Rational vs Irrational Numbers അഭിന്നകസംഖ്യകൾ
Every number you used until Class 8 could be written as a fraction. Then you met √2 and discovered a number that cannot. This page makes that discovery precise: what "rational" and "irrational" mean, why √2 is irrational, and how irrational numbers still sit at exact points on the number line.
1. Rational numbers: the fractions ഭിന്നസംഖ്യകൾ
A number is rational if it can be written as p/q with p and q whole numbers (q not zero). 3 = 3/1, 0.75 = 3/4, −2½ = −5/2, 0.333… = 1/3 are all rational. When you divide p by q the decimal either stops (3/8 = 0.375) or repeats a block forever (2/7 = 0.285714 285714…). This is not an accident – watch the remainders.
When dividing by q the remainder is always one of 0, 1, 2, …, q − 1. With only q possibilities, a remainder must eventually repeat (or become 0) – and from then on the digits repeat too. Every fraction has a decimal that stops or recycles.
2. Irrational numbers: decimals that never stop and never repeat
An irrational number (അഭിന്നകസംഖ്യ) is one that cannot be written as p/q. Its decimal goes on forever without any repeating block. √2, √3, √5, √10 and π are all irrational. Look as far as you like – no block ever repeats.
A fraction's digits must recycle within q digits. These digits have been computed to millions of places and never settle into a repeating pattern – because, as we prove below, no fraction equals √2.
3. Where √2 comes from: the diagonal of a square
√2 is simply "the length of the diagonal of a unit square" – a perfectly real length. Its square is exactly 2. Its decimal is 1.41421356…, found by squeezing (page 1).
4. Why √2 is NOT a fraction – the famous proof
Could some fraction, perhaps with huge numbers on top and bottom, equal √2 exactly? The Greeks showed 2500 years ago that the answer is no. The argument only uses the fact that an even number squared is even and an odd number squared is odd.
We assumed √2 = p/q in lowest terms and were forced to conclude that p and q are both even – which contradicts "lowest terms". An assumption that leads to nonsense must be false. So no fraction equals √2: it is irrational. The same idea works for √3, √5, √6, … – every √n where n is not a perfect square.
5. Not every square root is irrational
√4 = 2, √9 = 3, √(1/4) = ½, √0.25 = 0.5, √(16/25) = 4/5 are rational – they are square roots of perfect squares. And some tricky-looking expressions are rational too: √50 ÷ √2 = √25 = 5 and √2 × √8 = √16 = 4 (you will learn these rules in this chapter). The number π = 3.14159… is irrational as well, though it is not a square root at all.
6. Irrational numbers live on the number line too
A number with a never-ending decimal still has an exact home on the number line. With a ruler and compass you can mark √2, √3 and √5 exactly, using right triangles.
Each new triangle adds a leg of length 1 at right angles to the previous hypotenuse, so the hypotenuses are √2, √3, √4, √5, … Every square root is an exact length you can draw.
Why do we care? Because measurements like the diagonal of a tile, the distance across a field, or the side of a square plot of 50 square metres are irrational – no measuring tape ever gives them exactly. Engineers write them as √2, √50 = 5√2, and only round off at the very end. That is the skill this chapter teaches.
7. Check yourself
- Rational = can be written p/q; decimal stops or repeats (because remainders must repeat).
- Irrational = cannot be written p/q; decimal never stops and never repeats: √2, √3, √5, π.
- √2 is the diagonal of a unit square; the parity argument proves it is not a fraction.
- √(perfect square) is rational: √9 = 3, √(1/4) = ½, √0.25 = 0.5. Everything else under a √ is irrational.
- Irrational numbers are exact points on the number line, constructible with right triangles.