Odd & Even Numbers, Lowest Terms, Square Identities ഒറ്റയും ഇരട്ടയും, വർഗസമവാക്യങ്ങളും
The chapter ends with a famous proof: no fraction has square 2. The proof uses only four small facts – what odd and even numbers look like in algebra, that the square of an odd number is odd, that a fraction can be written in lowest terms, and the habit of "supposing the opposite". This page also revises the Class 8 square identities that two of the practice questions need.
- Even numbers 2, 4, 6, … divide by 2 exactly; odd numbers 1, 3, 5, … leave remainder 1.
- Expanding brackets: a(b + c) = ab + ac; squaring means multiplying by itself.
- Equivalent fractions: 2/4 = 1/2 (page 2).
1. Even is 2n, odd is 2n + 1
Every even number (ഇരട്ടസംഖ്യ) is 2 × something: 2n. Every odd number (ഒറ്റസംഖ്യ) is one more than an even number: 2n + 1. Dots in pairs show it.
2. The square of an odd number is odd; of an even number, even
| × | even | odd |
|---|---|---|
| even | even (2m × 2n = 4mn) | even (2m × (2n + 1) = 2(2mn + m)) |
| odd | even | odd |
Only odd × odd gives odd. In particular a square copies the parity of its root: 3² = 9, 5² = 25, 7² = 49 are odd; 2² = 4, 6² = 36 are even.
Turn the fact around. Suppose a natural number x has an even square. Could x be odd? No – an odd x would have an odd square. So if x² is even, then x itself is even. (Likewise: if x² is odd, x is odd.) Remember this sentence; the whole proof on page 10 turns on it.
3. Fractions in lowest terms ലഘുരൂപം
6/4 and 3/2 are the same number; 3/2 is the lowest-terms form – numerator and denominator have no common factor (പൊതുഘടകം) other than 1. Every fraction has exactly one such form, found by dividing out the highest common factor.
Important for the proof: in lowest terms the numerator and the denominator can never both be even – if they were, 2 would be a common factor and you could reduce further.
4. The square identities (Class 8)
And (a − b)² = a² − 2ab + b²
Inside a square of side a, the square of side a − b (green) is what remains after removing two a × b strips – but the b × b corner was removed twice, so add it back once: (a − b)² = a² − ab − ab + b² = a² − 2ab + b². For example 9² = (10 − 1)² = 100 − 20 + 1 = 81.
5. Every odd number is a difference of two squares
Go from an n × n square to an (n + 1) × (n + 1) square: you add an L-shaped border of n + n + 1 = 2n + 1 unit squares. So (n + 1)² − n² = 2n + 1 – every odd number is the difference of two consecutive squares. Practice set 1 uses 7 = 4² − 3² and 11 = 6² − 5².
6. The identity (x + 1)² − (x − 1)² = 4x
Expand both squares: (x² + 2x + 1) − (x² − 2x + 1) = 4x. The textbook (page 26) divides by 4 and writes x = (½(x + 1))² − (½(x − 1))² – which says: x is the difference of two squares, so a right triangle with hypotenuse ½(x + 1) and one side ½(x − 1) has third side √x. Page 7 of this guide turns that into a drawing.
7. Proving by supposing the opposite
Some facts are proved by assuming they are false and showing that leads to nonsense. A tiny example before the big one on page 10:
8. Check yourself
- Even = 2n, odd = 2n + 1; (2n + 1)² = 2(2n² + 2n) + 1 is odd, (2n)² = 4n² is even. So if x² is even, x is even.
- Every fraction has a lowest-terms form; in it, numerator and denominator are not both even.
- (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², (a + b)(a − b) = a² − b².
- (n + 1)² − n² = 2n + 1 (every odd number); (x + 1)² − (x − 1)² = 4x.
- To prove something is impossible: suppose it happens, and show that leads to a contradiction.