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NCERT Solutions Ganita Prakash (Part 1) Chapter 3 .4 The Hindu Number System — Figure it Out

Book page 803 Updated on2026-09-05

Q1.
Why do you think the Chinese alternated between the Zong and Heng symbols? If only the Zong symbols were to be used, how would 41 be represented? Could this numeral be interpreted in any other way if there is no significant space between two successive positions?
Answer

Why they alternated: because the rod numerals were actual sticks laid out on a counting board, and a blank space between two places is invisible. Turning the rods through a right angle at every place makes the boundary between places visible instead.

  • Zong (upright rods) — units, hundreds, ten-thousands …
  • Heng (flat rods) — tens, thousands, hundred-thousands …

So in the numeral for 2634 on page 77 you read 2 (Heng), 6 (Zong), 3 (Heng), 4 (Zong), and there is never any doubt where one place ends and the next begins.

41 with Zong only: 41 = 4 tens and 1 one, so you would put down Zong-4 followed by Zong-1:

| | | |   |   — five upright rods in a row

Yes, it can be read in many other ways if the gaps are not clear. Five rods can be split up as:

How the rods are splitReads as
| | | | |  (no split)5
|   | | | |14
| |   | | |23
| | |   | |32
| | | |   |41
|   | |   | |122
|   |   | | |113
| |   |   | |212
| | |   |   |311
Why it happens: alternating the orientation is a clever partial cure, but only a partial one. It marks where the places begin, yet it still cannot say that a place is empty, so a blank had to be left for a missing place and blanks are hard to count. As the book notes, a symbol for zero would have turned the rod numerals into a fully developed place value system. It is exactly the step the Indian system took.
Q2.
Form a base-2 place value system using ‘ukasar’ and ‘urapon’ as the digits. Compare this system with that of the Gumulgal’s.
Answer

A base-2 system needs exactly two digits, one of which must stand for zero. Let

urapon = 0     ukasar = 1
landmark numbers: 1, 2, 4, 8, 16, 32, …
NumberGrouped into powers of 2Base-2 place value systemGumulgal system
11ukasarurapon
22ukasar uraponukasar
32 + 1ukasar ukasarukasar-urapon
44ukasar urapon uraponukasar-ukasar
54 + 1ukasar urapon ukasarukasar-ukasar-urapon
64 + 2ukasar ukasar uraponukasar-ukasar-ukasar
74 + 2 + 1ukasar ukasar ukasarukasar-ukasar-ukasar-urapon
88ukasar urapon urapon uraponukasar-ukasar-ukasar-ukasar

Comparison.

  • In the Gumulgal system the position of a word means nothing — you simply add: each ukasar contributes 2 and each urapon contributes 1. Its only landmark numbers are 1 and 2.
  • In the base-2 place value system the position is everything: the same word ukasar means 1, 2, 4 or 8 depending on where it stands. Its landmark numbers go on for ever: 1, 2, 4, 8, 16, …
  • Length: to write N, the Gumulgal need about N ÷ 2 words; the place value system needs only about log2N digits. For 1024 that is 512 words against 11 digits.
  • The place value version needs a digit for zero; the Gumulgal system has no use for one, because a place that is empty is simply not spoken.
Why it happens: both systems count in 2s, and yet they behave completely differently. The Gumulgal use the group size only to make names; the base-2 system uses it to make places, and each new place doubles the range. Watch out for one trap: 2 is ukasar in the Gumulgal system but ukasar urapon in the base-2 system — the same words, read by two different rules.
Q3.
Where in your daily lives, and in which professions, do the Hindu numerals, and 0, play an important role? How might our lives have been different if our number system and 0 hadn’t been invented or conceived of?
Answer

In daily life: prices and change at a shop, dates on a calendar, the time on a clock, phone numbers, PIN codes, bus and train numbers, house numbers, cricket scores and run rates, marks in a report card, the reading on a weighing scale, page numbers in this book.

In professions:

  • Shopkeepers and accountants — bills, ledgers, GST calculations.
  • Bankers — interest, instalments, account balances.
  • Engineers, architects and surveyors — measurements, scale drawings, load calculations.
  • Doctors and chemists — dosages in milligrams, blood reports.
  • Farmers — land area in hectares, yields, market rates.
  • Scientists — very large and very small numbers written as 1023 or 10-9, which is place value taken to its natural conclusion.
  • Programmers — every computer works in base 2, using only 0 and 1.

Without them, life would look very different:

  • Arithmetic would need a device. Recall page 60: people using Roman numerals had to use an abacus, and only specially trained people could work it. Ordinary shopkeepers could not have checked their own bills.
  • There would be no column addition, no long multiplication, no long division on paper.
  • Without 0 as a number there would be no negative numbers as Brahmagupta defined them, so no algebra of the modern kind, no coordinate geometry, no equations of motion.
  • There would be no decimals, no scientific notation, and no binary — so no computers, no digital payments, no mobile phones.
Why it happens: the Hindu system does not merely record numbers, it makes calculation possible for everybody. That is why Laplace called its significance so profound and yet so easily overlooked — it “placed arithmetic foremost among useful inventions” precisely by making it ordinary.
Q4.
The ancient Indians likely used base 10 for the Hindu number system because humans have 10 fingers, and so we can use our fingers to count. But what if we had only 8 fingers? How would we be writing numbers then? What would the Hindu numerals look like if we were using base 8 instead? Base 5? Try writing the base-10 Hindu numeral 25 as base-8 and base-5 Hindu numerals, respectively. Can you write it in base-2?
Answer

With 8 fingers we would almost certainly have settled on base 8. Everything else about the system would be unchanged — only the base would differ.

Landmark numbers: 1, 8, 64, 512, 4096, …
Digits needed: 0, 1, 2, 3, 4, 5, 6, 7 — eight of them, one fewer than the base
A number N would be written ak ak−1 … a1 a0, meaning
N = ak·8k + ak−1·8k−1 + … + a1·8 + a0,   with 0 ≤ ai ≤ 7

Writing 25:

BaseGroupingNumeral
825 = 3 × 8 + 131
525 = 1 × 25 + 0 × 5 + 0 × 1100
225 = 16 + 8 + 1 = 1×16 + 1×8 + 0×4 + 0×2 + 1×111001
1025 = 2 × 10 + 525

Checks: 3 × 8 + 1 = 25 ✓   1 × 25 = 25 ✓   16 + 8 + 0 + 0 + 1 = 25 ✓

Why it happens: the base decides two things at once. It fixes how many digits you need — always the base itself, counting 0 — and it fixes how long numerals are. A smaller base means fewer symbols to learn but longer numerals: 25 is “31” in base 8, “100” in base 5 and “11001” in base 2. Base 10 is not mathematically special in any way; ten fingers are the whole reason for it. What is special, and works in every base, is the structure — landmarks that are powers of the base, one digit per position, and a 0 for an empty position.
Check it yourself: 100 in base 5 is 25 in base 10, and 100 in base 8 is 64. In every base, the numeral “100” means the square of the base — which is exactly what place value promises.
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