Chapter 5 · Section 1Textbook pages 83–89

Multiplication ഗുണനം

In the chapter പുതിയ സംഖ്യകൾ you met numbers like √2 that cannot be written as fractions – irrational numbers (അഭിന്നകസംഖ്യകൾ). Joining lengths taught you to add them. Now areas will teach you to multiply them, and the big discovery is a very short rule: √2 × √3 = √6.

📖 What you already know (from pages 1–4 of this guide)

√x is the positive number whose square is x; √4 = 2, √9 = 3, but √2 = 1.414… never ends or repeats.
Pythagoras: in a right triangle, (hypotenuse)² = (leg)² + (leg)². Legs 1 and 1 give hypotenuse √2.
Decimals of √2: 1.4, 1.41, 1.414, 1.4142, … each one closer. Rounding rule: if the next digit is 5 or more, add 1.
Area of a rectangle = length × breadth (when the sides are ordinary numbers).

1. Four times √2 – where multiplication begins

Take a square of side 1 m. Its diagonal is √2 m (Pythagoras). Now draw a bigger square on that diagonal. What is its perimeter?

🎬 Animation: perimeter of the square on the diagonal

⭐ Writing without the × sign (page 84)

Four times √2 may be written 4 × √2 or √2 × 4, but usually we drop the sign and write 4√2:

4√2 = √2 + √2 + √2 + √2

In the same way, half of √2 is written ½√2, and ½√2 = 0.7071… (half of 1.4142…).

🎚️ Decimal lab: how many digits do you need?

Decimal places of √2

√2 = 1.41421356… To get the perimeter correct to the millimetre we only need 3 decimal places: 4 × 1.414 = 5.656 m. More digits make the answer more exact, but the exact value is simply 4√2.

🌍 Real life

A carpenter cutting a square tile of side 30 cm along its diagonal gets an edge of 30√2 ≈ 42.4 cm. A surveyor walking diagonally across a 100 m square field walks 100√2 ≈ 141.4 m. Engineers keep the √ in their formulas and only turn it into a decimal at the very end – exactly what you will learn here.

2. A rectangle from two triangles: area 2 × √3

Here is the key experiment of the section. Two equilateral triangles of side 2 m are rearranged into a rectangle. One side is 2 m, the other is the height of the triangle, √3 m. Is its area really 2 × √3?

🎬 Animation: cut, rearrange, and squeeze the area

💡 The squeeze idea

We cannot "count" the area of a 2 × √3 rectangle directly because √3 has no last digit. But rectangles 2 × 1.7, 2 × 1.73, 2 × 1.732, … fit inside it and get closer and closer. Their areas 3.4, 3.46, 3.464, … get closer and closer to the true area. So the true area is the number these approach – and that is what 2√3 means: 2 × 1.732… = 3.464…

3. The big question: √2 × √3 = ?

Now make a rectangle with sides √2 m and √3 m. Its area is written √2 × √3. Let us find its decimal value, and then discover something surprising.

🎚️ Multiply the approximations

Decimal places

Move the slider. The products 2.4, 2.44, 2.449, … approach √2 × √3. Now look at the last column: the squares of these products approach 6. A number whose square is 6 is √6. So √2 × √3 = √6 = 2.449…

📖 Side box: decimal rounding (ദശാംശക്കണക്ക്, page 86)

When we shorten a decimal to some number of places, we look at the next digit: if it is 5 or more we add 1 to the last digit we keep. 1.7 × 1.4 = 2.38, so to one decimal place the product is written 2.4. Likewise 1.41 × 1.73 = 2.4393 → 2.44.

💡 Why the squares head for 6 (side box ഗുണനവും വർഗമൂലവും, page 87)

(1.4 × 1.7)² = 1.4² × 1.7². As we add digits, 1.4², 1.41², 1.414², … approach 2 and 1.7², 1.73², 1.732², … approach 3. So the squares of the products approach 2 × 3 = 6, which means the products approach √6.

4. The rule: √x × √y = √(xy)

For perfect squares this was never in doubt: √4 × √25 = 2 × 5 = 10 and √(4 × 25) = √100 = 10. With fractions too: √¼ × √(9/25) = ½ × ⅗ = 3/10 = √(9/100). The experiment above shows it is also true for irrational square roots.

⭐ Key idea 1 (textbook page 87)

x, y എന്ന ഏതു രണ്ട് അധിസംഖ്യകളെടുത്താലും

√x × √y = √(xy)

For any two positive numbers x and y, the product of their square roots is the square root of their product.

🧪 Test the rule with any numbers

x y

Whatever x and y you choose, the two columns agree to every decimal place shown. When xy happens to be a perfect square (try x = 2, y = 8), the product of two irrational numbers is a whole number!

5. Simplifying square roots: √18 = 3√2

The rule also lets us write square roots in a simpler form. A right triangle with both legs 3 cm has hypotenuse √(9 + 9) = √18 cm. But 18 = 9 × 2, so √18 = √9 × √2 = 3√2. Here is the geometric picture from the textbook.

🎬 Animation: the square of area 18 is nine squares of area 2

🎚️ Simplify √n yourself

Look for the largest perfect square that divides n (4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, …). Pull its square root outside: √(k² × m) = k√m. If n itself is a perfect square the answer is a whole number; if no perfect square divides n, √n is already as simple as it gets.

🌍 Why simplify at all?

Compare √18 + √8 with 3√2 + 2√2. The first looks hopeless; the second is obviously 5√2. Simplified square roots are like terms you can add, compare and cancel. Builders, electricians and physicists use forms like 5√2 and 2√3 all the time – they are exact, short, and easy to combine.

6. Check yourself

🧭 Summary of Section 1

4√2 means √2 + √2 + √2 + √2; ½√2 is half of √2. Decimals: 4√2 ≈ 5.656, ½√2 ≈ 0.7071.
Rectangles with irrational sides have area = product of the sides, found by squeezing with decimals.
√x × √y = √(xy) for all positive x, y – e.g. √2 × √3 = √6, √5 × √20 = √100 = 10.
Simplify by pulling out perfect squares: √18 = √(9 × 2) = 3√2, √50 = 5√2, √108 = 6√3.

Next: Exercise 1 – all six questions solved with animations ▶