Why a base alone still runs out of symbols

What to assume they know

• What "base n" means, and why ten is a choice not a law — the two-clause definition of base-n, and why regrouping works the same way at every landmark
• The Egyptian system, and what a landmark number is for — the eight Egyptian signs and the powers they carry
• Reading 10⁷ as a crore, and the Indian names up to crore
• Division with remainder, for the base-4 construction

What they should be able to do

• State the largest number the eight printed Egyptian signs can write, and why
• Explain why no sign in a base-10 system is ever needed ten or more times
• Distinguish the length of a numeral from the size of the sign set it draws on, and say how each grows
• Build a base-4 system from the standard rule and write 1 to 16 in it
• Identify the point at which a new sign becomes necessary in any base
• Give the rule for multiplying by the base in a base-5 system, and say why it is the same rule as appending a zero
• State the problem that place value is about to solve, in your own words

Where it usually goes wrong

• "A base means you can write any number." Not in a system where each landmark needs its own sign. Base-10 grouping is fine; it is the sign per power that runs out.
• "Just invent more signs." That is the move the chapter rejects, and it rejects it on principle rather than on effort: there is no end to the powers, so there is no end to the signs, so the problem is not solved but postponed.
• "Ten of a sign is allowed, it's just untidy." It is not a numeral at all under the system's own rule. Ten of any sign is one of the next, and a writer who leaves ten standing has stopped halfway.
• "Longer numerals mean a bigger sign set." These are independent. 1111 has a four-mark numeral drawing on four signs; 9999 has a thirty-six-mark numeral drawing on the same four. What changes ninefold is the number of marks; the sign set is fixed by the number of places, which is the rule derived two bullets above.
• "The Egyptians ran out because they were early." They ran out because of the design. Any system that gives each power its own sign runs out, whenever it lives.
• "Base-4 will need fewer signs because 4 is smaller." It needs more, and sooner: base-4 reaches its third landmark at 16 where base-10 reaches its third at 100. Smaller base, shorter runs of the same sign, faster growth in the sign set. This trade is worth drawing.

Questions to check understanding

• State the largest number writable in a given sign-per-power system, and justify the bound
• Explain why no sign can appear as many times as the base
• Given a base, say at which number the third landmark sign becomes necessary
• Construct a small base-n system and write a stated range of numbers in it
• Give the rule for multiplying by the base in a stated base, and connect it to appending a zero in decimal
• Explain in your own words what problem place value is going to solve

Examples worth working on the board

Values marked worked by me are worked out here on the chapter's stated inputs; the chapter prints no answers in this subsection.

• The stated verdict (Part I p.69): the Egyptian system managed reasonably efficient representation for numbers up to a crore, written there as 10⁷, along with relatively easy computation — and then had a drawback.
• The drawback, as stated (Part I p.69): to reach larger and larger numbers you must keep inventing signs for higher and higher powers of ten, an unending sequence of them. The chapter's own reading of this is that the original challenge of writing numbers down has reappeared in a new form.
• The ceiling, made concrete. Worked by me: with the eight printed signs and no sign usable ten or more times, the largest writable numeral is nine of each, that is 99,999,999 — one short of ten crore. Below that everything is fine; at that boundary the system simply has nothing to draw. Part I p.69 says the system serves "till a crore", which is the informal statement of the same fact — it describes the range the system handles comfortably, not its exact bound. Teach 99,999,999 as the bound and note the book's looser phrasing, so a student reading both is not left with two different ceilings.
• Figure it Out item 1 (Part I p.69): can there be a number whose Egyptian numeral has one of the signs occurring ten or more times, and why not. This is the question that pins down the first cost. As a check: the answer is the regrouping rule itself — ten of any sign is by construction one of the next, so a numeral written that way was simply not finished.
• Figure it Out item 2 (Part I p.70): create a base-4 system of your own and write the numbers 1 to 16 in it. Worked by me, as background: the landmarks are 1, 4, 16, so the task needs two signs to reach 15 and a third exactly at 16. That is the moment a student meets the symbol cost first-hand, and it is why 16 rather than 15 or 20 is the stopping point in the question.
• Figure it Out item 3 (Part I p.70): give a simple rule for multiplying a number by 5 in the base-5 system built earlier in the chapter. As a check: every sign moves up one landmark, which is the base-5 counterpart of writing a zero on the end of a decimal numeral — the same fact stated twice in two notations.
• The two growth rates, the heart of section 4. Worked by me, not printed: for a number N written in base 10, the count of marks is the digit sum, which never exceeds nine per place; the count of distinct signs needed is the number of places. The first is bounded per place, the second grows without limit as N grows. A base controls the first and leaves the second untouched.
• The forward promise (Part I p.69): the chapter states that the next idea both removes this problem and makes representation and computation markedly simpler. Quote the promise structurally, and let the next topic keep it.
• Numbers worth showing: 10⁷ written as 1 followed by seven zeros; 99,999,999 as the last writable numeral; 1, 4, 16 as the base-4 landmarks; and the count of distinct signs — one, two, three — needed as a base-4 numeral passes 3, then 15.

Figures to have open

• The two-counter comparison of section 4: numeral length against sign-set size, as N grows through 9, 99, 999, 9999. An added construction, and it is the argument of the topic.
• The base-4 count from 1 to 16 as a filling table, showing exactly where each new sign is forced. An added construction from the chapter's exercise.
• The Egyptian sign ladder with a wall drawn after 10⁷ (built from Part I p.62). Standard schematic.
• No page figure is strictly required; this subsection is a page of prose and three questions, and its content is an argument rather than a picture.

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 III. Shortcomings of the Egyptian System, Part I p.69, with its Figure it Out running from Part I p.69 onto Part I p.70, three items in all.
• Back-references: the eight Egyptian signs are at Part I p.62; the base-n definition and the base-5 system are at Part I p.63.
• Forward pointer: §3.4 "Place Value Representation" opens at Part I p.70 and is the promised answer; it belongs to the four topics of module m04.
• The chapter's SUMMARY (Part I p.81) does not restate this shortcoming; it states the solution instead, in its fifth bullet.