Counting came first
Notches on a bone, tallies of cattle. Where mathematics began.
The natural numbers $1, 2, 3, \ldots$ answer one question: how many?
↳ Each new family of numbers exists because the previous one could not say something.
📚 Chapter Learning Kit
The World of Numbers · Class 9 Mathematics
English 3 topics 178 leveled MCQs 60 flashcards 4 games Free
Shared by a Veda teacher · Generated with Veda AI · Oct 2026
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Brahmagupta and shunya
Treating nothing as a number to calculate with.
Adding zero to the counting numbers gives the whole numbers.
↳ Zero as a working number, not merely an empty column, is an Indian contribution.
The integers
Debts and fortunes, in Brahmagupta’s own framing.
The whole numbers together with the negatives give $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$
↳ Integers let you count backwards past zero as well as forwards.
Only two things, ever
Carry out the division a fraction asks for. What can happen?
$\tfrac{3}{8} = 0.375$ stops after three digits.
↳ Every rational number either terminates or repeats, with no third option.
A remainder of zero
What makes a division stop?
Dividing 3 by 8, the remainders run 3, 6, 4, then 0.
↳ Reaching a zero remainder is exactly what terminating means.
A remainder that comes back
And what makes it loop?
Dividing 5 by 11, the remainders run 5, 6, 5, 6 and never reach zero.
↳ A repeated remainder forces a repeated block of digits.
📖 Smart notes
What you'll study, topic by topic
1
Rational Numbers on the Number Line
Counting gave the natural numbers, zero and the negatives gave the integers, and sharing gave the rationals: every number writable as one integer over another. Each has its place on the number line, and between any two o…
~40 min · full explanation, examples & memory tricks in the app
2
Decimal Expansions of Rational Numbers
Divide out any fraction and only two things can happen: the division reaches a zero remainder and stops, or a remainder recurs and the digits loop for ever. Which one you get is decided entirely by the prime factors of t…
~45 min · full explanation, examples & memory tricks in the app
3
Irrational Numbers and the Real Line
The diagonal of a unit square has a length that no fraction can express, and proof by contradiction shows exactly why. Such numbers are irrational, their decimals never end and never repeat, and they can still be constru…
~45 min · full explanation, examples & memory tricks in the app
❓ Leveled MCQ practice
Try the smart MCQs from this kit
178 questions laddered from warm-up to topper-level, each with an explanation. A taste:
What is the average of $\tfrac{1}{3}$ and $\tfrac{1}{2}$?
Intermediate
A $\tfrac{2}{5}$
B $\tfrac{5}{6}$
C $\tfrac{5}{12}$
D $\tfrac{1}{5}$
Show answer & explanation
$\tfrac{5}{12}$
$\tfrac{1}{3} + \tfrac{1}{2} = \tfrac{5}{6}$, and half of that is $\tfrac{5}{12}$.
Find a rational number lying between $\tfrac{2}{7}$ and $\tfrac{4}{7}$.
Beginner
A $\tfrac{3}{7}$
B $\tfrac{6}{7}$
C $\tfrac{1}{7}$
D $\tfrac{8}{7}$
Show answer & explanation
$\tfrac{3}{7}$
Their average is $\tfrac{6}{7} \div 2 = \tfrac{3}{7}$, which sits neatly between them.
What is the average of the integers 3 and 4?
Beginner
A $\tfrac{3}{4}$
B $7$
C $\tfrac{7}{2}$
D $12$
Show answer & explanation
$\tfrac{7}{2}$
$(3 + 4) \div 2 = \tfrac{7}{2}$, which is a rational number between them.
Which rational number lies exactly halfway between $-\tfrac{3}{4}$ and $-\tfrac{1}{4}$?
Intermediate
A $-1$
B $\tfrac{1}{2}$
C $-\tfrac{1}{2}$
D $-\tfrac{3}{8}$
Show answer & explanation
$-\tfrac{1}{2}$
Their sum is $-1$, and half of that is $-\tfrac{1}{2}$.
🃏 Flashcards
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60 flashcards in this kit — the app reviews them with spaced repetition so the right card returns on the right day.
🎮 Learning games
Play your way through this kit
Every game is built from this kit's own content — scores feed your mastery, so playing counts as studying.
Word Match
True False
Fill Blank
Word Scramble
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