Squares, Square Roots & Perfect Squares വർഗവും വർഗമൂലവും
This chapter asks one question over and over: "a square has this much area – how long is its side?" To answer it you need to be fluent with squaring (വർഗം), with square roots (വർഗമൂലം) and with the perfect squares (പൂർണവർഗം) 1, 4, 9, 16, … Here is everything you need, with pictures.
- Multiplying a number by itself: 3 × 3 = 9, 7 × 7 = 49.
- The area (പരപ്പളവ്) of a rectangle is length × breadth; a square is a rectangle with equal sides.
- Fractions such as 3/2 and mixed numbers such as 1½ (page 2 revises them in detail).
1. Squaring is drawing a square
"n squared", written n², means n × n – and it is exactly the area of a square whose side is n. Count the little squares.
Move the slider: n rows of n unit squares make n × n = n² squares. So 4² = 16 is not just a number – it is a 4 × 4 square.
2. The perfect squares പൂർണവർഗങ്ങൾ
A number that is the square of a natural number is called a perfect square. Click any one to see it as a square. Learn these fifteen by heart – they appear in every problem of the chapter.
3. The reverse question: the square root വർഗമൂലം
Squaring goes from side to area. The square root goes back from area to side: "a square has area 16 – what is its side?" The answer is the number that squares to 16, and it is written with the root sign: √16 = 4.
For any number A that is an area, √A (read "the square root of A", A യുടെ വർഗമൂലം) is the side of the square whose area is A. So √A × √A = A, always.
√1 = 1, √4 = 2, √9 = 3, √16 = 4, √25 = 5, … √100 = 10, √144 = 12, √225 = 15.
4. Square roots of fractions and decimals
If the numerator and the denominator of a fraction are both perfect squares, its square root is easy: √(9/4) = 3/2, because (3/2) × (3/2) = 9/4. In mixed numbers that reads √(2¼) = 1½ – a fact the textbook uses on page 22.
Decimals work the same way once you write them as fractions: 2.25 = 225/100 = (15/10)², so √2.25 = 1.5; 0.25 = 25/100 = (5/10)², so √0.25 = 0.5.
5. When the area is not a perfect square
What about √10? There is no natural number whose square is 10 – but we can say between which two it lies: 3² = 9 is less than 10 and 4² = 16 is more, so the side of a square of area 10 is between 3 and 4. Nesting the squares shows it.
The green square (area x) fits between the amber square (area n²) and the blue square (area (n + 1)²), so its side √x is between n and n + 1. Chapter 2 pushes this idea further: tenths, hundredths, thousandths…
6. Mini-game: which side gives this area?
7. Check yourself
- n² = n × n is the area of a square of side n. The perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
- √A is the side of the square of area A: √A × √A = A. √49 = 7, √(9/4) = 3/2 = 1½, √(2¼) = 1½, √2.25 = 1.5.
- If x is between n² and (n + 1)², then √x is between n and n + 1: 3 < √10 < 4.
Next: Fractions, mixed numbers and decimals – and how to square them ▶