Why No Fraction Squares to 2 അനുബന്ധം
Trying fractions forever can never prove that none of them squares to 2 – the next one might work! This page gives the textbook's proof: suppose such a fraction exists, follow the consequences, and arrive at something impossible. Then the supposition must have been false.
- Odd squares are odd. (2n + 1)² = 4n² + 4n + 1 = 2(2n² + 2n) + 1. So if x² is even (ഇരട്ടസംഖ്യ), x cannot be odd: x must be even.
- Lowest terms. Every fraction can be written in its simplest form, where the numerator and denominator have no common factor (പൊതുഘടകം). In that form they cannot both be even, because then 2 would be a common factor.
1. The proof, one step at a time
ഇതിൽ നിന്ന് ഒരു ഭിന്നസംഖ്യയുടെയും വർഗം 2 അല്ലെന്ന് മനസ്സിലാക്കാം.
We found that the denominator y is odd, and then that it is even. Both cannot be true. The only thing we assumed was that a fraction with square 2 exists – so that assumption is false. No fraction has square 2.
If a fraction in its simplest form had square 2, its denominator would have to be odd. Thinking a little further, the same denominator would have to be even. These cannot both happen. This style of argument – suppose the opposite and reach an impossibility – is called proof by contradiction.
2. Try to beat the proof
Try any x and y. The two sides never match. Notice also: whenever x² is even, x itself is even – the first tool in action. Good near misses: 7/5 (49 vs 50), 17/12 (289 vs 288), 41/29 (1681 vs 1682), 99/70 (9801 vs 9800).
3. The same idea for 3 beyond the textbook
Section 1 said no fraction squares to 3 either. The proof is the same with "multiple of 3" in place of "even".
About 2500 years ago the Pythagoreans believed every length was a ratio of whole numbers. This little argument destroyed that belief. Numbers like √2 are now called irrational numbers (അഭിന്നകസംഖ്യകൾ); you will name and multiply them in Chapter 5.
4. Check yourself
- Suppose x/y (lowest terms) squares to 2. Then x² = 2y², so x is even, so y is odd.
- Write x = 2z: then y² = 2z², so y is even too – impossible.
- Therefore no fraction has square 2; the same argument works for 3.