3. A Story of Numbers

Part I, printed Chapter 311 topics2 h

Chapter 3 of NCERT Mathematics for Class 8 (Part I, printed Chapter 3): A Story of Numbers. Four sections — What counting actually requires, Some early number systems, The idea of a base and Place value. 11 videos, 2 h in all.

Worked answers to the 23 exercise questions in this chapter →

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Part 1

What counting actually requires

10 minWhy any number system needs a fixed, ordered sequence of symbolsBefore a number system can do anything, it needs symbols in a fixed order that never runs short. That is a harder demand than it sounds.10 min · 13 partsहिंदी

Part 2

Some early number systems

10 minBody parts and tally marks: counting before numeralsThe two oldest counting systems are made from the only things every human already has: a body, and something to scratch. Each solves half the problem.10 min · 13 partsहिंदी10 minCounting in twos, and what number-names reveal about a culture's baseSix words for one to six, in a language you do not speak. Read them carefully and they tell you the culture counted in twos.10 min · 13 partsहिंदी10 minRoman numerals: grouping a number into tens, fives and onesEverything easy about Roman numerals and everything awful come from the same property of the sizes they group by.10 min · 13 partsहिंदी

Part 3

The idea of a base

10 minThe Egyptian system, and what a landmark number is forRome chose its landmark numbers. Egypt generated them — and that one difference decides everything that follows.10 min · 13 partsहिंदी11 minWhat "base n" means, and why ten is a choice not a lawTen is nowhere in the mathematics. The rule that builds the sizes says: start with one, then take ten of whatever you just made.11 min · 13 partsहिंदी10 minWhy a base alone still runs out of symbolsA base bounds one cost and leaves the other alone. Keeping those two apart is what makes the next thousand years of number systems make sense.10 min · 12 partsहिंदी

Part 4

Place value

10 minMesopotamian base-60: place value in a sexagesimal systemStop drawing the landmark numbers and let position say which one you mean. Mesopotamia got there more than three thousand years ago.10 min · 12 partsहिंदी10 minThe Mayan system, and a placeholder symbol for zeroPlace value and a base are two different ideas, and the Maya are the proof — they had the first without the second.10 min · 12 partsहिंदी9 minChinese rod numerals: base-10 place value, one symbol shortThis is the system you write in, with exactly one part missing. Sticks laid on a ruled board, base ten, place value, and no zero.9 min · 12 partsहिंदी10 minThe Hindu number system, and why treating 0 as a digit changed everythingZero was promoted twice, and the first promotion bought almost nothing. Only the second one made arithmetic possible.10 min · 12 partsहिंदी

What you will be able to do

  • Separate what the chapter actually requires: one defining condition on the sequence (a fixed order), one procedure for using it (pairing its terms one-to-one with the objects, in that order), and one property it then goes looking for (that the sequence be unending and convenient to say)
  • Trace the printed body-part sequence from 1 to 27 and identify the point of symmetry at 14
  • Decode a Gumulgal number name into an addition, and build the name for a given number by the same rule
  • Convert a number below 40 to Roman form by taking tens, then fives, then ones
  • State the Egyptian rule for generating landmark numbers and apply it to produce the first five
  • Build the landmarks of a base-5 system from the same rule that builds the Egyptian ones
  • State the largest number the eight printed Egyptian signs can write, and why
  • State the two marks the Mesopotamians used, and build any number from 1 to 59 from them
  • Build any number from 1 to 19 out of dots and bars
  • Write 1 to 9 in both rod forms and say which form belongs to which place
  • Read a Hindu numeral as a sum of digits times powers of ten, using the chapter's own layout

What this chapter assumes you already know

  • Reading, writing and comparing whole numbers in the Indian place value system
  • That a number can be written in figures and also said in words
  • Multiplication and division of whole numbers, for the arithmetic the chapter asks students to redo without Hindu numerals
  • Powers of ten written as 10², 10³, and read aloud
  • No prior history is assumed; the chapter supplies its own
  • That a system can be unending yet inconvenient, or convenient yet finite
  • Reading a labelled diagram in which the numbers are printed inside the artwork
  • Familiarity with tally marks from primary school, which the chapter assumes

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 one, two, three."
  • "You need numbers to compare two collections."
  • "The digits 0–9 are the numbers."
  • "They are Arabic numerals, so they came from Arabia."
  • "Roman numerals are a primitive scribble."
  • "A number system has to be written."
  • "Any list of symbols will do."
  • "Body counting is just holding up fingers."