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NCERT Solutions Ganita Prakash (Part 1) Chapter 3 – 613.2 IV. The Roman Numerals — Figure it Out

Book page 60 Updated on2026-09-05

Q1.
A group of indigenous people in a Pacific island use different sequences of number names to count different objects. Why do you think they do this?
Answer

Because in such languages the number word is still tied to the thing being counted — the count has not yet been separated from what is counted.

  • The name carries extra information. One series for coconuts, another for canoes, another for fish: the word tells the listener both how many and of what, so nothing has to be said twice.
  • Different goods are handled in different natural groups. Fish may come in pairs, coconuts in bunches, yams by the basket. A counting series that grew out of the way a good is actually stacked and traded is the most convenient one for that good.
  • Status and ceremony. A separate series was often kept for people, or for objects used in rituals, marking them out as different from ordinary trade goods.
Why it happens: the mathematically important point is what such a system cannot do. If “three” for fish and “three” for coconuts are different words, then 3 fish + 3 coconuts cannot be added inside the language — the system has no idea of “three” by itself. Realising that the threeness of three fish and three coconuts is one and the same thing is the step that makes arithmetic possible. Our own language still keeps a trace of the older habit: we say a pair of shoes, a brace of birds, a dozen eggs.
Q2.
Consider the extension of the Gumulgal number system beyond 6 in the same way of counting by 2s. Come up with ways of performing the different arithmetic operations (+, –, ×, ÷) for numbers occurring in this system, without using Hindu numerals. Use this to evaluate the following: (i) (ukasar-ukasar-ukasar-ukasar-urapon) + (ukasar-ukasar-ukasar-urapon) (ii) (ukasar-ukasar-ukasar-ukasar-urapon) – (ukasar-ukasar-ukasar) (iii) (ukasar-ukasar-ukasar-ukasar-urapon) × (ukasar-ukasar) (iv) (ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar) ÷ (ukasar-ukasar)
Answer

In this system urapon = 1 and ukasar = 2, and a number is simply a string: as many ukasar as there are 2s in it, followed by one urapon if it is odd. Extending past 6 needs no new word at all — just longer strings.

The four rules, stated inside the system:

  • Add: write the two strings one after the other; if two urapon now appear, replace them by one ukasar. (A string never needs more than one urapon.)
  • Subtract: cancel ukasar against ukasar and urapon against urapon. If the larger number has no urapon to cancel, first split one of its ukasar into two urapon.
  • Multiply: replace each ukasar of the second number by a full copy of the first number, and each urapon by half a copy — or more simply, write the first number out as many times as the second number says, then tidy up the urapons.
  • Divide: break the string into equal bunches, each bunch matching the divisor; the number of bunches is the answer.
PartIn 2s and 1sWorkingAnswer
(i)(2+2+2+2+1) + (2+2+2+1)
= 9 + 7		7 ukasar and 2 urapon; the two urapon make one more ukasar → 8 ukasarukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar  (16)
(ii)(2+2+2+2+1) − (2+2+2)
= 9 − 6		Cancel three ukasar; one ukasar and one urapon remainukasar-urapon  (3)
(iii)(2+2+2+2+1) × (2+2)
= 9 × 4		Write the 9-string four times: 16 ukasar and 4 urapon; the four urapon make 2 more ukasar → 18 ukasarukasar repeated 18 times  (36)
(iv)(8 ukasar) ÷ (2 ukasar)
= 16 ÷ 4		Break the 8 ukasar into bunches of 2 ukasar: 4 bunchesukasar-ukasar  (4)
Why it happens: every rule here is just the tally rule with one extra exchange — “two urapon make one ukasar”. That single exchange halves the length of a numeral, which is a real gain, but no further exchange is ever available: 100 still needs fifty ukasar. Counting in a single group size takes you exactly one step beyond tally marks and no further.
Q3.
Identify the features of the Hindu number system that make it efficient when compared to the Roman number system.
Answer

Five features, and each of them fixes a specific Roman weakness.

FeatureHindu number systemRoman number system
Value of a symbolPlace value — the 3 in 375 means 3 × 102, the 3 in 32 means 3 × 10Fixed value — X is 10 wherever it stands
A symbol for nothing0, used as a digit and as a numberNo symbol for zero at all
How many symbolsTen symbols write every number, however largeA new symbol needed for each new landmark: I V X L C D M …
Landmark numbersAll powers of 10, so landmark × landmark is again a landmarkRatios jump ×5, ×2, ×5, ×2 — no pattern, so no multiplication rule
Length of a numeralGrows very slowly — 1888 needs 4 digitsGrows fast — 1888 = MDCCCLXXXVIII, 13 symbols
Why it happens: the deepest of these is place value together with 0. Because each position already announces which power of 10 it counts, no symbol has to be invented for 108 or 1015; and because 0 can hold an empty position, 305 cannot be misread as 35. Everything else — short numerals, column addition, long multiplication and division — follows from those two ideas.
Check it yourself: add 1888 and 1976 in Roman numerals, then in Hindu numerals. The second takes a few seconds; the first takes several exchanges and a lot of care.
Q4.
Using the ideas discussed in this section, try refining the number system you might have made earlier.
Answer

Take the system from page 54 (● = 1, ■ = 10, ★ = 100) and put it through the two refinements the chapter has introduced so far.

Refinement 1 — make the landmark numbers powers of one number. Ours already are: 1, 10, 100. So extend the same way instead of inventing unrelated jumps: the next landmarks must be 1000, 10000, 100000, and so on.

Refinement 2 — check the two tests the chapter has set.

TestBeforeAfter refining
Is the sequence unending?No — stops below 1000Yes, once every power of 10 has its own symbol
Is regrouping simple?Yes — ten ● make one ■Yes, and now the same rule holds at every level
Is a landmark × a landmark a landmark?Yes — ■ × ■ = ★Yes, so multiplication has a rule
Does a symbol ever repeat 10 times?No — ten of any symbol become one of the nextNo — so at most 9 of each

One problem survives: an unending supply of new symbols is still needed. That is exactly the Egyptian shortcoming of page 69, and the fix is on page 73 — stop drawing the landmark symbol and let the position say which landmark you mean. Since no symbol ever repeats more than 9 times, exactly nine marks plus a mark for “none” — ten symbols in all — are enough. Refined all the way, your system has turned into the Hindu number system.

Why it happens: this is the whole argument of the chapter in miniature. Grouping shortens numerals; a base makes regrouping and multiplying mechanical; place value removes the need for endless symbols; and 0 is what place value needs to be unambiguous.
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