3. The World of Numbers

17 topics3 h

Chapter 3 of NCERT Mathematics for Class 9: The World of Numbers. Six sections — Before there were numerals, Zero, and the integers it made possible, Rational numbers fill the line — almost, The gaps rationals cannot fill, The decimal expansion as a signature and The real line, and what lies past it. 17 videos, 3 h in all.

Worked answers to the 43 exercise questions in this chapter →

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Part 1

Before there were numerals

9 minOne-to-one correspondence: counting without number wordsA herder with no word for any number still has to know, before dark, that every animal is back. One pebble out, one pebble in — and it works.9 min · 12 partsहिंदी10 minWhy India needed names for powers of tenA name for a power of ten is not vocabulary. It is infrastructure, and a warehouse keeping its accounts in tally strokes is what happens without it.10 min · 12 partsहिंदी

Part 2

Zero, and the integers it made possible

10 minFrom śhūnyatā to śhūnya: turning nothing into a number you can compute withA placeholder and a number are not the same thing, and what separates them is rules. 205 needs nothing in the tens column — and nothing had to become a number.10 min · 12 partsहिंदी10 minDebts and fortunes: negative numbers close subtractionNegative numbers were not invented to be strange. They were invented because subtraction was broken.10 min · 12 partsहिंदी10 minWhy a debt times a debt is a fortuneTwo negatives make a positive is the most memorised and least understood rule in school arithmetic. It is not a convention anybody agreed to.10 min · 12 partsहिंदी

Part 3

Rational numbers fill the line — almost

9 minWhat "rational" means, and why the denominator cannot be zeroEveryone can recite that you cannot divide by zero. Almost nobody can say why — and “it is not allowed” is a rule with the reason removed.9 min · 12 partsहिंदी9 minPlacing a rational number on the line, and distance as |a − b|Where exactly is a fraction on the number line? Not roughly — exactly. There is a construction, and it has two instructions and a direction.9 min · 12 partsहिंदी9 minDensity: averaging always finds another rational in betweenBetween any two fractions there is a third, and you never have to hunt for it. The average is a machine that manufactures one on demand.9 min · 12 partsहिंदी

Part 4

The gaps rationals cannot fill

9 minProof by contradiction: why √2 cannot be a ratio of integersDraw a square one unit on a side and draw its diagonal. That length cannot be written as one whole number over another — not hard, impossible.9 min · 12 partsहिंदी9 minConstructing an irrational length and marking it on the number lineYou cannot reach √2 by cutting the unit into equal parts, because every such cut gives a fraction. A compass reaches it in one arc.9 min · 12 partsहिंदी10 minπ: from Āryabhaṭa's approximation to Mādhava's infinite seriesEvery circle gives back the same number when you divide its circumference by its diameter, and nothing about a circle promises that.10 min · 12 partsहिंदी

Part 5

The decimal expansion as a signature

10 minWhy long division must either stop or loopThree eighths stops. Five elevenths never does. The usual question is which fractions do which — the better one is why those are the only two options.10 min · 12 partsहिंदी10 minPredicting the expansion from the denominator's prime factors0.375 is not a different kind of object from a fraction. It is a fraction whose bottom is a power of ten, and that fact decides everything.10 min · 12 partsहिंदी11 minConverting a terminating or repeating decimal back to p/qTurning 0.35 into a fraction takes no cleverness — it stopped, so it has a last place to name. Now try 0.666…, which has no last place at all.11 min · 12 partsहिंदी9 minCyclic numbers: the hidden symmetry inside 1/7One seventh repeats every six digits. Double it, treble it, and the same six digits come back in the same order, just starting somewhere else.9 min · 12 partsहिंदी9 minIrrational decimals: an expansion with no stop and no repeating blockOne third never stops either, and one third is about as rational as a number gets. So “never ends” cannot be what makes a number irrational.9 min · 12 partsहिंदी

Part 6

The real line, and what lies past it

10 minUniting rationals and irrationals into an unbroken lineFill the line with every fraction you can think of and they crowd in impossibly tightly. It feels full. It is riddled with holes.10 min · 12 partsहिंदी

What you will be able to do

  • Explain how pairing one token to one object decides whether a collection is complete, without either quantity being counted or named
  • Explain why standardised weights and long-distance trade generate quantities that a tally cannot record usefully
  • Distinguish a positional placeholder from a number, and say which the Babylonians and Mayans had
  • Explain why the natural numbers, and even the whole numbers, cannot answer every subtraction question
  • State the two printed product rules for signed numbers and give the chapter's instances of each
  • State what makes a number rational, and write integers and whole numbers in that form
  • Set up a number line with an origin and equally spaced integers
  • State what it means for a set of numbers to be dense
  • Derive the length of a unit square's diagonal from the Baudhāyana–Pythagoras Theorem
  • Explain why the subdividing method of §3.4.1 cannot locate √2
  • State what π measures about a circle
  • Carry out long division past the decimal point and record the remainder at each step

What this chapter assumes you already know

  • Counting a small collection aloud, and the idea that two collections can be the same size
  • Addition and subtraction within the whole numbers
  • What a prime number is, and the ability to test small numbers for primality
  • Doubling
  • Set notation of the kind {1, 2, 3, …} — braces, commas, and an ellipsis meaning "and onwards"
  • Reading and writing large numbers, and the Indian grouping into thousands, lakhs and crores
  • What a power means, and that 10³ is 10 × 10 × 10
  • That each place in a written numeral is worth ten of the place to its right

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "Counting means saying the number names."
  • "The pot tells the herder how many cows there are."
  • "Tally marks are just primitive counting, with nothing mathematical in them."
  • "29 notches means someone counted to 29 and stopped."
  • "11, 13, 17, 19 — they go up by 2 then 4 then 2, so the next is 21."
  • "Subtraction always works, we just have not learned the answer yet."
  • "Everyone has always counted in tens."
  • "Big numbers were named because ancient people liked big numbers."