Crores and millions: one number, two naming systems

What to assume they know

• What it actually takes to make one lakh — one lakh as 1 followed by five zeros, and the Indian comma after the thousands group
• The Land of Tens: each place is ten of the one before — that every place is worth ten of the one below it
• Reading a 6-digit number aloud
• Comparing two whole numbers using < and >

What they should be able to do

• State how many zeros follow the 1 in a lakh, a crore, a million, a billion and an arab
• Insert commas into an unpunctuated digit string in both systems and read the result aloud in each
• Convert a number given by its Indian name into digits, and by its American name into digits
• Explain why the grouping, not the number, is what differs between the systems
• Compare a quantity named in one system with a quantity named in the other by converting both to a common unit first
• State the "hundred times" chain that links thousand, lakh, crore and arab, and the "thousand times" chain that links thousand, million and billion

Where it usually goes wrong

• "A lakh is the Indian word for a million." It is not: a million is ten lakhs. This single substitution wrecks every cross-system comparison a student will meet.
• "A crore is the Indian word for a billion." A crore is ten million; a billion is a hundred crores. The chapter's own comparison item, 640 crore against 60 billion, is built on exactly this error.
• "Changing the commas changes the number." It changes only how the number is read aloud. The digits never move.
• "Indian commas go every three digits, like the American ones, just shifted." No — the first comma is after three digits, and every comma after that is after two.
• "Ten lakhs and a million are different quantities because they have different names." They are the same quantity. The chapter's table prints them on one row for exactly this reason.
• "You can compare by comparing the names." You compare by converting both to one unit. Names are labels attached to different-sized groups.

Questions to check understanding

• Punctuate a given digit string in the Indian system and write its number name
• Write the same number's name in the American system
• Convert a number given in words to digits, in either system
• Fill <, > or = between quantities named in different systems
• Convert between a unit of one system and a unit of the other — the chapter sets exactly this at Part I, §1.6, p.20, question 8, asking how many lakhs go into a billion; verified, the answer is ten thousand
• State the number of zeros in a named unit
• Competency-style: explain why a headline reading "5 million" and one reading "50 lakh" describe the same crowd

Examples worth working on the board

Values marked verified are worked out here against the printed page; the chapter prints the paired table but leaves the exercises blank.

• Where crore arrives (Part I, §1.2, p.8). The chapter asks what appears if the +10,00,000 button is pressed ten times, and answers in running text: a hundred lakhs, called a crore, written 1,00,00,000 with seven zeros. This is the one place in the section where the book supplies its own answer, and it is the bridge from §1.2 into §1.3.
• The paired table (Part I, §1.3, p.8). Seven rows. Each row carries one quantity twice over: on the left its digits and name as India writes them, on the right its digits and name as America does.
  • Row 1 — 1,000 on both sides, and a thousand either way.
  • Row 2 — 10,000 on both sides, ten thousand either way.
  • Row 3 — on the left 1,00,000, a lakh; on the right 100,000, said as a hundred thousand.
  • Row 4 — on the left 10,00,000, ten lakhs; on the right 1,000,000, said as a million.
  • Row 5 — on the left 1,00,00,000, a crore; on the right 10,000,000, said as ten million.
  • Row 6 — on the left 10,00,00,000, ten crores; on the right 100,000,000, said as a hundred million.
  • Row 7 — on the left 1,00,00,00,000, an arab, which the same cell also calls a hundred crores; on the right 1,000,000,000, said as a billion.

The digits are identical down every pair. Only the commas and the names move.

• The two grouping rules (Part I, §1.3, p.8). Indian: three digits, then two at a time, reading right to left. American: three at a time throughout, reading right to left.
• The zero counts (Part I, §1.3, p.8, and repeated in the SUMMARY, p.22). Lakh 5 zeros, million 6, crore 7, arab and billion 9.
• The hundred-times chain (Part I, §1.3, p.8). A lakh is a hundred thousands, a crore is a hundred lakhs, an arab is a hundred crores. Verified: on the American side the step is a thousand each time — thousand, million, billion.
• Two questions in running text (Part I, §1.3, pp.8–9): how many zeros a thousand lakh has, and how many a hundred thousand has. Verified: eight and five.
• One number read both ways (Part I, §1.3, p.9). 9876501234 is punctuated as 9,87,65,01,234 and read as 9 arab 87 crore 65 lakh 1 thousand 234, or equivalently as 987 crore 65 lakh 1 thousand 234; and as 9,876,501,234, read as 9 billion 876 million 501 thousand 234.
• Numbers to punctuate and name (Part I, §1.3, p.9, "Figure it Out" 1): 4050678, 48121620, 20022002, 246813579, 345000543, 1020304050. Verified Indian punctuation: 40,50,678; 4,81,21,620; 2,00,22,002; 24,68,13,579; 34,50,00,543; 1,02,03,04,050.
• Names to convert to digits (same page, "Figure it Out" 2). Four number names, three of them built almost entirely out of ones. Verified: one crore, one lakh, one thousand and ten gives 1,01,01,010; one billion, one million, one thousand and one gives 1,00,10,01,001; ten crore, twenty lakh, thirty thousand and forty gives 10,20,30,040; and nine billion, eighty million, then seven hundred thousand and six hundred gives 9,08,07,00,600.
• Four comparisons (same page, "Figure it Out" 3). Verified: 30 thousand is less than 3 lakhs (30,000 against 3,00,000); 500 lakhs is greater than 5 million (5,00,00,000 against 50,00,000); 800 thousand is less than 8 million (8,00,000 against 80,00,000); 640 crore is less than 60 billion (6,40,00,00,000 against 60,00,00,00,000). Two of the four — (b) and (d) — turn on the false-friend trap of section 10. (a) and (c) do not: neither sets a lakh against a million or a crore against a billion.
• Where the names travel (Part I, §1.3, p.8). The Indian scheme is used across the neighbourhood as well: Nepal and Bhutan, Bangladesh and Myanmar, Sri Lanka and the Maldives, Pakistan and Afghanistan. The chapter adds that the American scheme is used in many countries too.

Figures to have open

• The seven-row paired table from Part I, §1.3, p.8. It is the topic's central evidence and must be redrawn rather than reproduced — same rows, same order, with the identical digit strings made visually identical across the two halves.
• Two identical digit strips with different comma positions marked. Standard schematic, and the single most important picture here.
• A map of South and Central Asia marking India together with Nepal, Bhutan, Bangladesh, Myanmar, Sri Lanka, the Maldives, Pakistan and Afghanistan. Standard schematic; the chapter prints the list as running text, not as a map.
• No photograph is required.

Where this sits in the book

• NCERT Ganita Prakash Class 7, Part I, printed Chapter 1, "Large Numbers Around Us", §1.3 "Of Crores and Crores!", pp.8–9. §1.3 carries no subheadings.
• The paragraph introducing crore sits at the top of Part I, p.8, still inside §1.2, and is treated here as this topic's opening because it is where the word is defined.
• Part I, §1.6, p.20, question 8 — the lakhs-to-a-billion conversion — is the chapter's own late test of this material and is cited in the assessment hooks.
• Part I, SUMMARY, p.22 restates the four zero counts.