Figure it Out — The Mechanism of Counting — A Story of Numbers

1. 13 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 54

   Suppose you are using the number system that uses sticks to represent numbers, as in Method 1. Without using either the number names or the numerals of the Hindu number system, give a method for adding, subtracting, multiplying and dividing two numbers or two collections of sticks.

   Hint. In this system a number simply *is* a collection of sticks, so every operation has to be described as something you physically do to the collections.

   In this system there are no numerals at all — a number is nothing more than a collection of sticks. So each operation has to be carried out physically on the collections themselves.

   Addition. Take the two collections and push them together into a single heap. That combined heap is the sum, because adding is exactly the act of counting two collections together as one.

   Subtraction. Starting from the larger collection, remove one stick for every stick in the smaller collection, pairing them off one to one. Whatever is left over is the difference, since the paired sticks are precisely the part being taken away.

   Multiplication. Make as many equal collections as the second number tells you, each one the size of the first collection, then push them all together. Multiplication is repeated addition of equal groups, so the total heap is the product.

   Division. From the given collection, keep pulling out groups of the required size until you cannot form another full group. The number of complete groups formed is the quotient, and any sticks left over that cannot fill a group form the remainder.

   Notice that every one of these is done by one-to-one matching and grouping alone — no number name or written symbol is needed anywhere.

   ✦ Add by combining the heaps; subtract by pairing off and keeping the leftover; multiply by making that many equal heaps and combining; divide by splitting into groups of the given size and counting the groups.

2. 22 marksGanita Prakash Cl-8 Part 1, Figure it Out, page 54

   One way of extending the number system in Method 2 is by using strings with more than one letter — for example, we could use 'aa' for 27. How can you extend this system to represent all the numbers? There are many ways of doing it!

   Hint. The alphabet runs out after 26. The trick is to agree on a rule that keeps producing fresh, never-repeating letter strings forever.

   Step 1 — See where the problem lies. The English alphabet has only 26 letters, so single letters run out at 26. To go further we need longer strings, and the only requirement is that the rule never produces the same string twice and never stops.

   Step 2 — One simple extension (the book's own). After a, b, c, …, z, continue with the doubled letters aa, bb, cc, …, zz; then the tripled letters aaa, bbb, ccc, …, zzz; and so on. This gives 1–26 as single letters, 27–52 as doubled letters, 53–78 as tripled letters, and so on without end. Since every fresh block of 26 uses one more repetition than the last, no string can ever be repeated, and every number gets a name.

   Step 3 — Another way worth noticing. We could instead run through all two-letter strings in dictionary order — aa, ab, ac, …, az, ba, bb, …, zz — which alone covers 26 × 26 = 676 more numbers, then all three-letter strings, and so on. This version is far more economical, because it reuses the alphabet in every position instead of only repeating one letter.

   Either rule works; the second is closer in spirit to how a place value system behaves, since the position of each letter carries meaning.

   ✦ Continue a, b, …, z with aa, bb, …, zz, then aaa, bbb, …, zzz, and so on — each fresh block of 26 uses one more repetition, so every number gets a unique name.

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.