Summary, Formula Sheet & Quick Quiz പുനരവലോകനം
The seven key boxes of the chapter with pictures, the squeeze tables for √2 and √3, the fan of triangles, the perimeter facts, the appendix proof in five lines, the mistakes to avoid, flashcards, and a scored quiz to find out what needs another look.
1. The key results
No fraction has square 2. 1¼, 1⅓, 1½, … all miss: their squares are 1 9/16, 1 7/9, 2¼ – close to 2, never 2.
The diagonal of the unit square is not a fraction. The square on it has area 2 (four half-squares), so its side would have to square to 2.
Give the length a name: the side of the square of area 2 is written √2. Just as √4 = 2 and √(2¼) = 1½.
The squeeze: 1.4² < 2 < 1.5², 1.41² < 2 < 1.42², 1.414² < 2 < 1.415², … the squares creep up to 2, one decimal place at a time.
√2 = 1.41421… The dots mean the digits go on without end or repetition. √2 ≈ 1.4, √2 ≈ 1.41 – the sign ≈ (ഏകദേശം) means "approximately".
√3 = 1.73205… – the height of the equilateral triangle of side 2 (its square is 4 − 1 = 3). 1.7² = 2.89, 1.8² = 3.24; 1.73² = 2.9929, 1.74² = 3.0276.
For any positive x, √x is the side of the square of area x. Sometimes a natural number or a fraction (√9 = 3, √(1 9/16) = 1¼); otherwise a decimal that never ends.
2. The squeeze tables
√2 = 1.41421…
| places | below | its square | above | its square |
|---|---|---|---|---|
| 0 | 1 | 1 | 2 | 4 |
| 1 | 1.4 | 1.96 | 1.5 | 2.25 |
| 2 | 1.41 | 1.9881 | 1.42 | 2.0164 |
| 3 | 1.414 | 1.999396 | 1.415 | 2.002225 |
| 4 | 1.4142 | 1.99996164 | 1.4143 | 2.00024449 |
| 5 | 1.41421 | 1.9999899241 | 1.41422 | 2.0000182084 |
Side box അടുത്തടുത്ത്: 2 − 1.4² = 0.04, 2 − 1.41² = 0.0119, 2 − 1.414² = 0.000604, 2 − 1.4142² = 0.00003836, 2 − 1.41421² = 0.0000100759.
√3 = 1.73205…
| places | below | its square | above | its square |
|---|---|---|---|---|
| 0 | 1 | 1 | 2 | 4 |
| 1 | 1.7 | 2.89 | 1.8 | 3.24 |
| 2 | 1.73 | 2.9929 | 1.74 | 3.0276 |
| 3 | 1.732 | 2.999824 | 1.733 | 3.003289 |
The same method gives any square root: find the two consecutive whole numbers, then tenths, then hundredths… Each step keeps the one whose square is just below and the one just above.
Handy values: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √6 ≈ 2.449, √7 ≈ 2.646, √10 ≈ 3.162, √11 ≈ 3.317, √12 ≈ 3.464.
3. The fan of triangles and the perimeter facts
The fan (spiral) of right triangles
Each triangle has one side 1; the hypotenuse of each is a side of the next. The nth triangle has sides 1, √n, √(n + 1). Natural-number hypotenuses appear at the 3rd (√4 = 2), 8th (√9 = 3), 15th (√16 = 4) triangles.
Adding and subtracting
| 1, 1, √2 triangle | perimeter 2 + √2 = 3.4142… → 3.41 m (cm), 3.414 m (mm) |
| 1, √2, √3 triangle | perimeter 1 + √2 + √3 = 4.146… (1.414 + 1.732 = 3.146, then + 1) |
| 1, √3, 2 triangle | perimeter 3 + √3 = 4.732… |
| differences | √3 − 1 ≈ 0.732; 2 − √2 ≈ 0.586; √3 − √2 ≈ 0.318 |
| how to add | add the approximations digit by digit: 1.4 + 1.7 = 3.1; 1.41 + 1.73 = 3.14; 1.414 + 1.732 = 3.146 |
| keep √ to the end | (3 + √3) − (1 + √2 + √3) = 2 − √2 – no decimals needed until the last step |
4. Sum and square root · Decimal forms
√x + √y > √(x + y)
Squares of areas x and y corner to corner; the square on the third side of the triangle between them has area x + y. The triangle's sides are √x, √y and √(x + y), and two sides of a triangle together exceed the third. Example: 3 + 4 = 7 > 5 = √25.
Decimal forms ദശാംശരൂപങ്ങൾ
| terminating | 7/10 = 0.7, 21/100 = 0.21, ½ = 0.5, ¼ = 0.25, 1.414 = 1 414/1000 |
| repeating | ⅓ = 0.333…, ⅙ = 0.1666…, 1/11 = 0.0909… (a block of digits repeats forever) |
| never repeating | √2 = 1.41421356…, √3 = 1.7320508… – no block ever repeats; that is what makes them new numbers |
| rounding | look at the next digit: 5 or more → add 1. 3.4142 → 3.41 (cm) / 3.414 (mm); 4.146 → 4.15 |
5. The appendix proof in five lines അനുബന്ധം
- Suppose a fraction has square 2. Write it in lowest terms as x/y – x and y have no common factor.
- Then x²/y² = 2, so x² = 2y²: x² is even. An odd number has an odd square, so x is even.
- x and y have no common factor and x is even, so y is odd (otherwise 2 would divide both).
- Write x = 2z. Then 4z² = 2y², so y² = 2z²: y² is even, so y is even.
- y is odd AND y is even – impossible. So the supposition was wrong: no fraction has square 2. ∎ (The same idea with "multiple of 3" works for √3.)
6. Common mistakes
- Writing √2 = 1.41. It is √2 ≈ 1.41. The decimal is a rounded value; 1.41² = 1.9881, not 2.
- Thinking a longer decimal is exact. 1.41421 is closer, but 1.41421² = 1.9999899241 – still not 2. No decimal ever is.
- Adding under the root. √2 + √3 is NOT √5: 1.414 + 1.732 = 3.146, while √5 ≈ 2.236. In fact √x + √y > √(x + y) always.
- Rounding to the wrong place. "Correct to a millimetre" means 3 decimal places of a metre: 3.414 m, not 3.41 m.
- Calling every square root a new number. √4 = 2, √(2¼) = 1½ and √(1 9/16) = 1¼ ARE fractions. Only when no fraction squares to the number is it new.
- Confusing "square" with "double". The square on the diagonal has twice the area, but the diagonal is √2 ≈ 1.414 times the side, not 2 times.
- Forgetting the half. The height of an equilateral triangle of side 2 uses the half-base 1: height² = 2² − 1² = 3, not 2² − 2².
7. Flashcards പദങ്ങൾ
Click a card to flip it. Shuffle to test yourself in a new order.
8. Scored quiz (10 questions)
Answer all ten, then press Check my score. Each wrong answer tells you which page to revisit.
9. Your study checklist
Pages you have marked as completed (tick the box at the bottom of each page). Aim to complete all of them before the model test.
If you scored 8 or more in the quiz and the checklist is complete, go to the Model Test Paper. Attempt it in 1½ hours without looking at the solutions.
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