A Peek Beyond the Point

Why This Matters

Think about buying things at a shop. A bus ticket costs ₹12.50. A bottle of cold drink is ₹40.75. Your height is 1.6 metres. The thickness of a single hair is 0.1 mm.

Look at those numbers. They all have a small dot in the middle. That dot is the decimal point, and the numbers are called decimals.

Why do we even need them? Because whole numbers like 1, 2, 3 are not enough. Many things in real life are not a clean whole number. A screw is a little longer than 2 cm but not quite 3 cm. A piece of paper is half a centimetre thick. Money has paise, which are smaller than a rupee.

So we need a way to write the bits that are smaller than one. That is exactly what this chapter is about. By the end, you will read, write, compare, add, and subtract decimals with confidence — and you will understand why each rule works, not just that it works.

The Big Idea

A decimal is just a clever way to write a fraction whose bottom number is 10, 100, 1000, and so on. The decimal point marks the boundary: everything to the left is the whole part, and everything to the right is the part smaller than one. As you move right, each place is one-tenth of the place before it — tenths, then hundredths, then thousandths. That single idea — “split into ten, again and again” — explains everything about decimals.

Let’s Break It Down

Before we begin, let’s quickly refresh two ideas from earlier classes that this whole chapter leans on.

Tenths, hundredths and thousandths — and why each is 1/10 of the one before

Imagine one whole thing — say, a 1-cm length on a ruler. Now cut it into 10 equal pieces. Each piece is one-tenth, written as 1/10. We can write this as the decimal 0.1.

That answers “how do we measure something smaller than 1?” But what if one-tenth is still too big? Easy — do the same trick again. Cut each tenth into 10 equal pieces. Now the whole is in 100 equal pieces. Each tiny piece is one-hundredth, written 1/100, or the decimal 0.01.

Here is the key question: why is a hundredth exactly one-tenth of a tenth? Because we made it by cutting one tenth into ten equal parts. One out of those ten parts is, by definition, one-tenth of that tenth. So 10 hundredths fit inside 1 tenth, and 100 hundredths fit inside 1 whole.

The picture below shows this splitting happen step by step. Figure 3.1 shows one whole, then the same whole cut into 10 tenths, then each tenth cut again into hundredths.

Tap to zoom

Keep going and you get even smaller pieces. Cut each hundredth into 10 parts and you get thousandths (1/1000, or 0.001). A thousand of these make one whole. You can keep splitting by 10 forever, getting smaller and smaller places — but for now, tenths, hundredths, and thousandths are all we need.

A decimal is a fraction

This is the most important sentence in the chapter: a decimal is just a fraction in disguise.

• 0.7 means 7 tenths, which is the fraction 7/10.
• 0.25 means 25 hundredths, which is the fraction 25/100.
• 0.003 means 3 thousandths, which is 3/1000.

So whenever a decimal looks confusing, read it back as a fraction over 10, 100, or 1000. The number of digits after the point tells you the bottom number: one digit means tenths (/10), two digits means hundredths (/100), three digits means thousandths (/1000).

For example, 0.45 has two digits after the point, so it is 45/100. And 0.6 has one digit, so it is 6/10.

This also explains a fact that surprises many students: 0.5 and 0.50 are the same number. Why? Because 0.5 = 5/10 and 0.50 = 50/100, and 50/100 simplifies to 5/10. They are equal. Adding a zero at the end of a decimal does not change its value — it just splits the same amount into smaller pieces.

Reading and writing decimals

To write a decimal, you put the whole-number part, then the point, then the fraction part. To read it, you say the whole part, the word “point”, and then each digit after the point one at a time.

So:

• 70.5 is read “seventy point five” (short for seventy and five-tenths).
• 7.05 is read “seven point zero five” (seven and five-hundredths).
• 0.274 is read “zero point two seven four” — not “zero point two hundred seventy-four”.

Why do we say the digits one by one after the point? Because the point already tells you the places. In 0.274, the 2 is tenths, the 7 is hundredths, and the 4 is thousandths. Saying “two seven four” keeps each digit in its own place. Saying “two hundred seventy-four” would wrongly suggest a whole number.

Notice how badly we need that point. Without it, 705 could mean three completely different things: seven hundred and five (705), or seventy and five tenths (70.5), or seven and five hundredths (7.05). The point is the separator that tells us exactly where the whole part ends and the fraction part begins.

Figure 3.2 lays this out as a place value chart, so you can see each digit drop into its correct slot.

Tap to zoom

We write a decimal in expanded form by giving each digit its place value. For 7.05:

7.05 = (7 × 1) + (0 × 1/10) + (5 × 1/100)

And for 70.5:

70.5 = (7 × 10) + (0 × 1) + (5 × 1/10)

This is exactly the same expanded-form idea you used for whole numbers like 456 = (4 × 100) + (5 × 10) + (6 × 1). Decimals just add more places on the right.

Decimals on the number line

A decimal is a number, so it has a home on the number line. Finding it is easy once you remember the “split into ten” idea.

To place a decimal between 0 and 1, take the gap from 0 to 1 and cut it into 10 equal steps. Each step is one-tenth. The first mark is 0.1, the next is 0.2, and so on up to 1.0. So 0.7 sits at the 7th mark — it is 7 tenths along the way from 0 to 1.

What about a hundredth like 0.74? It lives between 0.7 and 0.8. To find it, zoom into that small gap and cut it into 10 equal parts. Then 0.74 is at the 4th of those tiny marks. Figure 3.3 shows both the main line and this zoom.

Tap to zoom

This is the big picture: zooming in and splitting by ten again is exactly how you locate any decimal, no matter how many places it has.

Comparing and ordering decimals

To compare two decimals, do not just look at the digits as if they were whole numbers. Compare them place by place, starting from the left — ones first, then tenths, then hundredths.

Here is the rule and the reason. The leftmost place that differs decides the winner. Why? Because one whole tenth is bigger than any number of hundredths put together (you would need 10 hundredths just to make 1 tenth). So once a higher place is settled, the lower places cannot overturn it.

Let’s compare 0.45 and 0.5:

• Ones: both have 0. Tie. Move right.
• Tenths: 0.45 has 4 tenths, but 0.5 has 5 tenths. Since 5 > 4, 0.5 is bigger. We can stop here.

The 5 in the hundredths place of 0.45 looks big, but it cannot help — it is only 5 hundredths, far less than the one extra whole tenth that 0.5 has. Figure 3.4 shows this lined up neatly.

Tap to zoom

A handy trick: to compare decimals with different numbers of digits, you may add zeros at the end so they match. Comparing 0.5 and 0.45 becomes comparing 0.50 and 0.45. Now both have hundredths, and 50 hundredths clearly beats 45 hundredths. (Remember, adding an end-zero does not change the value.)

To order a list of decimals, just compare them in pairs using this method, then line them up from smallest to largest.

Adding and subtracting decimals — line up the points

Adding decimals uses one golden rule: line up the decimal points, one directly under the other. Then add each column just like ordinary whole numbers.

Why must the points line up? Because only then do tenths sit under tenths, hundredths under hundredths, and ones under ones. You can only add things of the same kind: tenths with tenths, not tenths with hundredths. The lined-up point guarantees that. If a column adds up to 10 or more, you carry to the next place on the left — exactly like normal addition, because 10 tenths make 1 whole, 10 hundredths make 1 tenth, and so on.

Let’s see it in action. The picture in Figure 3.5 shows 2.7 + 3.5 worked out with the points lined up.

Tap to zoom

Subtraction works the same way: line up the points and subtract column by column, starting from the right. If a top digit is too small, borrow from the place to its left — and remember 1 whole becomes 10 tenths, 1 tenth becomes 10 hundredths.

Here is a full example with both adding and a small carry.

Now a subtraction with borrowing.

Common Mistakes

Quick Check

Practice Problems

Easy

Medium

Challenge

Summary

• A decimal is a way to write numbers smaller than one. The decimal point separates the whole part (left) from the fraction part (right).
• Each place to the right of the point is one-tenth of the place before it: tenths (1/10), then hundredths (1/100), then thousandths (1/1000). This is because we split each part into 10 equal pieces, again and again.
• A decimal is just a fraction over 10, 100, or 1000. The number of digits after the point gives the bottom number: 0.7 = 7/10, 0.25 = 25/100, 0.003 = 3/1000.
• Adding a zero at the end of a decimal does not change its value: 0.5 = 0.50.
• Read decimals by saying each digit after the point one at a time: 0.274 is “zero point two seven four”.
• To place a decimal on the number line, split the gap into 10 equal steps; zoom in and split by 10 again for hundredths.
• Compare decimals place by place from the left; the first place that differs decides the winner, because one whole tenth outweighs any number of hundredths.
• To add or subtract, line up the decimal points, then work column by column — carrying or borrowing exactly like whole numbers.

What’s Next

You now have a complete toolkit for decimals. Next, in Chapter 4 — Expressions Using Letter-Numbers, we move from fixed numbers to using letters to stand for numbers. Just as decimals extended the place value system to handle smaller quantities, letter-numbers extend arithmetic to describe patterns and rules that work for any number at once.