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CBSE · Class 9 · Mathematics

Chapter 3: The World of Numbers

5 concepts · 40 practice questions with answers · Free to read

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Integers and their arithmetic

From daily life

Temperature 5 °C below zero is −5; a lift going up 3 floors from −2 reaches 1. Gains and losses, floors above and below ground, all use integers.

Simple definition

Integers are …, −3, −2, −1, 0, 1, 2, 3, …. Adding a negative is subtracting; subtracting a negative is adding. Product or quotient of two numbers with the same sign is positive, with different signs it is negative.

Formula: (+)(+) = +, (−)(−) = +, (+)(−) = −, a − (−b) = a + b

−4−3−2−101234-21

In the figure: Start at −2 and move 3 steps right to reach 1: −2 + 3 = 1.

Rule 1: Add and subtract — Think of moves on the number line; subtracting a negative moves right.

  • Easy: 5 + (−8) = −3.
  • Medium: −4 − (−9) = −4 + 9 = 5.
  • Exam-type: −7 − 6 = −13.

Rule 2: Multiply and divide — Same signs give positive, different signs give negative.

  • Easy: (−6) × (−3) = 18.
  • Medium: (−20) ÷ 4 = −5.
  • Exam-type: (−2) × (−3) × (−4) = −24.
Remember

An even count of negative factors gives a positive product; an odd count gives negative.

Common mistake

Writing −4 − (−9) as −13 instead of 5.

Memory clue

"Two minuses make a plus."

Practice questions

Easy
  1. 1Which collection contains only counting numbers 1, 2, 3, …?
  2. 2Evaluate (−9) × (−6).
  3. 3Evaluate −3 − (−4).
Medium
  1. 4Evaluate (−20) ÷ 5.
  2. 5At night the temperature of a hill station is −8 °C. By noon it rises by 16 °C. What is the temperature at noon?
  3. 6How many integers x satisfy −8 < x < 5?
Challenging
  1. 7Evaluate (2 − 13) × (−4) + 8.
  2. 8What is the largest integer x for which 6x + 9 < 28?
Show answers (try them yourself first)
  1. 1natural numbers
  2. 254
  3. 31
  4. 4−4
  5. 58
  6. 612
  7. 752
  8. 83
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Integers and their arithmetic

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Rational numbers

From daily life

Sharing 3 chapatis among 4 children gives 3/4 each. Any such sharing is a ratio of whole numbers.

Simple definition

A rational number can be written as p/q with integers p, q and q ≠ 0. Every integer is rational (5 = 5/1). Between any two rational numbers there are infinitely many more.

Formula: p/q, q ≠ 0; a/b + c/d = (ad + bc)/bd

−2−1012012

In the figure: Fractions such as 1/2, 3/4 and −3/2 sit between the marked integers.

Rule 1: Compare and add — Make denominators equal before adding or comparing.

  • Easy: 1/2 + 1/3 = 5/6.
  • Medium: 3/4 > 2/3 because 9/12 > 8/12.
  • Exam-type: −2/5 + 3/10 = −1/10.

Rule 2: A number between two numbers — Average the two numbers; repeat to get as many as needed.

  • Easy: Between 1/4 and 1/2: 3/8.
  • Medium: Between 0 and 1: 1/2, 1/4, 1/8, … never ends.
  • Exam-type: Between 2 and 3: 5/2.
Remember

Every terminating or repeating decimal is a rational number.

Common mistake

Adding fractions as (a + c)/(b + d), e.g. 1/2 + 1/3 = 2/5.

Memory clue

"Same bottom first."

Practice questions

Easy
  1. 1Is 0 a rational number?
  2. 2Fill in the blank: 3/5 = □/20.
  3. 3Which is greater: 3/9 or 5/7?
Medium
  1. 4Which rational number lies exactly halfway between 1/7 and 3/7?
  2. 5Which is greater: −1/4 or −1/6?
  3. 6Find 4/7 + 1/5.
Challenging
  1. 7What is the additive inverse of −6/10?
  2. 8Find the rational number x such that 3/3 + x = 1/5.
Show answers (try them yourself first)
  1. 1Yes, because 0 = 0/1
  2. 212
  3. 35/7
  4. 42/7
  5. 5−1/6
  6. 627/35
  7. 76/10
  8. 8−4/5

Irrational numbers

From daily life

The diagonal of a square tile of side 1 m is √2 m. No fraction measures it exactly, yet it is a real length.

Simple definition

An irrational number cannot be written as p/q. Its decimal never ends and never repeats. Examples: √2, √3, √5, π.

Formula: If n is not a perfect square, √n is irrational

11√2√2 = 1.41421…never ends,never repeats

In the figure: A unit square has diagonal √2 = 1.41421…, a decimal that never ends or repeats.

Rule 1: Spot irrationals — Roots of perfect squares are rational; other square roots are irrational.

  • Easy: √9 = 3 is rational.
  • Medium: √7 is irrational.
  • Exam-type: 0.101001000… (pattern never repeats) is irrational.

Rule 2: Operations — Rational + irrational is irrational. Two irrationals can give a rational.

  • Easy: 2 + √3 is irrational.
  • Medium: √2 × √2 = 2 is rational.
  • Exam-type: (3 + √5) + (3 − √5) = 6 is rational.
Remember

Between any two numbers there are both rational and irrational numbers.

Common mistake

Treating 22/7 or 3.14 as exactly π; they are only approximations.

Memory clue

"Never ends, never repeats: irrational."

Practice questions

Easy
  1. 1Identify the irrational number among these.
  2. 2Is √9 rational or irrational?
  3. 3Simplify √4 × √64.
Medium
  1. 4Simplify √6 × √96.
  2. 5Which expression gives an irrational number?
  3. 6Between which two consecutive integers does √55 lie?
Challenging
  1. 7Between which two consecutive integers does √44 lie?
  2. 8Evaluate (8 + √7)(8 − √7).
Show answers (try them yourself first)
  1. 1√2
  2. 2rational, since √9 = 3
  3. 316
  4. 424
  5. 54 + √12
  6. 67 and 8
  7. 76 and 7
  8. 857
Live figure

Irrational numbers

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Decimal expansions and recurring decimals

From daily life

1/4 = 0.25 stops, but 1/3 = 0.333… keeps going. A calculator shows only some of the 3s.

Simple definition

A rational number has a terminating decimal (denominator has only 2 and 5 as prime factors) or a recurring decimal. A recurring decimal like 0.3333… repeats a block of digits for ever. Every recurring decimal can be converted back into p/q.

Formula: x = 0.ddd… ⇒ 10x − x = d ⇒ x = d/9

x = 0.333…→10x = 3.333…→9x = 3→x = 1/3

In the figure: Subtract x from 10x to remove the repeating tail: 9x = 3, so x = 1/3.

Rule 1: Fraction to decimal — Divide numerator by denominator; stop when remainder is 0 or digits repeat.

  • Easy: 3/8 = 0.375 (terminating).
  • Medium: 1/6 = 0.1666… (6 repeats).
  • Exam-type: 1/7 = 0.142857 repeating.

Rule 2: Decimal to fraction — Multiply by 10 for one repeating digit, 100 for two, then subtract.

  • Easy: 0.666… = 6/9 = 2/3.
  • Medium: 0.4545… (45 repeats) = 45/99 = 5/11.
  • Exam-type: 0.999… = 9/9 = 1.
Remember

Terminating or recurring means rational; non-terminating non-recurring means irrational.

Common mistake

Calling 0.121212… irrational because it goes on forever.

Memory clue

"Repeats: rational. Random: irrational."

Practice questions

Easy
  1. 1Express 11/25 as a decimal.
  2. 2Without dividing, say whether 5/11 gives a terminating decimal.
  3. 3Express the repeating decimal 0.222… (that is, 0.2 with the digit repeating) as a fraction.
Medium
  1. 4Express the repeating decimal 0.888… (that is, 0.8 with the digit repeating) as a fraction.
  2. 5How many digits are there in the repeating block of the decimal expansion of 1/9?
  3. 6Express 0.696969… (the block 69 repeats) as a fraction in lowest terms.
Challenging
  1. 7Write 0.4888… (the digit 8 repeats forever, after the digit 4) as a fraction in lowest terms.
  2. 8Find 0.333… + 0.222… as a fraction in lowest terms.
Show answers (try them yourself first)
  1. 10.44
  2. 2no
  3. 32/9
  4. 48/9
  5. 51
  6. 623/33
  7. 722/45
  8. 85/9

Real numbers and surds

From daily life

Every length you can measure, such as 2, 3/4 or √5 cm, is a point on a ruler. Rational and irrational numbers together fill the whole ruler.

Simple definition

Rational and irrational numbers together make the real numbers. A surd is an irrational root like √3 or ∛5. Key laws: √a × √b = √(ab), √a ÷ √b = √(a/b). To rationalise 1/√a multiply top and bottom by √a.

Formula: 1/√a = √a/a; (a + √b)(a − √b) = a² − b

Natural→Integers→Rational→Real

In the figure: Each set sits inside the next. Irrational numbers fill the gaps in the real line.

Rule 1: Simplify surds — Pull out perfect-square factors from under the root.

  • Easy: √50 = √(25 × 2) = 5√2.
  • Medium: √12 × √3 = √36 = 6.
  • Exam-type: √18 + √8 = 3√2 + 2√2 = 5√2.

Rule 2: Rationalise — Multiply by the conjugate when the denominator has two terms.

  • Easy: 1/√3 = √3/3.
  • Medium: 2/√5 = 2√5/5.
  • Exam-type: 1/(2 + √3) = 2 − √3.
Remember

Every real number is a point on the number line, and every point is a real number.

Common mistake

Writing √a + √b = √(a + b); there is no such law.

Memory clue

"Square factors jump out; roots multiply under one roof."

Practice questions

Easy
  1. 1Simplify √72.
  2. 2Which is greater: √10 or 4?
  3. 3Simplify 3√2 + 6√2.
Medium
  1. 4Simplify √180 − √80.
  2. 5Rationalise the denominator of 1/√12.
  3. 6Simplify (4√3) × (4√3).
Challenging
  1. 7Expand (√7 + 3)².
  2. 8Rationalise 1/(√6 − √5).
Show answers (try them yourself first)
  1. 16√2
  2. 24
  3. 39√2
  4. 42√5
  5. 5√12/12
  6. 648
  7. 716 + 6√7
  8. 8√6 + √5