Naming the powers of ten: the Lalitavistara list, the million-to-decillion names, and the googol

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What to assume they know

The laws of exponents, and where each one comes from — that multiplying powers of one base adds the exponents
Scientific notation, and why the standard form is 1 ≤ x < 10 — scientific notation and standard form
Why the nearest power of ten is the only handle on a quantity too big to picture — lakh, crore, arab, kharab, neel, padma as they are used on the animal ladder, one page before they are defined
Reading a long number in both the Indian and the international grouping

What they should be able to do

Read a naming ladder and state the step it uses, in exponent terms
Convert between a name and a power of ten, in both systems
Show why a hundred lakhs is a crore using the addition of exponents
Compare the Indian and the international ladders and say where they agree and where they part
Continue either ladder for one or two further names, given its step
Identify the counting prefix inside an international name and match it to the exponent
State what a googol and a googolplex are, in powers of ten
Place a historically named number on a ladder of powers of ten

Where it usually goes wrong

"Indian and international names are two ways of saying the same thing." They step differently. A hundred of one Indian name gives the next; a thousand of one international name gives the next. Only where the ladders happen to coincide — arab and billion, both 10⁹ — do the two systems name the same number.
"A billion is a lakh crore, or some fixed multiple I should memorise." Convert through exponents and no memorising is needed: 10⁹ against 10⁷ is two rungs, so an arab is a hundred crore, and an arab is a billion. Do it once with exponents and the relations stop needing storage.
"These old names are curiosities." They encode the same exponent arithmetic being taught in this chapter, and three of them — kharab, neel, padma — are used as working units on the previous page.
"A googolplex is a googol times a googol." It is 10 raised to a googol, an exponent tower, not a product. The distinction is the whole reason the name exists.
"A googol is the biggest number with a name." The chapter prints asaṅkhyeya at 10¹⁴⁰, which is larger, and a googolplex, which is beyond comparison with either.
"A hundred-trillion-dollar note means the country was rich." It means each dollar was worth very little. The chapter gives the exchange figure — about thirty US dollars — precisely so the point is unmissable.
"The names go up smoothly." The Lalitavistara list names odd powers only. Different lists choose different steps, and reading the step is the skill.

Questions to check understanding

Convert a number given in Indian names into a power of ten, and back
Convert between the Indian and the international name for one quantity
Given a naming ladder, state its step and continue it by one or two names
Write a quantity such as 30 kharab in scientific notation
Say which of two named numbers is larger and by what factor
Given an international name, identify the counting prefix and the exponent it implies
Express a googol and a googolplex as powers, and say which of two very large named numbers is bigger
Explain why a hundred lakhs is a crore, using the addition of exponents rather than counting zeros

Examples worth working on the board

The Lalitavistara (Part I p.42). A Buddhist treatise of the first century BCE, carrying names for the odd powers of ten as far as 10⁵³. The chapter quotes a passage from a dialogue between a mathematician named Arjuna and Prince Gautama, the Bodhisattva. Its named steps, as printed: ayuta at 10⁹, niyuta at 10¹¹, kankara at 10¹³, then an ellipsis, then visamjna-gati at 10⁴⁷, sarvajna at 10⁴⁹, vibhutangama at 10⁵¹ and tallakshana at 10⁵³. The construction each time is a hundred of the previous name. Do not read the quoted passage aloud; describe the pattern.
Three more lists (Part I p.42). Mahaviracharya's Ganita-sara-sangraha gives 24 terms, reaching 10²³. An anonymous Jaina treatise, the Amalasiddhi, names every power of ten as far as 10⁹⁶, the top one called dasha-ananta. A Pali grammatical treatise of Kāccāyana reaches 10¹⁴⁰, the top name being asaṅkhyeya.
Names built from a base word (Part I p.43). Jaina and Buddhist texts build high powers on base words: sahassa for a thousand and koṭi for ten million. The worked instance printed is prayuta, 10⁶, expressed as ten hundred thousand.
The Indian ladder, as printed in a four-row table (Part I p.43). Each row gives the name, the multiplication in digits, and the same statement in powers of ten:
A hundred thousand is a lakh: 100 × 1000 = 1,00,000, that is 10² × 10³ = 10⁵.
A hundred lakhs is a crore: 100 × 1,00,000 = 1,00,00,000, that is 10² × 10⁵ = 10⁷.
A hundred crores is an arab: 100 × 1,00,00,000 = 1,00,00,00,000, that is 10² × 10⁷ = 10⁹.
A hundred arab is a kharab: 100 × 1,00,00,00,000 = 1,00,00,00,00,000, that is 10² × 10⁹ = 10¹¹.
The ladder continued in prose (Part I p.43): a hundred kharab makes a neel at 10¹³, a hundred neel a padma at 10¹⁵, a hundred padma a shankh at 10¹⁷, and a hundred shankh a maha shankh at 10¹⁹.
The international ladder, as printed in a three-row table (Part I p.43), laid out the same way:
A thousand thousand is a million: 1000 × 1000 = 1,000,000, that is 10³ × 10³ = 10⁶.
A thousand million is a billion: 1000 × 1,000,000 = 1,000,000,000, that is 10³ × 10⁶ = 10⁹.
A thousand billion is a trillion: 1000 × 1,000,000,000, printed as 1,00,000,000,000, that is 10³ × 10⁹ = 10¹². See the note below — the digit string in this row is wrong.
The rest of that ladder (Part I p.43): a thousand trillion is a quadrillion, 10¹⁵, and the pattern goes on. The chapter then prints the run of names with their powers, from million at 10⁶ up through billion, trillion, quadrillion, quintillion, sextillion, septillion, octillion and nonillion to decillion at 10³³, setting the leading syllable of each in bold, and asks what that first part of each name denotes. Left open, and it is the best question on the page.
Two named giants (Part I p.43): 10¹⁰⁰ is a googol, and 10 raised to a googol is a googolplex. The chapter adds that the number of atoms in the universe is put somewhere between 10⁷⁸ and 10⁸².
Banknotes (Part I p.43). The largest denomination currently issued in India is the 2000-rupee note. The largest numerical value ever printed on a banknote anywhere was a Hungarian note of 1946 for one sextillion pengő — 10²¹, also written as one milliard bilpengő — which was never issued. In 2009 Zimbabwe printed a note for one hundred trillion Zimbabwean dollars, 10¹⁴, worth about thirty US dollars at the time.
A cross-check. The animal ladder of the previous section used kharab, neel and padma one page before this section defines them. Showing the two together — the trees at 30 kharab, the mosquitoes at 11 neel, the beetles at 1 padma — turns the naming table into something with a use rather than a list to learn.

Figures to have open

One exponent axis carrying both naming ladders, Indian names on one side and international on the other, so the two-step and three-step rhythms are visible against each other and the coincidences stand out. The chapter prints two separate tables and never sets them side by side; this figure is added here and is what the topic is for.
A historical strip placing the Lalitavistara, the Ganita-sara-sangraha, the Amalasiddhi and the Kāccāyana treatise against the highest power each reaches. Standard schematic. Note that the chapter gives a date only for the first.
A prefix-decomposition card for the international names, so that the bi-, tri-, quad- pattern can be seen rather than told. Not in the book, and it answers the chapter's own open question.
A power-tower diagram for the googolplex — 10, then 100 zeros, then that many zeros. Not in the book.
No photograph is needed. Do not reproduce images of the banknotes.

Where this sits in the book

NCERT Ganita Prakash Class 8, Part I, printed Chapter 2, "Power Play", the section printed as "2.5 A Pinch of History", Part I pp.42–43, running to the end of that section.
The Indian and international names are first used a page earlier, at Part I pp.36 and 38 (Why the nearest power of ten is the only handle on a quantity too big to picture), before they are defined here.
The exponent addition every row of both tables depends on was established at Part I p.24 (The laws of exponents, and where each one comes from).
The chapter-end "Figure it Out" at Part I p.45 item 12 turns on exactly this naming — two populations of about 10⁹ each, and six candidate totals.