Measures and Numbers അളവുകളും സംഖ്യകളും
Section 1 left us with a length – the side of a square of area 2 – that no fraction can name. A length we can draw but cannot write is a problem. This section solves it in two moves: first we give the length a name, √2; then we find out how big √2 is by squeezing it between decimals that get closer and closer.
- A square of area 4 has side √4 = 2; a square of area 2¼ has side √(2¼) = 1½ (വർഗമൂലം).
- Decimals are fractions: 1.4 = 1 4/10, 1.41 = 1 41/100 (ദശാംശരൂപം); 1.4² = 1.96 and 1.41² = 1.9881 by hand.
- No fraction has square 2, so the side of a square of area 2 is a new kind of number (Section 1).
1. First, a name
For squares whose sides are old numbers we already have a way of writing the side from the area. We simply use the same way for the new length.
പരപ്പളവ് 2 ചതുരശ്രമീറ്റർ ആയ സമചതുരത്തിന്റെ വശത്തിന്റെ നീളം √2 മീറ്റർ എന്നെഴുതാം.
The side of a square of area 2 sq m is written √2 m. Read it "root two". It means exactly one thing: the length whose square is 2.
2. A name is not enough – how big is √2?
If somebody says a rope is √2 m long, you want to know roughly how long that is. The textbook's idea: draw squares with ordinary sides inside the √2 square and see how close they come.
1.1, 1.2, 1.3 and 1.4 fit inside the √2 square (their squares are less than 2); 1.5 sticks out (2.25 is more than 2). So √2 is between 1.4 and 1.5.
With a calculator (textbook page 23)
| side | side² | compared with 2 |
|---|---|---|
| 1.1 | 1.21 | less |
| 1.2 | 1.44 | less |
| 1.3 | 1.69 | less |
| 1.4 | 1.96 | less – the last one below 2 |
| 1.5 | 2.25 | more – the first one above 2 |
So 1.4² < 2 < 1.5², which means 1.4 < √2 < 1.5. Now do the same between 1.4 and 1.5 with hundredths, then thousandths…
3. The squeeze അടുത്തടുത്ത്
The whole squeeze on one page (textbook page 24)
| below √2 | its square | above √2 | its square | |
|---|---|---|---|---|
| 1.4 = 1 4/10 | 1.96 | < 2 < | 1.5 | 2.25 |
| 1.41 = 1 41/100 | 1.9881 | < 2 < | 1.42 | 2.0164 |
| 1.414 = 1 414/1000 | 1.999396 | < 2 < | 1.415 | 2.002225 |
| 1.4142 = 1 4142/10000 | 1.99996164 | < 2 < | 1.4143 | 2.00024449 |
| 1.41421 = 1 41421/100000 | 1.9999899241 | < 2 < | 1.41422 | 2.0000182084 |
1.4, 1.41, 1.414, 1.4142, 1.41421 എന്നിങ്ങനെ തുടരുന്ന ഭിന്നസംഖ്യകളുടെ വർഗങ്ങൾ 2 നോട് അടുത്തടുത്തു വരുന്നു.
The squares of the fractions 1.4, 1.41, 1.414, 1.4142, 1.41421, … come closer and closer to 2. None of them is √2 – but they creep up on it, one decimal place at a time, and the process never has to stop.
The dots say "the digits go on, and we can find as many as we like". When we stop after a few digits we write ≈ (ഏകദേശം തുല്യം, approximately equal): √2 ≈ 1.4 to one decimal place, √2 ≈ 1.41 to two, √2 ≈ 1.414 to three.
2 − 1.4² = 0.04, 2 − 1.41² = 0.0119, 2 − 1.414² = 0.000604, 2 − 1.4142² = 0.00003836, 2 − 1.41421² = 0.0000100759. Drawn to the same scale, the later bars are too thin to see – magnify them by 10, 100, 1000, 10 000 times and they appear one by one. Each extra decimal place shrinks the gap by roughly a hundred times.
4. Squeeze any square root yourself
The lower number is the largest decimal with that many places whose square is still below x; the upper one is the next decimal. For a perfect square like x = 9 or 16 the squeeze stops: √x is a natural number.
5. The same for √3
The height of the triangle with sides 2 m (page 5) is the side of a square of area 3, so we write it √3 m. Squeezing works exactly as before.
| below √3 | its square | above √3 | its square | |
|---|---|---|---|---|
| 1.7 | 2.89 | < 3 < | 1.8 | 3.24 |
| 1.73 | 2.9929 | < 3 < | 1.74 | 3.0276 |
| 1.732 | 2.999824 | < 3 < | 1.733 | 3.003289 |
Try x = 3 with 4, 5 and 6 decimal places in the lab above to continue the table.
6. √x for any positive number x
x ഏത് അധിസംഖ്യ ആയാലും, പരപ്പളവ് x ആയ സമചതുരത്തിന്റെ വശത്തിന്റെ നീളമാണ് √x.
For any positive number x, √x is the length of the side of a square of area x. Sometimes √x is a natural number or a fraction (√4 = 2, √(2¼) = 1½); otherwise we find fractions whose squares approach x and write √x as a decimal that never ends.
Slide through the quarters: 2.25 = (1½)² and 6.25 = (2½)² give fractions, 4, 9, 16, 25, 36, 49 give natural numbers, and every other value gives a new number with an endless decimal.
7. Shattered beliefs തകരുന്ന വിശ്വാസങ്ങൾ
About 2500 years ago the followers of Pythagoras believed that any two lengths can be compared using natural numbers – that one length is always "a : b" of another. Then came the diagonal of a square. If diagonal : side were a : b, the square on the diagonal would be (a/b)² times the square on the side; but it is exactly twice, so (a/b)² = 2 – and we know no fraction can do that. Legend says Hippasus, who found this, was thrown overboard by the others for destroying their belief. Pairs of lengths like the side and diagonal of a square are called incommensurable magnitudes: no unit, however small, measures both exactly.
A calculator shows √2 as 1.4142135 and stops – not because the digits end, but because the screen does. A tailor cutting a 1 m square scarf along its diagonal needs √2 to the centimetre: 1.41 m. A machinist milling a steel plate needs it to the millimetre or finer: 1.414 or 1.4142 m. A surveyor laying out a 100 m square plot uses 100√2 ≈ 141.42 m. Everybody uses an approximation good enough for the job; nobody ever uses – or needs – all the digits.
Draw a right triangle with 1 cm sides in GeoGebra, build squares on its sides with the Regular Polygon tool and read their areas with the Area tool. Which of the sides are fractions? Only the two 1 cm ones – the third is √2.
8. Check yourself
- √x means "the side of a square of area x", for any positive x. √4 = 2, √(2¼) = 1½, and √2 is the new length of Section 1.
- To find how big √2 is, squeeze it: 1.4² < 2 < 1.5², 1.41² < 2 < 1.42², 1.414² < 2 < 1.415², … The squares creep towards 2.
- √2 = 1.41421… and √3 = 1.73205…; the dots mean the digits never end. √2 ≈ 1.41 means "approximately".
- Each extra decimal place shrinks the gap between the square and 2 about a hundred times.
Next: Practice set 1 – squares of area 7, 10 and 11, the fan of triangles, fractions between √2 and √3 ▶