Lengths and Numbers നീളങ്ങളും സംഖ്യകളും
Every length you have measured so far was a number you could write: 3 m, 2½ m, 1.75 m – natural numbers or fractions. This section finds a length that is none of these: the diagonal of a square whose sides are 1 metre. It is a real length you can draw with a ruler, yet no fraction, however clever, is exactly right.
- The area of a square is side × side; a square of area 4 sq m has side √4 = 2 m (വർഗം, വർഗമൂലം).
- A mixed number like 1½ is a fraction (ഭിന്നസംഖ്യ), and (1½)² = 1 + 1 + ¼ = 2¼ – the square can be seen as four pieces.
- Pythagoras: the square on the hypotenuse of a right triangle equals the two squares on the perpendicular sides together.
- Any side of a triangle is shorter than the other two sides added together.
1. The square on the diagonal
Draw a square, then draw its diagonal (വികർണ്ണം) and build a second square on that diagonal. How do the two areas compare?
The small square has area 1 × 1 = 1 sq m, and the big square has area 2 sq m. What is the length of its side? It is the diagonal of the small square – longer than 1 m. It is also the third side of a triangle whose other two sides are 1 m and 1 m, so it is shorter than 1 + 1 = 2 m. Somewhere between 1 and 2, then. Which number between 1 and 2 has square 2?
2. Hunting for the fraction
Let us do what the textbook does: try fractions between 1 and 2 and square them. The picture shows why (1½)² = 1 + 1 + ¼: the square of side 1½ is a 1 × 1 square, two 1 × ½ strips and a ½ × ½ corner.
Whatever n and d you choose, the square comes out a little below 2 or a little above 2 – never exactly 2. Some fractions get very close (try 7/17: 1 7/17 squared is 1.99…), but close is not equal.
The textbook's three tries (page 20)
| Fraction | Square, piece by piece | Value | Compared with 2 |
|---|---|---|---|
| 1½ | 1 + ½ + ½ + ¼ = 1 + 1 + ¼ | 2¼ | too big |
| 1¼ | 1 + ¼ + ¼ + 1/16 = 1 + ½ + 1/16 | 1 9/16 | too small |
| 1⅓ | 1 + ⅓ + ⅓ + 1/9 = 1 + ⅔ + 1/9 | 1 7/9 | too small, but closer than 1¼ |
Trying fractions one after another can go on forever. Algebra settles the matter once and for all – that is the appendix of the chapter, page 10 of this guide.
ഒരു ഭിന്നസംഖ്യയുടെയും വർഗം 2 അല്ല.
No fraction has square 2.
വശങ്ങളുടെയെല്ലാം നീളം 1 ആയ സമചതുരത്തിന്റെ വികർണ്ണത്തിന്റെ നീളം ഒരു ഭിന്നസംഖ്യയായി പറയാൻ കഴിയില്ല.
If the diagonal were some fraction of a metre, that fraction would have square 2 (because the side of a square of area 2 must square to 2). No fraction does. So the diagonal of a square with sides 1 cannot be written as a fraction – a length that is not a number we know!
Lay a measuring tape across a square tile from corner to corner. The tape's marks are all fractions of a metre (centimetres, millimetres), and the corner never lands exactly on a mark – it is always a little past 1.41 and a little short of 1.42. Carpenters, tilers and architects simply use a rounded value, 1.414 for example, and that is why the next section learns to find such decimals.
3. Another such length: the height of a triangle
Take an equilateral triangle (സമഭുജത്രികോണം) with sides 2 m and cut it along its height. Is the height a fraction of a metre?
Hunting again – this time for a square of 3
| Try | Square | Compared with 3 |
|---|---|---|
| 1.7 | 2.89 | too small |
| 1.8 | 3.24 | too big |
| 1.73 | 2.9929 | too small |
| 1.74 | 3.0276 | too big |
Just as with 2, no fraction has square 3 (the appendix method works for 3 too). So the height of this triangle is another length that cannot be written as a fraction.
4. And a cube of volume 2
A cube (ഘനം) of side s has volume (വ്യാപ്തം) s × s × s. 1.2³ = 1.728 is too small and 1.3³ = 2.197 is too big; and – you guessed it – no fraction has cube exactly 2. Lengths that are not fractions turn up everywhere once you start looking.
5. Where do numbers come from? സംഖ്യകൾ ഉണ്ടാകുന്നത്
Numbers were invented as people needed them. Counting sheep, days or people needs only the natural numbers 1, 2, 3, … Measuring – land in the Nile valley after the yearly floods (around 3000 BC), cloth, grain – needs parts of a unit, so fractions appeared. Now we have met lengths that even fractions cannot name, so new numbers are needed; that is this chapter. Much later, physics needed negative numbers and even "complex numbers". Every kind of number was born from a problem that the old numbers could not handle.
Nothing is wrong with the length: you can draw the diagonal and cut a stick to match it. What is missing is a number for it. The next section fixes that by giving the length a name, √2, and then finding out how big √2 is.
6. Check yourself
- The square drawn on the diagonal of a square has twice the area (four half-squares).
- For a square of side 1 m the diagonal square has area 2 sq m, so the diagonal is a number between 1 and 2 whose square is 2.
- Trying fractions – 1½, 1¼, 1⅓, … – gives squares close to 2 but never 2. No fraction has square 2, so the diagonal is not a fraction.
- The height of an equilateral triangle of side 2 (its square is 3) and the side of a cube of volume 2 are more such lengths.
- New numbers are needed to name these lengths – that is the job of Section 2.
Next: Section 2 – naming the length √2 and finding its decimals ▶