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NCERT Solutions Ganita Prakash (Part 1) Chapter 3 .3 II. Variations on the Egyptian System and the Notion of Base — Figure it Out

Book page 633 Updated on2026-09-05

Q1.
Write the following numbers in the above base-5 system using the symbols in Table 2: 15, 50, 137, 293, 651.
Answer

The symbols are △ = 1, □ = 5, ⬡ = 25, ○ = 125, ∿ = 625, ↑ = 3125. Each time, take as many of the largest possible landmark as you can, then move down.

NumberGroupingNumeralBase-5 digits
155 + 5 + 5□□□30
5025 + 25⬡⬡200
137125 + 5 + 5 + 1 + 1○ □□ △△1022
293125 + 125 + 25 + 5 + 5 + 5 + 1 + 1 + 1○○ ⬡ □□□ △△△2133
651625 + 25 + 1∿ ⬡ △10101

Checks: 15 = 3 × 5 ✓   50 = 2 × 25 ✓   125 + 10 + 2 = 137 ✓   250 + 25 + 15 + 3 = 293 ✓   625 + 25 + 1 = 651 ✓

Why it happens: taking the largest landmark first is not just habit — it is what guarantees that no symbol ever has to be drawn five times. If for 651 you began with 125s you would need five of them, and five ○ are one ∿, so you would be forced to regroup anyway. Being greedy from the top gets the shortest numeral straight away.
Tip: 651 is striking — a number in the six hundreds written with just three symbols, because 651 = 625 + 25 + 1 sits almost exactly on the powers of 5.
Q2.
Is there a number that cannot be represented in our base-5 system above? Why or why not?
Answer

Yes — zero. There is no symbol for it, and there is no way to build it, because a numeral here is a collection of symbols and zero would have to be an empty collection, which cannot be seen on the page.

Every counting number 1, 2, 3, … can be written, and here is the argument:

Take any number N.
Pick the largest landmark 5k with 5k ≤ N. Draw that symbol and subtract.
The leftover is smaller than before, so the process must stop.
When it stops, nothing is left over — every part has been drawn.

Provided you never use a symbol five or more times, that numeral is also the only one for N.

Why it happens: notice that the same objection applies to the Egyptian and Roman systems — none of them can write zero either, and none of them needed to, because they only ever recorded quantities that were actually there. Zero becomes indispensable only in a place value system, where an empty position must be shown so that 305 is not read as 35. That is why the invention of 0 belongs to the last stage of the story, not the first.
Did you know? Negative numbers and fractions cannot be written in this system either. Brahmagupta’s work of 628 CE, which gave 0 and the negative numbers the full status of numbers, is what opened all of that up.
Q3.
Compute the landmark numbers of a base-7 system. In general, what are the landmark numbers of a base-n system?
Answer

Start at 1 and multiply by 7 each time.

70 = 1
71 = 7
72 = 49
73 = 343
74 = 2401
75 = 16807  …

So the landmark numbers of a base-7 system are 1, 7, 49, 343, 2401, 16807, …

In general, the landmark numbers of a base-n system are the powers of n:

n0 = 1,   n,   n2,   n3,   n4, …
Why it happens: the definition on page 63 says the first landmark is 1 and each next one is the current one multiplied by n. Starting at 1 and multiplying by n repeatedly is exactly what taking powers of n means. So “base-n” and “landmarks are the powers of n” are two ways of saying the same thing — and this is the property that will make landmark × landmark = landmark on page 67.
Check it yourself: 7 × 343 = 2401 and 49 × 49 = 2401 as well. Two different pairs of landmarks give the same landmark, because 71 × 73 and 72 × 72 both come to 74.
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