What "base n" means, and why ten is a choice not a law

What to assume they know

• The Egyptian system, and what a landmark number is for — the Egyptian rule, and landmarks as powers of ten
• Roman numerals: grouping a number into tens, fives and ones — why regrouping in the Roman system needs a different rule at each landmark
• Index notation and the law that 10^a x 10^b = 10^(a+b)
• The distributive law in the form (a + b) x n = an + bn
• Column addition with carrying, in the Hindu system

What they should be able to do

• Build the landmarks of a base-5 system from the same rule that builds the Egyptian ones
• State the two-clause definition of a base-n number system and test a given system against it
• Write a number in base-5 signs, and read one back
• Add two numerals in a base system by collecting like signs and regrouping at the base
• Show that the product of two landmark numbers in a base system is again a landmark, and use it to multiply
• Explain, using the distributive law, why multiplying a whole numeral by the base shifts every sign up one landmark
• Read a number off a decimal counting board where a raised counter is worth five

Where it usually goes wrong

• "Base 10 is the correct base." The chapter builds base-5 from the identical rule and every property carries over. Ten is a decision about symbols, not a fact about numbers.
• "Changing the base changes the number." It changes only the writing. 143 is the same quantity whether it is written as three digits in base 10 or as seven signs in base 5.
• "Carrying is a rule you memorise." It is one instruction — n of a sign becomes one of the next — and it is the same instruction at every place precisely because consecutive landmarks stand in a constant ratio. The whiteboard on Part I p.66 shows the identical move in three systems.
• "Multiplication is repeated addition, so nothing new happens here." What is new is that a product of two landmarks is a landmark. That single closure property is what turns multiplication into bookkeeping, and it fails in Rome.
• "Any system with symbols for big numbers has a base." Rome has symbols for 1, 5, 10, 50, 100, 500, 1000 and no base: the ratios alternate between five and two, so clause (b) of the definition fails.
• "A base means the numerals never run out." They still do. The Egyptian sign set stops at 10⁷, and one of the chapter's own products on Part I p.67 lands at 10¹⁰. That is the next topic's problem.
• "On the abacus every counter is worth one." A counter above a line is worth five. Six ones on the board is one raised counter plus one on the line, not six counters.

Questions to check understanding

• Convert a number to base-5 signs and back
• Test a given landmark list against the two-clause definition and say which clause fails
• Compute the landmark numbers of base-7, base-4 and base-n
• Add two numerals in a stated base, showing the regrouping
• Multiply two landmark numbers in a base system and name the resulting landmark
• Multiply a multi-sign numeral by the base, justifying the step with the distributive law
• Read a number off a drawn abacus, including a raised counter, and set a given number on a blank board

Examples worth working on the board

Items marked counted were read from the printed page; the sign-based numerals do not appear in the extracted text.

• The base-5 construction (Part I pp.62–63). First landmark 1; bundle five of it for 5; bundle five of those for 5 x 5 = 25; bundle five of those for 5 x 25 = 125. The chapter then prints the powers 5⁰ = 1, 5¹ = 5, 5² = 25, 5³ = 125, 5⁴ = 625, 5⁵ = 3125 with a sign under each: triangle, square, hexagon, circle, wave, upward arrow, in that order. The pebble artwork above it shows one counter, a group of five, and a group of five such groups.
• 143, worked on the page (Part I p.63): 143 = 125 + 5 + 5 + 5 + 1 + 1 + 1, written as one circle, three squares and three triangles. Both the sum and the numeral are printed.
• The definition (Part I p.63), in two clauses: (a) the first landmark is 1; (b) each next landmark is the current one multiplied by some fixed number n. A system satisfying both is a base-n system. The Egyptian system is base-10; the constructed one is base-5; a base-10 system is also called decimal.
• The general result (Part I p.63): in any base-n system every landmark is a power of n, with the list opening at n⁰ = 1.
• Figure it Out (Part I p.63), three items: write 15, 50, 137, 293 and 651 in the base-5 signs; say whether any number cannot be written in base-5 and why; compute the landmarks of base-7 and then of base-n. Hand over the five numbers and the two questions.
• The Egyptian addition worked over two pages (Part I pp.64–65). The two addends, counted off the page: the first is eight arches and seven strokes, the second is seven arches and eight strokes. The page states the collected totals as fifteen arches and fifteen strokes. Ten arches then make one coil, and ten strokes make one arch. A printing slip lives here — see Notes.
• Figure it Out (Part I p.65), item 1, two Egyptian additions. Counted: (i) a nine-sign block of hooked stems, six coils and eight strokes, to be added to five coils and seven strokes; (ii) one hooked stem with eight arches, to be added to four arches and six strokes. Item 2, one base-5 addition: circle, hexagon, hexagon, square, triangle, triangle (six signs), added to circle, circle, circle, hexagon, square, square, triangle, triangle (eight signs). Both addends verified on the printed page — the fourth rounded shape in the second addend is a hexagon, not a circle, which is easy to misread at page scale and changes that addend's value. Hand the numerals over as described; the totals are the exercise.
• The whiteboard (Part I p.66). One sum, 47 + 56 = 103, set out three times: as Hindu columns with the carried 1 ringed; as Egyptian signs with the carry drawn as an arrow; and as base-5 signs with five squares ringed and replaced by one hexagon. Verified: 47 is one hexagon, four squares and two triangles; 56 is two hexagons, one square and one triangle; the sum is four hexagons and three triangles, with the squares vanishing entirely. This panel is the single best asset in the chapter for showing that carrying is one idea, not three.
• Multiplying by a landmark (Part I pp.66–68). The chapter's own examples, described by their powers: multiply each landmark by 10, then by 10², then a mixed set. Read off Part I p.67: the four products set there are 10 x 10⁵, 10² x 10³, 10³ x 10³ and 10⁴ x 10⁶. The last of these is 10¹⁰, which the eight-sign Egyptian set cannot write — a fact the chapter does not remark on.
• The distributive step (Part I p.67), worked twice. First, 200 x 10 treated as (100 + 100) x 10, giving 1000 + 1000. Then a three-term case: 121 x 10 treated as (100 + 20 + 1) x 10, with the chapter noting that the law extends from two terms to three for the same reason it holds for two.
• Two more products (Part I p.68), counted: (i) a numeral of five coils, two arches and two strokes, multiplied by one arch; (ii) a numeral of one hooked stem and one arch, multiplied by one arch. The second is worth flagging: its answer needs the 10⁴ sign, which neither factor uses.
• The abacus (Part I pp.68–69). Around the 11th century, users of Roman numerals took up a board built on the decimal system: horizontal lines, the lowest standing for 1 and each line above it for the next power of ten. A counter on a line counts once; a counter placed above a line counts five. Fig. 3.1 shows 3426, and the page invites the reader to notice how the six ones are done. Counted on the figure: three counters on the 1000 line, four on the 100 line, two on the 10 line, and for the ones, one counter floating above the 1 line together with one counter on it.
• The board addition (Part I p.69): 2907 + 43, the two numbers set on either side of a vertical divider. Counted on the left board: left of the divider, two counters on the 1000 line; one counter above the 100 line and four on it; the 10 line empty; one counter above the 1 line and two on it. Right of the divider, four counters on the 10 line and three on the 1 line. The chapter's hint is that the seven ones and the three ones make ten ones, which sends one counter to the 10 line. Counted on the right-hand board: two counters on the 1000 line, one above the 100 line and four on it, one above the 10 line, and nothing on the 10 or 1 lines — which reads as the completed sum. Anyone who wants to leave the exercise open should show only the left board.

Figures to have open

• The two-column bundling figure for section 1, running the identical rule at ten and at five. An added construction, and it is the argument.
• The six base-5 signs with their powers (Part I p.63). Simple shapes; redraw.
• The three-panel whiteboard, 47 + 56 in Hindu columns, Egyptian signs and base-5 signs, with the carry synchronised across all three. This is the chapter's own figure (Part I p.66) and section 6 cannot be taught without it. Redraw larger; the printed panel is small and the base-5 column is hard to read.
• The abacus board with four labelled lines and the raised-counter convention shown explicitly (Part I p.68, Fig. 3.1). Standard schematic, but the raised counter must sit visibly between two lines.
• The two boards of the addition (Part I p.69) with the divider drawn. Show the left board at minimum.
• A power axis for section 8 marked 1 through 10⁷ with a dotted extension to 10¹⁰, so the product that falls off the end is visible.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 3, "A Story of Numbers", §3.3 "The Idea of a Base", subsection II. Variations on the Egyptian System and the Notion of Base (Part I pp.62–63), and the two unnumbered subheadings that follow it, "Advantages of a Base-n System" (Part I pp.64–68) and "Abacus that Makes Use of the Decimal System" (Part I pp.68–69). These two can be named but not cited by number.
• Figure it Out sets at Part I p.63 (three items) and Part I p.65 (two items).
• Fig. 3.1 at Part I p.68 is the only numbered figure in the chapter.
• The chapter's SUMMARY (Part I p.81) states the base-n definition in its fourth bullet.
• Forward pointer: the symbol-set ceiling this topic keeps brushing against is argued at Part I pp.69–70 and belongs to Why a base alone still runs out of symbols.