Fractions, Mixed Numbers & Decimals ഭിന്നസംഖ്യകളും ദശാംശരൂപങ്ങളും
In this chapter you will square fractions like 1½ and 1⅓, square decimals like 1.41 by hand, turn fractions into decimals, compare 1.41 with 1.414, and round a length to the centimetre or the millimetre. Each of these is a small skill; this page sharpens all of them.
- A fraction (ഭിന്നസംഖ്യ) such as 3/4 means 3 parts out of 4 equal parts of a unit.
- Decimals: 0.7 = 7/10, 0.25 = 25/100, and the place values after the point are tenths, hundredths, thousandths.
- Squaring (page 1): n² = n × n, the area of a square of side n.
1. Mixed numbers are fractions too
1½ means 1 + ½. As a single fraction it is 3/2: three halves. The textbook calls all of these fractions (ഭിന്നസംഖ്യകൾ) – whole numbers included, since 2 = 2/1.
2. Squaring a mixed number – two ways
Way 1: turn it into an improper fraction and square top and bottom: (1½)² = (3/2)² = 9/4 = 2¼. Way 2 is the textbook's picture: a square of side 1½ is cut into a 1 × 1 square, two 1 × ½ strips and a ½ × ½ corner.
Both ways always agree. The picture explains why (1 + f)² = 1 + 2f + f²; the improper fraction is quicker for calculating.
3. Decimals are fractions with denominators 10, 100, 1000, …
1.4 is 1 and 4 tenths: 1 4/10. 1.41 is 1 41/100, and 1.414 is 1 414/1000. Every decimal that stops is a fraction – this matters, because the chapter's whole argument is about which lengths are fractions.
4. Squaring a decimal by hand
A calculator is not always allowed. Multiply as if there were no decimal point, then put the point back: the answer has as many decimal places as the two factors together.
The squares the textbook uses (page 24)
| Decimal | Without the point | Product | Decimal places | Square |
|---|---|---|---|---|
| 1.4 | 14 × 14 | 196 | 1 + 1 = 2 | 1.96 |
| 1.5 | 15 × 15 | 225 | 2 | 2.25 |
| 1.41 | 141 × 141 | 19881 | 2 + 2 = 4 | 1.9881 |
| 1.42 | 142 × 142 | 20164 | 4 | 2.0164 |
| 1.414 | 1414 × 1414 | 1999396 | 3 + 3 = 6 | 1.999396 |
| 1.415 | 1415 × 1415 | 2002225 | 6 | 2.002225 |
5. The decimal form of a fraction ദശാംശരൂപം
Dividing the numerator by the denominator gives the decimal form. Some divisions stop (7/10 = 0.7, ¼ = 0.25); others never stop but repeat a block of digits (⅓ = 0.333…, 1/11 = 0.0909…). Watch the remainders: as soon as a remainder comes back, the digits repeat.
A fraction's decimal either stops or repeats – there is no third possibility, because the remainder must eventually repeat (it is always smaller than the denominator). Keep this in mind: the decimal of √2 does neither.
6. Comparing decimals: 1.41, 1.414, 1.42
Which is bigger, 1.41 or 1.414? Compare place by place: both have 1 unit and 4 tenths and 1 hundredth; then 1.41 has 0 thousandths and 1.414 has 4. So 1.41 < 1.414 < 1.42. Zoom in on the number line to see it.
7. Rounding a length to centimetres and millimetres
A length in metres with many decimals is rounded to 2 places for centimetres and 3 places for millimetres. Rule: look at the next digit – if it is 5 or more, add 1 to the last digit you keep (ഏകദേശം = approximately, written ≈).
The textbook's example: a perimeter of 2 + √2 = 3.4142… m is 3.41 m correct to the centimetre and 3.414 m correct to the millimetre.
8. Check yourself
- 1½ = 3/2; (1½)² = 9/4 = 2¼ = 1 + 1 + ¼ (one square, two strips, one corner).
- 1.41 = 1 41/100 – every terminating decimal is a fraction. To square 1.41: 141 × 141 = 19881, four decimal places, 1.9881.
- A fraction's decimal stops or repeats a block: ⅓ = 0.333…, ⅙ = 0.1666…, 1/11 = 0.0909….
- Compare decimals place by place: 1.41 < 1.414 < 1.42. Round to 2 places for cm, 3 for mm; next digit ≥ 5 rounds up.
Next: Right triangles, Pythagoras and the squares on the sides ▶