Figure it Out — Chinese Rod Numerals, Base-2 and Other Bases — A Story of Numbers

1. 13 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 80

   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?

   Hint. In a place value system written with rods, the danger is not knowing where one place ends and the next begins.

   Why they alternated. The rod numerals are a base-10 place value system, so the reader must be able to tell where one position ends and the next begins. By writing the units, hundreds and ten-thousands with vertical (Zong) rods and the tens, thousands and hundred-thousands with horizontal (Heng) rods, adjacent positions always look different. The alternation acts as a built-in separator, so the boundary between places is visible even without careful spacing.

   Representing 41 with Zong only. 41 = (4 × 10) + (1 × 1), so it would be written as four vertical rods followed by one vertical rod — that is, five vertical rods in a row: |||| |

   Is it ambiguous? Yes, badly so. With no clear space, the same five rods could be split between the two positions in several ways, and they could also be read as a different number of positions altogether — for instance as 23, 32, or 122, depending on where the reader imagines the gaps to fall. A run of identical rods carries no information about where one place stops.

   This is the same ambiguity that troubled the Mesopotamian system, and it shows that place value alone is not enough: a system also needs an unmistakable way to mark position, which the Hindu system achieves by using exactly one distinct digit per place, including 0.

   ✦ They alternated so that adjacent positions always look different, making the boundary between places visible. Using Zong only, 41 is five identical vertical rods, which could equally be read as 23, 32 or 122.

2. 24 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 80

   Form a base-2 place value system using 'ukasar' and 'urapon' as the digits. Compare this system with that of the Gumulgal.

   Hint. A base-2 place value system needs digits for 0 and 1 only, and the positions stand for 1, 2, 4, 8, …

   Step 1 — Set up the system. A base-2 place value system needs exactly two digits, one for 0 and one for 1. Take urapon = 0 and ukasar = 1. The positions then stand for 2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, and so on.

   Step 2 — Write the first few numbers both ways.

   Number	Base-2 place value (ur = 0, uk = 1)	Gumulgal system
   1	uk	urapon
   2	uk ur	ukasar
   3	uk uk	ukasar-urapon
   4	uk ur ur	ukasar-ukasar
   5	uk ur uk	ukasar-ukasar-urapon
   6	uk uk ur	ukasar-ukasar-ukasar
   7	uk uk uk	ukasar-ukasar-ukasar-urapon
   8	uk ur ur ur	ukasar-ukasar-ukasar-ukasar

   Step 3 — Compare the two. Both systems use only two words, but they use them in completely different ways.

   • The Gumulgal system simply repeats ukasar once for every 2 in the number and adds an urapon if there is a leftover 1. So the name for a number grows in direct proportion to its size — the name for 100 would need fifty ukasars.

   • The base-2 place value system uses position instead, so each new digit doubles the range covered. Eight is reached with four digits, and a number as large as 1000 needs only ten. The growth is far slower because each position is worth twice the one to its right.

   The decisive difference is that the base-2 system needs a symbol for nothing (urapon = 0) in order to mark an empty position, whereas the Gumulgal system never needs one, since it never uses position at all. This is exactly the step that turns counting into a place value system — and it is the same step that makes modern computers possible, since they store all numbers in base 2.

   ✦ With urapon = 0 and ukasar = 1: 1 = uk, 2 = uk ur, 3 = uk uk, 4 = uk ur ur, 5 = uk ur uk, 6 = uk uk ur, 7 = uk uk uk, 8 = uk ur ur ur. The base-2 system uses position so its numerals grow far more slowly than the Gumulgal names, but it needs a symbol for zero, which the Gumulgal system never does.

3. 32 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 80

   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?

   Hint. Think of anything that involves money, measurement, time, or a machine.

   In daily life. Prices and bills, dates and clock times, phone numbers, PIN codes and passwords, bus and train numbers, page numbers, marks and percentages, measurements while cooking or building, and every reading on a weighing scale or thermometer.

   In professions. Shopkeepers and accountants for money and interest; engineers and architects for measurement and design; doctors for dosages and test readings; scientists for experimental data; bankers for balances and rates; programmers and computer engineers, whose machines represent everything in base 2 and rely on 0 absolutely.

   If zero and the system had never been conceived. Writing large numbers would need endless new symbols, as in the Egyptian and Roman systems, so recording large quantities would be clumsy and error-prone. Multiplication and division would be so difficult that ordinary people could not do them without an abacus and special training, which is exactly how it was in medieval Europe.

   More seriously, the absence of zero would have blocked whole areas of mathematics from developing. Zero is needed as a number to write equations, to define negative numbers, to work with the decimal point, and to express the very idea of 'nothing' as a quantity. Since algebra, calculus and eventually computing were all built on that foundation, modern science and technology as we know them would not have been possible.

   This is why the chapter calls the Indian number system one of the greatest inventions in human history.

   ✦ In money, time, measurement, data and computing — and without zero, large numbers would need endless new symbols, calculation would need an abacus, and algebra, calculus and computing could never have developed.

4. 44 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 80

   The ancient Indians likely used base 10 because humans have 10 fingers. But what if we had only 8 fingers? What would the Hindu numerals look like in base 8? In base 5? Write the base-10 numeral 25 in base 8 and base 5. Can you write it in base 2?

   Hint. In base n you need n digits, 0 to n−1, and the positions stand for 1, n, n², n³, …

   Step 1 — What changes with the base. In a base-8 system we would need exactly eight digits, 0 to 7, and the positions would stand for 8⁰ = 1, 8¹ = 8, 8² = 64, 8³ = 512, … So a number would be written as n = aₖ·8ᵏ + aₖ₋₁·8ᵏ⁻¹ + … + a₁·8 + a₀, with each digit aᵢ between 0 and 7, and the numeral is just the string of digits aₖaₖ₋₁…a₁a₀. Base 5 works identically with the digits 0 to 4 and the positions 1, 5, 25, 125, …

   Step 2 — Write 25 in base 8. The largest power of 8 not exceeding 25 is 8¹ = 8, and 25 contains three 8s with 1 left over: 25 = (3 × 8) + (1 × 1) 25 in base 8 = 31

   Step 3 — Write 25 in base 5. Here 25 = 5², which fits exactly once with nothing left over at all: 25 = (1 × 25) + (0 × 5) + (0 × 1) 25 in base 5 = 100 This is the same reason 100 in base 10 means 10² — a number that is exactly the square of the base always comes out as 1 followed by two zeros.

   Step 4 — Write 25 in base 2. The powers of 2 are 1, 2, 4, 8, 16, 32. The largest not exceeding 25 is 16, and 25 = 16 + 8 + 1: 25 = (1 × 16) + (1 × 8) + (0 × 4) + (0 × 2) + (1 × 1) 25 in base 2 = 11001 Check: 16 + 8 + 0 + 0 + 1 = 25 ✓

   Notice that the smaller the base, the longer the numeral — 25 takes two digits in base 8, three in base 5 and five in base 2. A small base means few different symbols to learn but longer numbers, and base 10 sits at a convenient middle point (which our ten fingers probably decided for us).

   ✦ 25 in base 8 = 31; 25 in base 5 = 100; 25 in base 2 = 11001

Solutions written by the tuition.in editorial team and checked against the NCERT Class 8 Mathematics textbook Ganita Prakash Part 1, Reprint 2026-27 (hegp103.pdf). This chapter is a history of number *systems* — tally marks, Roman, Egyptian, base-5, Mesopotamian, Mayan, Chinese rod and Hindu numerals — not a chapter on rational numbers. Questions appear as numbered 'Figure it Out' blocks plus 'Math Talk'/'Try This' prompts in the running text; every numeric answer here is checked against the book's own printed answer key at the end of the chapter. Four sub-parts whose questions exist only as printed Egyptian/base-5 glyph images (the addition drills on pages 65 and the products on page 68) are deliberately omitted rather than guessed at, since the operands cannot be recovered from the text.. Questions are referenced from the NCERT textbook for identification.