NCERT Solutions Ganita Prakash (Part 1) Chapter 3 – 583.2 III. Number Names Obtained by Counting in Twos — In-text Questions

Book page 57 Updated on2026-09-05

Q1.

Quickly count the number of objects in each of the following boxes:

Answer

Reading the nine boxes on page 57, row by row:

Box	Objects	Could you see it at a glance?
Hens	2	Yes, instantly
Bunch of yellow flowers	too many to say	No — you would have to pull the bunch apart
Nesting dolls	4	Yes, just about
Steps of the staircase	about 10	No — you have to run your eye up them
Dog	1	Yes, instantly
Bunch of grapes	too many to say	No
Apples	6	No — most people have to count
Pencils	about 8	No — you count them one by one
Pyramids	3	Yes, instantly

Why it happens: the boxes have been chosen deliberately. The ones you can read off — 1 dog, 2 hens, 3 pyramids, 4 dolls — are all small. The moment a box holds 5 or more you stop seeing the number and start counting it. The grapes and flowers make the point at the other extreme.

Q2.

Up to what group size could you immediately see the number of objects without counting?

Answer

Up to about 4. Most humans find it difficult to take in a group of 5 or more objects in a single glance.

Why it happens: this limit of perception is not a mathematical fact but a fact about human eyes and brains — and it has left its fingerprints all over the history of numerals. Once a tally reached five marks nobody could read it at sight, so people replaced every group of five marks by a single new symbol. That is exactly the Roman V standing in place of IIIII, and X in place of two Vs.

Did you know? This is why the group sizes that keep reappearing in number systems are small and human-sized: 2 (the Gumulgal), 5 (the Roman V), 10 (fingers of two hands — the Egyptian and Hindu systems) and 20 (fingers and toes — the Mayan system).

Q3.

What could be the difficulties with using a number system that counts only in groups of a single particular size? How would you represent a number like 1345 in a system that counts only by 5s?

Answer

The difficulty is that the numeral stays almost as long as the number itself.

1345 ÷ 5 = 269 (since 5 × 269 = 1345)
So 1345 = 5 + 5 + 5 + … + 5 (269 times)

You would have to write the “five” symbol 269 times. Grouping by 5 has shortened the tally from 1345 marks to 269 — a saving, but not nearly enough. Writing it, reading it and comparing two such numerals are all still impossible in practice.

The cure is not a bigger group size but a sequence of group sizes, each one 5 times the last:

1 → 5 → 25 → 125 → 625 → 3125
1345 = (2 × 625) + (0 × 125) + (3 × 25) + (4 × 5) + (0 × 1)
= 1250 + 0 + 75 + 20 + 0 = 1345 ✓
as a base-5 numeral: 20340

Now the same number needs only 9 symbols in the picture-form (∿∿ ⬡⬡⬡ □□□□) instead of 269.

Why it happens: with one group size, the length of a numeral grows in proportion to the number — double the number and you double the writing. With landmark numbers that are powers of 5, each new symbol you add multiplies the range by 5, so the length grows only as fast as the number of powers needed. That is the whole reason a base is worth having.

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