Numbers like 1, 2, 3, 45, 100 are whole numbers. But what happens when we need to express something between two whole numbers — like half a metre, or ₹12 and 50 paise?
We use a decimal number! The small dot . called the decimal point, separates the whole part from the fractional part.
🔢 The Decimal Point
The dot (.) in a decimal number is called the decimal point. It is the "point" in the chapter title!
3.75
⬆ Whole Part
The digits to the left of the decimal point form the whole number part.
3.75
⬇ Decimal Part
The digits to the right of the decimal point form the decimal (fractional) part.
3.75
| Situation | Decimal Number | Meaning |
|---|---|---|
| Price of a pen | ₹12.50 | 12 rupees and 50 paise |
| Length of a board | 1.5 m | 1 metre and 5 tenths of a metre |
| Weight of mangoes | 2.75 kg | 2 kg and 75 hundredths of a kg |
| A race time | 9.58 s | 9 seconds and 58 hundredths |
| Body temperature | 37.2°C | 37 degrees and 2 tenths |
Decimals and fractions are closely related. Any fraction whose denominator is 10, 100, 1000 (or any power of 10) can be directly written as a decimal.
1/10 = 0.1
One-tenth = zero point one
3/10 = 0.3, 7/10 = 0.7
1/100 = 0.01
One-hundredth = zero point zero one
25/100 = 0.25, 7/100 = 0.07
1/1000 = 0.001
One-thousandth = zero point zero zero one
125/1000 = 0.125
Denominator is 10 → 1 decimal place
3/10 = 0.3
Denominator is 100 → 2 decimal places
47/100 = 0.47
Denominator is 4. Make it 100: multiply top and bottom by 25
3/4 = (3×25)/(4×25) = 75/100 = 0.75
First convert 3/5: multiply by 2 → 6/10 = 0.6
So 2 3/5 = 2 + 0.6 = 2.6
| Decimal | Read as | Fraction | Simplified |
|---|---|---|---|
| 0.5 | 5 tenths | 5/10 | 1/2 |
| 0.25 | 25 hundredths | 25/100 | 1/4 |
| 0.75 | 75 hundredths | 75/100 | 3/4 |
| 0.2 | 2 tenths | 2/10 | 1/5 |
| 0.125 | 125 thousandths | 125/1000 | 1/8 |
| 0.4 | 4 tenths | 4/10 | 2/5 |
0.7 → 1 digit → denominator 10 → 7/10
0.43 → 2 digits → denominator 100 → 43/100
0.125 → 3 digits → denominator 1000 → 125/1000
Just like whole numbers have place values (ones, tens, hundreds…), decimal numbers have place values after the decimal point: tenths, hundredths, thousandths.
| Thousands | Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|---|
| 1000 | 100 | 10 | 1 | . | 1/10 | 1/100 | 1/1000 |
| Tens | Ones | . | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|
| 3 | 5 | . | 7 | 4 | 2 |
Digit 3 → Tens place
Value = 3 × 10 = 30
Digit 5 → Ones place
Value = 5 × 1 = 5
Digit 7 → Tenths place
Value = 7 × 1/10 = 0.7
Digit 4 → Hundredths place
Value = 4 × 1/100 = 0.04
Digit 2 → Thousandths place
Value = 2 × 1/1000 = 0.002
Each place to the left is 10 times bigger.
Each place to the right is 10 times smaller.
Just like we write 345 = 300 + 40 + 5, we can expand decimal numbers using their place values.
35.742 = 30 + 5 + 7/10 + 4/100 + 2/1000
OR in decimal form:
35.742 = 30 + 5 + 0.7 + 0.04 + 0.002
204.05 = 200 + 0 + 4 + 0/10 + 5/100
204.05 = 200 + 4 + 0.05
Note: We skip zero-valued places when writing expanded form.
40 + 3 + 6/10 + 1/100 = ?
= 43 + 0.6 + 0.01 = 43.61
🔎 Interactive: Expand a Decimal
6.83 = 6 + 0.___ + 0.0___
Reading decimal numbers correctly is important. We do not read the decimal part as a number — we read each digit separately.
✓ Correct Way
- 3.14 → "three point one four"
- 25.06 → "twenty-five point zero six"
- 0.5 → "zero point five"
✗ Wrong Way
- 3.14 ✗ "three point fourteen"
- 25.06 ✗ "twenty-five point six"
- 0.5 ✗ "point five only"
0.06 has a 0 in the tenths place and 6 in the hundredths place.
0.6 has a 6 in the tenths place — it is ten times bigger than 0.06!
| Number in Words | Decimal |
|---|---|
| Five and three tenths | 5.3 |
| Zero point zero eight | 0.08 |
| Twelve and forty-five hundredths | 12.45 |
| One hundred and five thousandths | 100.005 |
| Zero point two five zero | 0.250 |
0.5 = 0.50 = 0.500 — all are equal!
✅ Like Decimals
Decimals with the same number of decimal places are called like decimals.
Examples: 3.25 and 7.81 (both 2 decimal places)
0.5 and 1.3 (both 1 decimal place)
⚡ Unlike Decimals
Decimals with different numbers of decimal places are called unlike decimals.
Examples: 3.5 and 2.75
0.4 and 0.125
Add zeros to the end of the decimal part to make the number of decimal places equal.
Maximum decimal places = 2 (from 2.45)
4.3 → 4.30 | 2.45 → 2.45 | 1.6 → 1.60
Now all have 2 decimal places — they are like decimals!
To compare two decimal numbers, follow this step-by-step method:
Compare Whole Parts
The number with the bigger whole part is greater.
7.34 > 5.98
Compare Tenths
If whole parts are equal, compare the tenths digit.
4.73 > 4.58
Compare Hundredths
If tenths are also equal, compare hundredths, and so on.
3.49 > 3.42
Step 1: Whole parts equal (both 4)
Step 2: Tenths equal (both 7)
Step 3: Hundredths — 9 > 3
∴ 4.79 > 4.73
Convert to like decimals: 2.30, 2.15, 2.07, 2.90
Compare: 2.07 < 2.15 < 2.30 < 2.90
Ascending: 2.07, 2.15, 2.3, 2.9
📊 Compare Two Decimals
Adding decimals is just like adding whole numbers — but you must line up the decimal points first!
Step 1
Write the numbers one below the other, lining up decimal points.
Step 2
Fill in missing places with zeros to make like decimals.
Step 3
Add column by column from right to left, carrying over as needed. Place the decimal point in the answer.
👆 Step-by-step Addition
Subtracting decimals follows the same rule: line up the decimal points first, then subtract column by column.
Ones: 30−84 requires borrowing → 130−84=46, borrow 1 from ones; 7−1−2=4
This is the most important shortcut! When you multiply a decimal by 10, 100 or 1000, the decimal point shifts to the right.
× 10 → Shift 1 place right
2.53 × 10 = 25.3
0.07 × 10 = 0.7
× 100 → Shift 2 places right
2.53 × 100 = 253
0.07 × 100 = 7
× 1000 → Shift 3 places right
2.53 × 1000 = 2530
0.007 × 1000 = 7
→ Decimal Point Shift Animation
Multiply normally, then count the total decimal places and put the point in the answer.
Ignore the decimal: 34 × 6 = 204
3.4 has 1 decimal place → answer has 1 decimal place
3.4 × 6 = 20.4
Ignore the decimal: 215 × 4 = 860
2.15 has 2 decimal places → answer has 2 decimal places
2.15 × 4 = 8.60
Ignore decimals: 12 × 4 = 48
1.2 has 1 decimal place, 0.4 has 1 decimal place → total = 2 decimal places
1.2 × 0.4 = 0.48
2.5 × 1.3 → (1 + 1) = 2 decimal places → 25 × 13 = 325 → 3.25
Decimal numbers are everywhere in our daily lives! Let's see where we use them.
💵 Money
- ₹45.75 = 45 rupees 75 paise
- 1 paise = ₹0.01
- 50 paise = ₹0.50
📏 Length
- 1 cm = 0.01 m
- 1 mm = 0.001 m
- Height: 1.65 m
⚖ Weight
- 1 g = 0.001 kg
- 500 g = 0.5 kg
- 2.75 kg of rice
⏱ Sports & Science
- 100m race: 9.58 s
- Temperature: 37.2°C
- Rainfall: 12.5 mm
25.50 + 12.75 + 8.00
= 25.50 + 12.75 + 8.00 = ₹46.25
8.50 − 3.75 = 4.75 m
1.5 × 6 = 15 × 6 ÷ 10 = 90 ÷ 10 = 9 L
12.8 × 100 = 1280 km