Real-World Applications പ്രയോഗങ്ങൾ
Numbers like √2 and √3 are not a classroom curiosity. They appear every time a square is measured corner to corner, every time a sheet of paper is folded in half, in every screen size and every ladder. This page shows where they turn up, how much of their decimal is really needed, and why a calculator can never show the whole number.
- The diagonal of a square of side 1 is √2, the side of the square of area 2; it is not a fraction (page 5).
- √2 = 1.41421…, √3 = 1.73205… – found by squeezing with decimals, one place at a time (page 6).
- √x is the side of the square of area x; the hypotenuse of a right triangle with perpendicular sides a and b is √(a² + b²) (page 6, page 7).
- Adding and subtracting such lengths: 2 + √2 = 3.4142…, and we round to the precision the job needs (page 8).
1. The diagonal of a square room
A square room with sides 3 m: how far is it from corner to corner? The diagonal is 3 × √2 = 4.2426… m. A tape measure shows centimetres and millimetres – fractions of a metre – and the far corner never sits exactly on a mark. Everyone in the building trade simply reads 4.24 m or 4.243 m, depending on how precise the job must be.
The diagonal is always side × √2 – the square on the diagonal has twice the area of the room, so its side is √2 times longer. Whatever the side, the decimal never ends.
2. The two set squares in your geometry box
One set square has angles 45°, 45°, 90°: its two short sides are equal, so its sides are in the ratio 1 : 1 : √2. The other has angles 30°, 60°, 90°: it is half an equilateral triangle, so its sides are 1 : √3 : 2. Measure yours – the long sides are never a "nice" number of centimetres.
3. A4 paper: the √2 rectangle
An A4 sheet is 210 mm by 297 mm, and 297 ÷ 210 = 1.4142… = √2. That is no accident: a rectangle whose sides are in the ratio 1 : √2 keeps the same shape when folded in half – the half sheet (A5) is again 1 : √2. Any other ratio would change shape at every fold.
4. Screen sizes: the diagonal is the number they sell
A "55-inch" television is 55 inches corner to corner. Its width and height are in the ratio 16 : 9, so the diagonal of a 16 × 9 rectangle is √(16² + 9²) = √337 = 18.357… units. Width = 55 × 16 ÷ 18.357 and height = 55 × 9 ÷ 18.357 – both involve √337, so they are never exact fractions of an inch.
5. A ladder against a wall
The foot of a ladder is a metres from the wall and its top is b metres up. The ladder is the hypotenuse: √(a² + b²). Sometimes that is a fraction (3, 4 → 5), usually it is not (1, 1 → √2; 2, 3 → √13).
Try 3 and 4, 1.5 and 2, 0.5 and 1.5: when a² + b² is a perfect square the ladder's length is a fraction; otherwise it is a new number like √13.
6. How many decimal places does the job need?
√2 = 1.41421356… Nobody uses all of it. A tailor marking cloth works to the centimetre, a carpenter to the millimetre, a machinist to a hundredth of a millimetre. Each needs a different number of decimal places – and each stops as soon as the error is smaller than what the tools can measure.
The error is the part of √2 that was thrown away. With 2 places (1.41) a 1 m diagonal is short by about 4 mm – fine for a tailor, not for a carpenter, who keeps 3 places (1.414, short by 0.2 mm).
7. What your calculator is hiding
Press √2 on a calculator and you see 1.4142135 or 1.414213562. That is a rounded value – the display is simply not wide enough for a number whose digits never end. Square the displayed number and you never get back exactly 2.
8. The square-root spiral
Right triangles with one side 1, each built on the hypotenuse of the last, give hypotenuses √2, √3, √4, √5, … They curl into a spiral that designers, jewellers and architects use as a motif, and that looks remarkably like a snail shell.
The nth triangle has sides 1, √n and √(n + 1). Only √4 = 2, √9 = 3 and √16 = 4 are natural numbers; all the others are new numbers.
9. Doubling the cube: the Delian problem
An ancient Greek story says the citizens of Delos were told to double the volume of their cubical altar. Doubling the side gives eight times the volume; the side they needed was ∛2 ≈ 1.26 times the old one – a number that, like √2, is not a fraction, and which cannot even be drawn with ruler and compass.
10. Fencing a triangular plot
Two plots: one is half a 30 m square cut along its diagonal (sides 30 m, 30 m, 30√2 m); the other has sides 30 m, 40 m, 50 m. Fencing is sold by the metre, so the first perimeter must be rounded before the bill can be written.
The followers of Pythagoras (Greece, about 2500 years ago) believed that any two lengths could be compared with whole numbers – that every ratio of lengths was a ratio a : b of natural numbers. The diagonal and the side of a square broke that belief: if diagonal : side = a : b then (a/b)² = 2, which is impossible. The story goes that Hippasus, who revealed this, was thrown into the sea. Today such pairs of lengths are called incommensurable, and the numbers that measure them – √2, √3, π – are the "new numbers" of this chapter.
11. Try these word problems
- Square roots appear wherever a diagonal, a slope or a fold is involved: rooms, tiles, paper sizes, screens, ladders.
- They are exact numbers (√2 is exactly the diagonal of the unit square) but their decimals never end, so every measurement uses a rounded value.
- Keep as many decimal places as the tool can measure – and remember that the calculator's display is a rounded value too.