NCERT Solutions Ganita Prakash (Part 1) Chapter 3 – 703.3 III. Shortcomings of the Egyptian System — Figure it Out

Book page 69 Updated on2026-09-05

Q1.

Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not?

Answer

No — not in the tidied-up form that the Egyptians actually wrote.

10 copies of a landmark 10^a = 10 × 10^a = 10^a+1
and 10^a+1 has a symbol of its very own.

So ten identical symbols would always be swapped for one symbol of the next kind. Each symbol can therefore appear at most 9 times.

Why it happens: this is the same fact that fixes the range of our digits. A digit in the Hindu system counts how many of that power of 10 there are, and the answer can never be 10 or more — so the digits run 0 to 9 and no further. Nine is not an arbitrary stopping point; it is one less than the base.

Tip: there is one place where the argument breaks. The topmost symbol is the sun, 10⁷. Ten suns would be 10⁸, and no symbol exists for that — so for numbers of ten crore and above, the sun really would have to be drawn ten or more times. That is exactly the shortcoming this section is about.

Q2.

Create your own number system of base 4, and represent numbers from 1 to 16.

Answer

Take three symbols for the first three landmarks of base 4:

∟ = 4⁰ = 1 △ = 4¹ = 4 □ = 4² = 16

Number	Grouping	Numeral	Base-4 digits
1	1	∟	1
2	1 + 1	∟∟	2
3	1 + 1 + 1	∟∟∟	3
4	4	△	10
5	4 + 1	△∟	11
6	4 + 2	△∟∟	12
7	4 + 3	△∟∟∟	13
8	4 + 4	△△	20
9	8 + 1	△△∟	21
10	8 + 2	△△∟∟	22
11	8 + 3	△△∟∟∟	23
12	4 + 4 + 4	△△△	30
13	12 + 1	△△△∟	31
14	12 + 2	△△△∟∟	32
15	12 + 3	△△△∟∟∟	33
16	16	□	100

Why it happens: read down the numeral column and you can see the base at work — no symbol ever appears four or more times, because four ∟ are exchanged for one △ and four △ for one □. That is why the base-4 digits are only 0, 1, 2, 3. It also explains why 16 is written with a single symbol while 15 needs six: 16 is a landmark and 15 is just below one.

Check it yourself: continue to 17 = □∟ and 20 = □△, and check that 16 + 4 = 20.

Q3.

Give a simple rule to multiply a given number by 5 in the base-5 system that we created.

Answer

Replace every symbol by the next symbol up, keeping the counts unchanged.

△ → □ → ⬡ → ○ → ∿ → ↑
(1 → 5 → 25 → 125 → 625 → 3125)

Worked example. Take ○ □□ △ = 125 + 5 + 5 + 1 = 136.

○ → ∿ □□ → ⬡⬡ △ → □
Result: ∿ ⬡⬡ □ = 625 + 25 + 25 + 5 = 680
Check: 136 × 5 = 680 ✓

Why it happens: break the number into its landmark parts (distributive law), and multiply each part by 5. Since 5 × 5^a = 5^a+1, every part becomes the next landmark, and nothing has to be regrouped because the counts have not grown. Written as base-5 digits, the rule is the familiar one: put a 0 at the end — 136 is 1021 in base 5, and 680 is 10210.

Tip: the same rule works in every base — multiplying by the base itself shifts everything up one place. That is why “multiply by 10, add a zero” works in base 10 and “multiply by 2, add a zero” works in binary.

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