The Algebra of Measures അളവുകളുടെ ബീജഗണിതം
Take a small rectangle and stretch every side by the same amount. How does its perimeter change? Instead of working it out again for every stretch, we find one rule that answers all the cases together. Writing that rule in algebra, and then in a new short form p(x), is the first step towards polynomials.
- Perimeter of a rectangle = 2 × (length + breadth) (ചുറ്റളവ്).
- A letter such as x can stand for “any number”, and 4x means 4 × x.
1. Stretching a rectangle
A rectangle (ചതുരം) has sides 3 cm and 2 cm. All its sides are lengthened by 1 cm each, giving a slightly bigger rectangle. What is the new perimeter? Step through the textbook's two ways of thinking.
The first way works out the new sides and starts again. The second way only asks “how much was added?” Four sides each grew by the same amount, so the perimeter grows by 4 times that amount. The second way gives a rule that works for every stretch.
2. One rule for every stretch
The blue parts are the old sides, 10 cm in all. The four red parts are the added lengths, each x cm. Whatever x is, the perimeter is 10 + 4 × x.
If every side is lengthened by x cm and the new perimeter is p cm, then
In words: the perimeter is four times the added length, plus 10 (കൂട്ടിയ നീളത്തിന്റെ നാലു മടങ്ങ് പത്തിനോട് കൂട്ടിയത്).
3. A machine that makes perimeters
The rule 4x + 10 is like a machine: put in the added length x, and it gives out the perimeter. The textbook tries 3, 3½ and 3¾ cm.
4. A shorter way to write it: p(x)
“If x = 3 then p = 22” is still long. Algebra has a shorter way: write p(3) = 22, read “p of 3 is 22”. It means: the perimeter when the added length is 3 is 22.
And for any added length x we write, all at once,
Writing p(x) shows clearly that p changes as x changes – the perimeter depends on the added length. The brackets do not mean multiplication here: p(3) is not “p times 3”; it is “the value of p when x is 3”.
The textbook puts the same fact side by side in ordinary language and in algebra:
🗣️ In ordinary language
If all the sides of a rectangle with sides two centimetres and three centimetres are lengthened equally, the perimeter of the bigger rectangle is ten added to four times the added length. For example, if all the sides are lengthened by one and a half centimetres, the perimeter becomes sixteen centimetres.
✏️ In algebra
If all the sides of a rectangle with sides 2 cm and 3 cm are lengthened by x cm, and the perimeter of the bigger rectangle is p(x) cm, then
p(x) = 4x + 10
For example, p(1½) = 16.
The algebra version is shorter, and it holds every case at once.
Say it in words too: “if every side grows by … cm, the perimeter is … cm”.
An auto-rickshaw charges ₹30 to start and ₹15 for every kilometre. The fare for x km is f(x) = 15x + 30; f(4) = 15 × 4 + 30 = ₹90. A mobile plan with ₹100 rent and ₹2 per GB used gives b(x) = 2x + 100. A fixed part plus a part that grows steadily – exactly the shape of p(x) = 4x + 10.
5. Check yourself
- A 3 cm × 2 cm rectangle has perimeter 10 cm. If every side grows by x cm, the four sides add 4x cm.
- So the new perimeter is p(x) = 4x + 10 (ബീജഗണിതം: letters for changing numbers).
- p(3) = 22 means “when x is 3, p is 22”. Also p(1½) = 16, p(2½) = 20, p(3½) = 24, p(3¾) = 25.
- The notation p(x) shows that p depends on x.