Why any number system needs a fixed, ordered sequence of symbols

What to assume they know

• Reading, writing and comparing whole numbers in the Indian place value system
• That a number can be written in figures and also said in words
• Multiplication and division of whole numbers, for the arithmetic the chapter asks students to redo without Hindu numerals
• Powers of ten written as 10², 10³, and read aloud
• No prior history is assumed; the chapter supplies its own

What they should be able to do

• Separate what the chapter actually requires: one defining condition on the sequence (a fixed order), one procedure for using it (pairing its terms one-to-one with the objects, in that order), and one property it then goes looking for (that the sequence be unending and convenient to say)
• Use a stick-per-cow pairing to decide which of two herds is larger, and by how many, without naming a single number
• Explain why a system built on the 26 English letters cannot count a collection of 30, and propose a repair
• Read and write the numbers 1 to 20 in the symbols of the chapter's Table 1
• Distinguish a number system (the sequence) from its numerals (the written signs of it)
• Say why a system may be unending yet unusable, and usable yet finite, and name a system of each kind
• Place the Indian numerals on a route — India, the Arab world, Europe — rather than at a single date

Where it usually goes wrong

• "Counting means saying one, two, three." The words are one available sequence among many. Sticks, letters and notches do the same job; the words are not what makes counting work.
• "You need numbers to compare two collections." You do not. Pair the herds stick for stick and the leftover sticks answer both "who has more" and "by how many". The chapter puts this before any arithmetic on purpose.
• "The digits 0–9 are the numbers." They are marks for numbers, and their shapes changed repeatedly on the way from Brahmi to print — the Part I p.50 chart is the evidence.
• "They are Arabic numerals, so they came from Arabia." The name records who Europe learnt them from, not who devised them. The chapter states plainly that Arab scholars themselves called them Hindu numerals.
• "Roman numerals are a primitive scribble." They are the chapter's own Method 3 and were Europe's working system for centuries. Their limits are specific — arbitrarily large numbers, and multiplication — not general.
• "A number system has to be written." Body parts and spoken names are number systems in exactly the chapter's sense; only some of them have numerals.
• "Any list of symbols will do." It must have a fixed order, or two people counting one herd will stop at different places.

Questions to check understanding

• Given two unlabelled collections, decide which is bigger and by how much using only a pairing, and write the reasoning without numerals
• State the two properties a standard sequence must have, and give a sequence that fails each
• Convert between 1–20 and the symbols of Table 1, both directions
• Explain why a 26-letter system cannot count 30 objects, and describe one way of extending it
• Distinguish "number system" from "numeral" with an example of a system that has no numerals at all
• Short-answer history: name the manuscript in which numbers first appear written with ten signs including a zero, and say why Europe called the numerals Arabic

Examples worth working on the board

Items marked counted were read off the printed page image, not from type.

• The opening strip (Part I p.48). Under the father's speech sits a bar of nine ruled cells, each holding a Mesopotamian numeral built from repeated wedges. Counted: the wedge counts run 1 through 9 across the nine cells. The internal stacking is not identical to the definitive table later in the chapter (Part I p.71) — on Part I p.48 I read 7 as three-plus-three-plus-one where Part I p.71 sets it as four-plus-three. Redraw from Part I p.71 and use Part I p.48 only as the hook.
• Reema's four questions (Part I p.48, thought bubbles): since when have humans been counting; what did they need counting for and what were they counting; since when have people written numbers in the modern form; how would a Mesopotamian have written 20, 50, 100. The last of these is answered in §3.4, not here — say so rather than leaving it hanging.
• The route of the Indian numerals (Part I pp.49–50). Dates as printed: number names on powers of ten in the Yajurveda Samhita, listed up to 10¹² and beyond, with eka, dasha, shata, sahasra, āyuta given as one, ten, hundred, thousand, ten thousand; the ten-digit way of writing numbers grew up in India roughly two thousand years back; the Bakhshali manuscript, dated to about the 3rd century CE, its zero written as a dot; Aryabhata c. 499 CE; transmission to the Arab world by about 800 CE; Al-Khwārizmī's book c. 825 and Al-Kindi's c. 830; onward to Europe and parts of Africa by about 1100 CE; Fibonacci arguing for them around 1200; general adoption only by the 17th century.
• The evolution-of-digits chart (Part I p.50, artwork). Eight labelled boxes linked by arrows. Five carry ten glyphs; Brahmi, West Arabic (Gubar) and 11th century (Apices) carry only nine, having no zero — which matters, because the chapter's argument is about when a zero glyph appears, so drawing a Brahmi zero would be an active error. In order: Brahmi → Hindu (Gwalior) → Sanskrit-Devanagari, then branching to West Arabic (Gubar) and East Arabic, then 11th century (Apices), then 15th century and 16th century (Dürer). Use the labels and the branching; do not trace the shapes.
• The herd (Part I p.51). Three questions posed with no numbers permitted: have all the cows come back; does the neighbour have more; if so how many more. The drawing beneath them shows counted eight cows above eight sticks, one stick per cow, the sticks planted in a row.
• The two representation tables (Part I p.52). Sticks: 1 to 5 shown as one to five strokes, then a vertical ellipsis. Letters: 1 to 5 shown as a to e, ellipsis, then 26 shown as z. The ceiling is 26 and the page says so.
• Table 1 (Part I p.53). Two banks of ten columns: 1–10 against I, II, III, IV, V, VI, VII, VIII, IX, X, and 11–20 against XI, XII, XIII, XIV, XV, XVI, XVII, XVIII, XIX, XX. This is the chapter's only numbered table with a printed caption, and later sections refer back to it by that name.
• Figure it Out (Part I p.54), three items, all open-ended: devise addition, subtraction, multiplication and division for collections of sticks; extend the letter system past 26 using strings such as aa for 27; invent a system of your own. Item 2 is the seed of place value.

Figures to have open

• The pairing diagram: a row of animals above a row of marks, with the joining lines drawn. This is the chapter's own picture (Part I p.51) and the argument of sections 5 and 6 lives in the joining lines, so redraw it as a schematic with the links visible rather than reproducing the artwork.
• Table 1 as a two-row strip, 1–20 over I–XX (Part I p.53). Standard schematic.
• The digit-evolution chart as a labelled tree with eight nodes and the branch after Sanskrit-Devanagari (Part I p.50). The labels carry the argument; the glyph shapes can be suggested rather than drawn accurately.
• A three-by-two verdict grid for section 10: sticks, letters, written signs against "goes on forever" and "short enough to use". Standard schematic.
• No photograph is needed. The Bakhshali manuscript can be named rather than shown.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 3, "A Story of Numbers", §3.1 "Reema's Curiosity", Part I pp.48–54. Inside §3.1 the chapter prints one unnumbered bold subheading, "The Mechanism of Counting" (Part I p.51), which can be named but not cited by number.
• Table 1 is at Part I p.53 and is referred to by that name at Part I pp.57, 58.
• The Figure it Out set for this section is at Part I p.54, three items.
• The chapter's SUMMARY (Part I p.81) states the definitions of number system and numeral in its first two bullets.
• Part I p.50 carries a quotation from Pierre-Simon Laplace (1749–1827) on the ten-symbol method. Reference it by speaker; do not reproduce it.
• Forward pointers inside the same chapter: the Mesopotamian numerals Reema is actually looking at are set out at Part I p.71 (§3.4 I); the Roman system of Table 1 is developed at Part I pp.58–60 (§3.2 IV).