NCERT Solutions Ganita Prakash (Part 1) Chapter 3 – 53The Mechanism of Counting — In-text Questions

Book page 52 Updated on2026-09-05

Q1.

How will you use such sticks to answer the other two questions (Q2 and Q3)?

Answer

By pairing one collection against the other and looking at what is left.

Step	What you do with the sticks	What it tells you
1	One stick for each of our cows; one stick for each of the neighbour’s cows	Two collections that stand for the two herds
2	Remove one stick from each pile at the same time, again and again	A one-to-one pairing is being built
3	Our pile empties first	We have fewer cows (Q2)
4	Count nothing — just keep the sticks still lying in his pile	That collection is how many more cows we need (Q3)

Why it happens: the sticks are a faithful copy of the herd, so any question about the herd can be turned into a question about the sticks — and sticks can be moved, paired and set aside, which cows cannot. This is why representing a number by objects is useful even before number names exist.

Q2.

How many numbers can you represent in this way using the sounds of the letters of your language?

Answer

Exactly as many as your language has letters — no more.

English: 26 letters a, b, c, …, z → numbers 1 to 26
Devanagari (Hindi): 11 vowels + 33 consonants = 44 letters → numbers 1 to 44
Tamil: 12 vowels + 18 consonants = 30 basic letters → numbers 1 to 30

Whichever language you take, the answer is a fixed, finite number. Count the letters of your own alphabet and that is your limit.

Why it happens: in this method each number is matched to one letter, following the letter-order. A one-to-one mapping can never reach further than the collection it maps into. So the alphabet, being finite, gives a finite standard sequence — and numbers are unending. That is the flaw: the method is convenient to count with but it stops.

Tip: compare this with the stick system. Sticks never run out of numbers but are clumsy for large collections; letters are easy to say but run out. The Hindu system is the one that manages both at once.

Q3.

Do you see a way of extending this method to represent bigger numbers as well? How?

Answer

Yes — the way the Roman system itself did it. Keep repeating the symbols, and bring in a fresh symbol whenever the repetition gets too long.

After XX, carry on: XXI, XXII, … XXIX, XXX (30), … XXXX or XL (40)
Instead of writing XXXXX for 50, give it a new symbol: L
Then LX, LXX, LXXX, XC … and again a new symbol for 100: C
and further: D = 500, M = 1000

So the extension works in two moves: (a) repeat the symbol you already have, and (b) when about five repetitions have piled up, replace that whole group by a brand-new symbol.

Why it happens: repetition alone is just tally marks — 1000 would need a thousand strokes. Introducing a new symbol at 5, 10, 50, 100, … keeps every numeral short. But the price is that each new landmark needs a new symbol, so however many symbols you invent, some number will still be out of reach. Section 3.4 shows the way out: use the position of a symbol instead of a new symbol.

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