.

Tenths#

When the bot­tom is $10$, the answer has one digit after the point.

$$\displaystyle \begin{aligned} \frac{1}{10} &= 0.1 \\ \frac{2}{10} &= 0.2 \\ \frac{3}{10} &= 0.3 \\ \frac{4}{10} &= 0.4 \end{aligned}$$

The top can be big­ger than $10$. Then there are whole ones too.

$$\displaystyle \begin{aligned} \frac{12}{10} &= 1.2 \\ \frac{125}{10} &= 12.5 \\ \frac{1235}{10} &= 123.5 \end{aligned}$$

Look at what hap­pened. Write the top num­ber, then put the point so that one digit stands after it. $\displaystyle \frac{12}{10}$ is $1.2$, read as one point two.

Hun­dredths#

When the bot­tom is $100$, the answer has two dig­its after the point.

$$\displaystyle \begin{aligned} \frac{1}{100} &= 0.01 \\ \frac{2}{100} &= 0.02 \\ \frac{5}{100} &= 0.05 \\ \frac{12}{100} &= 0.12 \\ \frac{125}{100} &= 1.25 \\ \frac{1235}{100} &= 12.35 \end{aligned}$$

In $\displaystyle \frac{1}{100}$ the top has only one digit, but we need two dig­its after the point. So we put a zero in front of it. Writ­ing $1$ as $01$ does not change it, just as $07$ is still seven. That gives $0.01$, read as zero point zero one.

How to read a dec­i­mal aloud#

Read $12.35$ as twelve point three five. Do not read it as twelve point thirty five.

Why? The $3$ and the $5$ live in dif­fer­ent places. The $3$ stands for 3 tenths and the $5$ stands for 5 hun­dredths. Say­ing "thirty five" joins them into one num­ber, and then nobody can tell which place each digit is in. So say the whole part as a num­ber, then say the dig­its after the point one at a time.

The grid shows $0.35$. Three full columns are 3 tenths. Five more squares are 5 hun­dredths. Alto­gether that is $35$ squares out of $100$, so $\displaystyle \frac{35}{100} = 0.35$.

Thou­sandths#

When the bot­tom is $1000$, the answer has three dig­its after the point.

$$\displaystyle \begin{aligned} \frac{1}{1000} &= 0.001 \\ \frac{2}{1000} &= 0.002 \\ \frac{5}{1000} &= 0.005 \\ \frac{12}{1000} &= 0.012 \\ \frac{125}{1000} &= 0.125 \\ \frac{1235}{1000} &= 1.235 \end{aligned}$$

Read $1.235$ as one point two three five. Read $0.001$ as zero point zero zero one.

Each time we cut the whole into ten times as many parts, each part becomes ten times smaller. That is why each place after the point is worth one tenth of the place before it.

The rule: count the zeros#

You have seen a pat­tern. The num­ber of zeros in the bot­tom tells you how many dig­its go after the point.

So here is the method.

1. Write the top num­ber.
2. Count the zeros in the bot­tom.
3. Start­ing from the right end of the top num­ber, count that many dig­its and put the point there.
4. If you run out of dig­its, put zeros in front to fill the gap. Then write $0$ before the point.

The rule keeps work­ing for big­ger bot­toms too.

$$\displaystyle \begin{aligned} \frac{1}{10000} &= 0.0001 \\ \frac{123}{1000000} &= 0.000123 \\ \frac{25}{10000000} &= 0.0000025 \\ \frac{256}{100000000} &= 0.00000256 \end{aligned}$$

Check the sec­ond one. $1000000$ has six zeros, so we need six dig­its after the point. The top, $123$, has only three dig­its. Three zeros go in front: $000123$. So the answer is $0.000123$.

Exam­ple 1#

Write $\displaystyle \frac{123}{10000}$ as a dec­i­mal.

$10000$ has four zeros, so four dig­its go after the point. $123$ has three dig­its, so one zero goes in front: $0123$.

$$\displaystyle \frac{123}{10000} = 0.0123$$

Exam­ple 2#

Write $\displaystyle \frac{2354}{1000000}$ as a dec­i­mal.

Six zeros, so six dig­its after the point. $2354$ has four dig­its, so two zeros go in front: $002354$.

$$\displaystyle \frac{2354}{1000000} = 0.002354$$

Exam­ple 3#

Write $\displaystyle \frac{23587}{1000}$ as a dec­i­mal.

Three zeros, so three dig­its after the point. Count three dig­its from the right of $23587$: that is $587$. The point goes just before them.

$$\displaystyle \frac{23587}{1000} = 23.587$$

This time the top had more dig­its than we needed, so there are whole ones in front of the point. No zeros were needed.

Exam­ple 4#

Write $0.07$ as a frac­tion.

There are two dig­its after the point, so the bot­tom is $100$. The dig­its are $07$, which is just $7$.

$$\displaystyle 0.07 = \frac{7}{100}$$

Remem­ber. A dec­i­mal is a frac­tion with bot­tom $10$, $100$, $1000$ and so on. The num­ber of zeros in the bot­tom is the num­ber of dig­its after the point.

Com­mon mis­takes#

• Writ­ing $\displaystyle \frac{7}{100}$ as $0.7$. Hun­dredths need two dig­its after the point, so it is $0.07$.
• Read­ing $12.35$ as twelve point thirty five. Say three five.
• For­get­ting to fill the gap with zeros. $\displaystyle \frac{23}{1000}$ is $0.023$, not $0.23$.
• Putting the point in the wrong place by count­ing from the left. Always count from the right end of the top num­ber.

Prac­tice#

1. Write $\displaystyle \frac{7}{10}$ as a dec­i­mal.
2. Write $\displaystyle \frac{25}{10}$ as a dec­i­mal.
3. Write $\displaystyle \frac{43}{100}$ as a dec­i­mal.
4. Write $\displaystyle \frac{506}{100}$ as a dec­i­mal.
5. Write $\displaystyle \frac{9}{1000}$ as a dec­i­mal.
6. Write $\displaystyle \frac{23}{1000}$ as a dec­i­mal.
7. Write $\displaystyle \frac{2405}{1000}$ as a dec­i­mal.
8. Write $\displaystyle \frac{3}{10000}$ as a dec­i­mal.
9. Write $\displaystyle \frac{125}{1000000}$ as a dec­i­mal.
10. How do you read $4.508$ aloud?
11. Write $0.048$ as a frac­tion.

Answers#