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NCERT Solutions Ganita Prakash (Part 1) Chapter 3 .1 Reema’s Curiosity — In-text Questions

Book page 483 Updated on2026-09-05

Q1.
Since when have humans been counting?
Answer

Since long before writing existed — at least from the Stone Age, about ten thousand years ago, and on the evidence of marked bones very much earlier still.

Lebombo bone (South Africa) — 29 notches — about 44,000 years old
Ishango bone (Democratic Republic of Congo) — notches in columns — 20,000 to 35,000 years old
Stone Age herders counting cattle — about 10,000 years ago
Why it happens: counting does not need writing, or even number names. A herder who keeps one stick for each cow already has a number system. So counting is far older than any of the numerals in this chapter — the bones only tell us how far back the record of counting goes, not when counting began.
Q2.
What was their need for counting? What were they counting?
Answer

They counted whatever they had to keep track of, and whatever they had to predict.

  • Food: the quantity of grain, fruit and vegetables in store.
  • Livestock: the number of animals in the herd, so a missing animal could be spotted.
  • Trade: details of goods exchanged with other groups.
  • Rituals: the number of offerings given.
  • Time: the passing days, so that the new moon, the full moon or the onset of a season could be known and predicted in advance.
Did you know? The Lebombo bone has 29 notches — very close to the 29 or 30 days of a lunar month. That is why many historians think it was used as a lunar calendar. Counting days was as important as counting cattle.
Q3.
Since when have people been writing numbers in the modern form?
Answer

About 2000 years ago, in India.

WhenWhat happened
Ancient (Yajurveda Samhita)Names of numbers built on powers of 10 — eka, dasha, shata, sahasra, āyuta … up to 1012 and beyond, almost as we say them today
c. 3rd century CEBakhshali manuscript — the first known writing of numbers with ten digits including 0, then notated as a dot
c. 499 CEAryabhata first fully explains the ten-symbol Indian system and computes with it
c. 628 CEBrahmagupta codifies 0 as a number, with negative numbers
c. 800 – 830 CETransmitted to the Arab world; Al-Khwārizmī and Al-Kindi popularise it
c. 1100 – 1200 CEReaches Europe; Fibonacci argues for adopting it
By the 17th centuryAdopted throughout Europe — not adopting it would have blocked scientific progress
Why it happens: notice that the spoken form came first. Indian texts had names based on powers of 10 thousands of years ago; the written form that matches those names — one digit per power of 10, with 0 for an empty power — took much longer to appear, because it needed the invention of 0.
Q4.
How would the Mesopotamians have written 20? 50? 100?
Answer

The Mesopotamian system is base 60. It has just two marks: a wedge for 1 (we write Y) and a corner wedge for 10 (we write <). Numbers up to 59 sit in a single position; from 60 onwards a second position to the left is used.

20 = 10 + 10 → <<
50 = 10 + 10 + 10 + 10 + 10 → <<<<<
100 = (1) × 60 + 40 → Y  |  <<<<

So 20 and 50 need only the ones position. But 100 is bigger than 60, so it needs two positions: one 60, and 40 left over.

Why it happens: in a base-60 system no power of 60 may occur 60 or more times — if it did, sixty of them would be regrouped into the next power. So the number of 1s is always between 0 and 59, which is exactly why marks for 1 and 10 are enough to fill one position.
Tip: you meet base 60 every day. 1 hour = 60 minutes and 1 minute = 60 seconds are the Mesopotamian system still in use.
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