NCERT Solutions Ganita Prakash (Part 1) Chapter 3 .3 Advantages of a Base-n System — In-text Questions

Book page 663 Updated on2026-09-05

Q1.

How to multiply two numbers in Egyptian numerals?

Answer

Split each numeral into the landmark numbers it is made of, multiply every landmark of the first by every landmark of the second, then pool the results and regroup.

(a + b) × (c + d) = ac + ad + bc + bd (the distributive law)

The whole method rests on one fact, established on the next two pages:

10^a × 10^b = 10^a+b — the product of two landmarks is again a landmark

So each of the small products is a single symbol, and the only work left is counting how many of each symbol you have ended up with.

Why it happens: in the Roman system the same plan collapses at the first step, because V × L = 250 is not a landmark and L × D = 25000 has no symbol at all. Having landmarks that are powers of one number is what turns multiplication from a special problem into a routine.

Q2.

What is any landmark number multiplied by ∩ (that is 10)? Find the following products — (i) ∩ × ∩ (ii) coil × ∩ (iii) lotus × ∩ (iv) finger × ∩

Answer

Multiplying a landmark by 10 gives the next landmark, because it raises the power of 10 by one.

10^a × 10 = 10^a+1

	Product	In powers of 10	Answer
(i)	∩ × ∩	10 × 10 = 10²	100 — the coil
(ii)	coil × ∩	10² × 10 = 10³	1000 — the lotus
(iii)	lotus × ∩	10³ × 10 = 10⁴	10,000 — the finger
(iv)	finger × ∩	10⁴ × 10 = 10⁵	1,00,000 — the tadpole

Why it happens: 10^a means 10 multiplied by itself a times. Multiplying by one more 10 makes it a + 1 tens multiplied together, which is 10^a+1. So the rule is not a coincidence of Egyptian drawing — it is the law of exponents, and it is the reason the symbol list is ordered the way it is.

Q3.

What is any landmark number multiplied by the coil (10²)? Find the following products — (i) ∩ × coil (ii) coil × coil (iii) lotus × coil (iv) finger × coil

Answer

Multiplying by 10² moves a landmark up by two steps.

10^a × 10² = 10^a+2

	Product	In powers of 10	Answer
(i)	∩ × coil	10 × 10² = 10³	1000 — the lotus
(ii)	coil × coil	10² × 10² = 10⁴	10,000 — the finger
(iii)	lotus × coil	10³ × 10² = 10⁵	1,00,000 — the tadpole
(iv)	finger × coil	10⁴ × 10² = 10⁶	10,00,000 — the kneeling man

Why it happens: multiplying by 10² is multiplying by 10 twice over, so each answer here is two symbols further along the list than the answers in the previous question. In Hindu numerals the same rule reads “multiplying by 100 puts two zeroes at the end”.

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