Chapter 2 · Section 3Textbook pages 26–29

Adding and Subtracting കൂട്ടലും കുറയ്ക്കലും

We can now name lengths like √2 and √3 and find their decimals. But a perimeter is a sum of lengths. What is 1 + 1 + √2? How much longer is one triangle's perimeter than another's? This section learns to add and subtract the new numbers – exactly, by keeping the √ signs, and approximately, by adding decimals digit by digit.

📖 What you already know (from pages 3, 5 and 6 of this guide)

Pythagoras: square on the hypotenuse (കർണ്ണം) = squares on the two perpendicular sides (ലംബവശങ്ങൾ) together.
√x is the side of a square of area x; √2 = 1.41421… and √3 = 1.73205… (the digits never end).
1.4, 1.41, 1.414, … are fractions whose squares come closer and closer to 2 – they are approximations (ഏകദേശം) of √2.
Rounding a length in metres: 2 decimal places give centimetres, 3 give millimetres.

1. The first triangle: area and perimeter

Start with the simplest right triangle of all – both perpendicular sides 1 metre.

🎬 Animation: area ½, perimeter 2 + √2

⭐ Writing a sum exactly (textbook page 27)

The perimeter is 2 + √2 m. This is an exact answer – nothing is rounded. When we need a decimal, we use the approximations of √2: 1.4, 1.41, 1.414, 1.4142, … and add 2 to each:

3.4, 3.41, 3.414, 3.4142, … so 2 + √2 = 3.4142…

2. How exact do you need it?

🎚️ Choose the decimal places – which unit is that good for?

Decimal places of √2

Correct to a centimetre the perimeter is 3.41 m; correct to a millimetre it is 3.414 m. A tailor is happy with centimetres, a carpenter wants millimetres, a machinist may want more – but nobody ever needs all the digits, which is lucky, because they never end.

3. A second triangle on the hypotenuse: 1 + √2 + √3

Use the hypotenuse √2 of the first triangle as a perpendicular side of a new right triangle, with the other perpendicular side 1 m (textbook figure, page 27).

🎬 Animation: the new hypotenuse √3, and adding decimals column by column

⭐ Adding the new numbers (textbook page 28)

To add √2 and √3, add their approximations with the same number of decimal places:

1.4 + 1.7 = 3.1 1.41 + 1.73 = 3.14 1.414 + 1.732 = 3.146 so √2 + √3 = 3.146…

Adding 1 gives 1 + √2 + √3 = 4.146… – the second perimeter is 4.146 m correct to a millimetre.

4. How much longer? Subtract first, then round

🎬 Animation: (1 + √2 + √3) − (2 + √2) = √3 − 1

💡 Two ways, same answer

By decimals: 4.146 − 3.414 = 0.732. Exactly: (1 + √2 + √3) − (2 + √2) = 1 + √2 + √3 − 2 − √2 = √3 − 1, and √3 − 1 ≈ 1.732 − 1 = 0.732 m. The exact way is shorter and makes no rounding mistakes on the way.

5. A third triangle: 2 − √2 without computing anything

🎬 Animation: hypotenuse exactly 2, and the difference 2 − √2

⭐ Subtracting the new numbers (textbook page 29)

Third perimeter − second perimeter = (3 + √3) − (1 + √2 + √3) = 3 + √3 − 1 − √2 − √3 = 2 − √2 ≈ 2 − 1.414 = 0.586 m, that is about 58.6 cm. We never needed the two perimeters themselves: the √3 simply cancels.

6. Build the fan yourself

🎚️ n right triangles, each with a new side of 1 m

Number of triangles n

The n-th triangle has sides 1, √n and √(n + 1). Its perimeter minus the previous one's is (1 + √n + √(n + 1)) − (1 + √(n − 1) + √n) = √(n + 1) − √(n − 1): the shared √n cancels every time. Watch for the triangles whose hypotenuse is a natural number: the 3rd (√4 = 2), the 8th (√9 = 3), the 15th (√16 = 4)…

7. Decimal forms ദശാംശരൂപങ്ങൾ

Fractions have decimal forms too: 7/10 = 0.7, 21/100 = 0.21, ½ = 5/10 = 0.5, ¼ = 25/100 = 0.25. For ⅓ the fractions 3/10, 33/100, 333/1000, … come closer and closer, so ⅓ = 0.333… – that looks just like √2 = 1.41421… What is the difference?

🔍 Compare the digits

Digits shown

In the decimal of a fraction a block of digits repeats for ever (highlighted): ⅓ repeats "3", ⅙ repeats "6" after the 1, 1/11 repeats "09", 2/7 repeats "285714". The decimals of √2 and √3 have no repeating block – they wander on without a pattern. That is the fingerprint of a number that is not a fraction.

8. Sum and square root തുകയും വർഗമൂലവും

Is √x + √y the same as √(x + y)? The textbook's picture on page 28 answers it with two squares.

🎬 Animation: squares of areas x and y, corner to corner

🎚️ Try any x and y

x y

√9 + √16 = 3 + 4 = 7, but √(9 + 16) = √25 = 5. The sum of two sides of a triangle is always more than the third side, so √x + √y > √(x + y) for every positive x and y. Never add under the root sign!

9. Subtraction വ്യവകലനം

🎬 Animation: √3 − √2 from the approximations

🌍 Real life

A farmer fencing a right-angled plot with sides 1 and 1 (in units of 10 m) needs 2 + √2 ≈ 3.414 units – 34.14 m of wire, and buying 34.2 m is safe. A tent rope cut to 2 − √2 m ≈ 58.6 cm needs millimetre accuracy; a garden path does not. Deciding how many decimals you need is part of the job.

10. Check yourself

🧭 Summary of Section 3

Sums like 2 + √2 and 1 + √2 + √3 are exact answers; their decimals come from adding approximations with equal numbers of decimal places: 2 + √2 = 3.4142…, √2 + √3 = 3.146….
To subtract, cancel first: (1 + √2 + √3) − (2 + √2) = √3 − 1 ≈ 0.732; (3 + √3) − (1 + √2 + √3) = 2 − √2 ≈ 0.586.
Round at the end: 2 decimal places of a metre for centimetres, 3 for millimetres.
The decimals of fractions repeat a block (⅓ = 0.333…, 1/11 = 0.0909…); those of √2, √3 never repeat.
√x + √y > √(x + y) – the two squares corner to corner show why.

Next: Practice set 2 – the four questions of pages 29–30 solved with animations ▶