Square Roots & Perfect Squares വർഗമൂലം
The whole chapter is about numbers like √2, √3 and √18. Before we can multiply or divide them we must be completely sure what the √ sign means, which square roots are ordinary numbers, and how to find the decimal value of the others. Take your time with every lab on this page.
1. Squaring a number വർഗം
To square a number means to multiply it by itself: 3² = 3 × 3 = 9, 7² = 49, (½)² = ¼, (1.5)² = 2.25. The little 2 is read "squared". The name comes from a real square: a square with side 3 has 3 × 3 = 9 unit squares inside it.
The numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, … are called perfect squares (പൂർണവർഗം) because that many dots can be arranged in a perfect square.
2. Square root: squaring in reverse വർഗമൂലം
The square root of a number x, written √x, is the positive number whose square is x. Because 5² = 25, √25 = 5. Because 12² = 144, √144 = 12. In general √(a²) = a for any positive a – the root undoes the square.
Green cells are perfect squares – their roots are whole numbers. All the others are not whole numbers, and (as the next page proves) not fractions either: their decimals go on forever.
A square room has an area of 36 square metres – how long is each wall? √36 = 6 m. A square plot has 250 square metres – each side is √250 ≈ 15.8 m. Square roots appear whenever you know an area and want a length.
3. Finding √2 by squeezing
√2 is the number whose square is 2. There is no whole number like that (1² = 1 is too small, 2² = 4 is too big), so we trap √2 between numbers that are closer and closer – a method you can do with nothing but multiplication.
Find two neighbouring numbers whose squares are just below and just above 2. Then add one more decimal place and repeat. Each round gives one more correct digit of √2 = 1.41421356… – and the digits never stop.
Pick n and increase the decimal places. The two trapping numbers close in on √n, and their squares close in on n from both sides. If n is a perfect square the trap closes completely.
4. Square roots of fractions and decimals
If the top and bottom of a fraction are both perfect squares, the root is easy: (3/5)² = 9/25, so √(9/25) = 3/5. Decimals work the same way once you write them as fractions over 100: 0.25 = 25/100, so √0.25 = 5/10 = 0.5.
5. Simplifying square roots: √18 = 3√2
√18 is not a whole number, but 18 = 9 × 2 and 9 is a perfect square. Taking the root of the 9 gives √18 = 3√2 (this is proved properly in Section 1 of the chapter; here we learn the skill). The trick is always the same: find the largest perfect square that divides the number.
Test 4, 9, 16, 25, 36, 49, 64, 81, 100, … in turn. Keep the largest one that divides exactly. √(k² × m) = k√m. A number whose largest square factor is 1 (like 2, 3, 5, 6, 7, 10) cannot be simplified at all.
6. Mini-game: is it a perfect square?
7. Check yourself
- n² = n × n. Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, …
- √x is the positive number whose square is x; √(a²) = a, so √49 = 7 and √(9/25) = 3/5.
- Other square roots are found by squeezing: 1.4 < √2 < 1.5, 1.41 < √2 < 1.42, … giving √2 = 1.41421…
- Simplify by pulling out the largest perfect-square factor: √18 = 3√2, √50 = 5√2, √108 = 6√3.