Figure it Out — Early Number Systems and Their Efficiency — A Story of Numbers

1. 12 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 60

   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?

   Hint. Think about what extra information a number name could carry besides 'how many'.

   In such systems the number name does two jobs at once: it says how many things there are, and it says what kind of thing is being counted.

   Some likely reasons:

   1. The name itself identifies what is being counted. If coconuts, canoes and people each have their own counting sequence, then simply hearing the number word tells a listener what the speaker is talking about, without the object having to be named separately. This makes speech shorter and removes ambiguity.

   2. Different things are naturally grouped differently. Objects that are always handled in pairs, bunches or bundles — fish on a line, coconuts in a cluster — are easier to count in those natural group sizes, so a separate sequence suited to each kind of object is more practical.

   3. Counting was tied to particular activities. Numbers grew out of real needs such as trade, fishing and ritual rather than as an abstract idea, so each activity developed its own vocabulary before any single general-purpose sequence emerged.

   The deeper point the chapter is making is that the idea of a number as something abstract — the same 'five' whether it counts fish or days — is itself an invention. It came later, and it is exactly what makes one universal number system possible.

   ✦ Because the number name also identifies what is being counted, different objects are naturally grouped in different sizes, and counting grew out of separate practical activities before the abstract idea of number emerged.

2. 24 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 60-61

   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 arithmetic operations (+, −, ×, ÷) in this system, without using Hindu numerals, and evaluate: (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)

   Hint. In this system urapon = 1 and ukasar = 2, and a number name is just a list of 2s with at most one 1 at the end.

   Step 1 — Read the system. urapon = 1 and ukasar = 2, and every number is written as a string of ukasars with at most one urapon at the end, since two urapons would simply be one ukasar.

   The method for each operation: • Add — write the two strings side by side, then tidy up: if two urapons appear, replace them by one ukasar. • Subtract — cross off ukasars from the larger string one for one against the smaller; if an urapon must be taken from an ukasar, split that ukasar into two urapons first. • Multiply — repeat the first string as many times as the second string counts, then tidy up. • Divide — split the string into equal parts of the given size and count how many parts are formed.

   Step 2 — Evaluate each.

   (i) First string = 2+2+2+2+1 = 9; second = 2+2+2+1 = 7. Sum = 16. The two urapons combine into one ukasar, so the answer is eight ukasars: ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar (8 ukasars = 16 ✓)

   (ii) 9 − 6 = 3. Removing three ukasars from four ukasars and an urapon leaves one ukasar and one urapon: ukasar-urapon (2 + 1 = 3 ✓)

   (iii) 9 × 4 = 36, and 36 is eighteen 2s with nothing left over, so: ukasar repeated 18 times (18 × 2 = 36 ✓)

   (iv) 16 ÷ 4 = 4. Splitting eight ukasars into groups of two ukasars gives four such groups, and 4 is written as two ukasars: ukasar-ukasar (2 + 2 = 4 ✓)

   Notice that the Gumulgal themselves called anything above 6 simply ras; this question asks what would happen if the counting-by-2s idea were carried on indefinitely.

   ✦ (i) ukasar × 8 (= 16) (ii) ukasar-urapon (= 3) (iii) ukasar × 18 (= 36) (iv) ukasar-ukasar (= 4)

3. 33 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 61

   Identify the features of the Hindu number system that make it efficient when compared to the Roman number system.

   Hint. Think about three things: what a symbol's position means, whether there is a symbol for nothing, and how easy calculation is.

   Hindu number system	Roman number system
   Place value — the same digit means different amounts in different positions (the 3 in 375 means 300, the 3 in 37 means 30)	Fixed value — X always means exactly 10, wherever it is written
   Has a symbol for zero, used both as a placeholder and as a number in its own right	No symbol for zero at all
   Calculation is easy — addition, subtraction, multiplication and division can all be done column by column	Calculation is extremely difficult, especially multiplication and division
   Only ten symbols are needed, and they suffice for every number however large	New symbols keep being needed (I, V, X, L, C, D, M …) as numbers grow

   Why these features matter:

   Because position carries meaning, ten digits are enough to write any number at all — there is no ceiling and no need to invent fresh symbols, which is exactly the wall the Roman system runs into above M.

   Because there is a digit for zero, an empty place can be marked unambiguously, so 375, 3075 and 30075 can never be confused.

   Because every regrouping is by the same amount — always ten — carrying works the same way at every column. In the Roman system the regrouping size keeps alternating between 5 and 2, which is what makes its arithmetic so laborious.

   ✦ Place value instead of fixed value; a symbol for zero; easy calculation; and only ten symbols needed for every number however large.

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.