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State Board · Class 9 · Mathematics Part 1

Lesson 2: Real Numbers

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

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Rational numbers and their decimal forms

From daily life

1 ÷ 4 = 0.25 stops quickly, but 1 ÷ 3 = 0.333... never stops. The denominator decides whether a fraction's decimal stops or repeats.

Simple definition

The decimal form of a rational number is either terminating (संपणारा) or non-terminating recurring (न संपणारा पुनरावृत्त).

Rule 1: Properties of rationals — For rational numbers a, b, c, addition and multiplication are commutative (a + b = b + a), associative, have an identity (0 for addition, 1 for multiplication) and an inverse (−a for addition; 1a for multiplication, a ≠ 0).

  • Easy : Easy: 3 + 5 = 5 + 3.
  • Medium : Medium: The additive inverse of 27 is −27.
  • Exam-type : Exam: The multiplicative inverse of 38 is 83; check 38 × 83 = 1.

Rule 2: Terminating or recurring? — Reduce the fraction and look at the denominator's prime factors. Only 2 or 5 → terminating: 720 = 0.35 (20 = 2 × 2 × 5), 316 = 0.1875. Any other prime factor → recurring: 512 = 0.41666... (12 = 2 × 2 × 3), 47 = 0.571428571428...

Fractions whose denominators have only the prime factors 2 and 5 give terminating decimals; others give recurring decimalsFraction → decimalTerminating decimals3/4 = 0.757/20 = 0.35Recurring decimals5/6 = 0.8333…4/11 = 0.3636…Denominator: 2, 5 only4 = 2×2, 20 = 2×2×5Other prime factors6 = 2×3, 11 is prime

In the figure : In the figure look at the prime factors of the denominator: only 2 or 5 means terminating; any other prime factor means non-terminating recurring.

  • Easy : Easy: 18 = 0.125 terminates (8 = 2 × 2 × 2).
  • Medium : Medium: 215 recurs, since 15 has the factor 3.
  • Exam-type : Exam: Which terminates: 940 or 942? 40 = 2 × 2 × 2 × 5, so 940 = 0.225.

Rule 3: Recurring decimal to fraction — Multiply by 10, 100 or 1000 by the number of recurring digits, then subtract. x = 0.444...: 10x − x = 4, so x = 49. x = 0.2323...: 100x − x = 23, so x = 2399. x = 2.1515...: 100x − x = 213, so x = 21399 = 7133.

  • Easy : Easy: 0.111... = 19.
  • Medium : Medium: 0.666... = 69 = 23.
  • Exam-type : Exam: Write 0.4545... as a fraction: 100x − x = 45, x = 4599 = 511.
Remember

Denominator only 2, 5 → terminating. Otherwise recurring.

Common mistake

Testing an unreduced fraction. 615 = 25 terminates even though 15 has 3.

Shortcut

Memory clue: "फक्त 2 आणि 5 = थांबतो".

Spot the difference

FractionDenominator factorsDecimal
7202, 2, 50.35 (terminating
5122, 2, 30.41666... (recurring
4770.571428... (recurring

Practice questions

Easy
  1. 1What are the two decimal forms of a rational number?
  2. 2Additive identity?
  3. 3Multiplicative identity?
Medium
  1. 4Decimal form of 316?
  2. 5Does 730 terminate?
  3. 6Write 0.333... as a fraction.
Challenging
  1. 7Write 0.8181... as a fraction.
  2. 8Write 3.7777... as a fraction.
Show answers (try them yourself first)
  1. 1Terminating and non-terminating recurring.
  2. 20.
  3. 31.
  4. 40.1875.
  5. 5No, 30 has the factor 3.
  6. 613.
  7. 7100x − x = 81, x = 8199 = 911.
  8. 810x − x = 34, x = 349.

Irrational and real numbers

From daily life

A square tile with side 1 m has a diagonal that cannot be written as a fraction of metres. Numbers like this need a new family.

Simple definition

An irrational number (अपरिमेय संख्या) cannot be written as pq; its decimal is non-terminating, non-recurring. Rational and irrational numbers together are real numbers (वास्तव संख्या).

Rule 1: √2 is irrational — Proof by contradiction: suppose √2 = pq with p, q having no common factor except 1. Then 2q2 = p2, so p is even; then q is even too, contradicting the assumption. Likewise √3, √5 and √7 are irrational. √2 = 1.41421..., √3 = 1.732...

Real numbers split into rational numbers such as 2/5 and irrational numbers such as √3 and πReal numbersRationalIrrationalIntegers: −3, 0, 5Fractions: 2/5, −7/4Decimals: 0.25, 0.333…(end or repeat)√3, √7, π0.1010010001…(never end, neverrepeat)

In the figure : In the figure real numbers are either rational or irrational.

  • Easy : Easy: √2 is irrational.
  • Medium : Medium: √9 = 3 is rational.
  • Exam-type : Exam: Is 0.101001000... rational? No, it does not repeat.

Rule 2: Real numbers and π — Every point on the number line is a unique real number, and every real number is a unique point. Every rational number is real, but not conversely (√2, π, 3 + √2 are not rational). π = circumference ÷ diameter is irrational, about 227 or 3.14. Aryabhat (499 CE) took π = 6283220000 = 3.1416.

A number line with the rational point 1/2 and the irrational points minus root 2, root 2, root 5 and piEvery point on the line is a real number−2−10123−√21/2√2√5πrationalirrational

In the figure : In the figure irrational numbers such as √2 and π also have their own places on the number line.

  • Easy : Easy: π is irrational.
  • Medium : Medium: 227 is only an approximation of π.
  • Exam-type : Exam: Who used π = 3.1416 in 499 CE? Aryabhat.

Rule 3: Rules — A rational ± an irrational is irrational (3 + √5). A non-zero rational × or ÷ an irrational is irrational (2√3). But two irrationals can give a rational: √2 × √2 = 2, √3 + (−√3) = 0; or an irrational: √2 + √3.

Live figure

Irrational and real numbers

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  • Easy : Easy: 5 + √2 is irrational.
  • Medium : Medium: 4 × √3 is irrational.
  • Exam-type : Exam: Give two irrationals whose product is rational. √5 and √5 give 5.
Remember

Irrational: non-terminating, non-recurring. Rational ± irrational = irrational.

Common mistake

Thinking 227 equals π. It is only close.

Shortcut

Memory clue: "परिमेय + अपरिमेय = वास्तव".

Spot the difference

TypeDecimalExample
RationalTerminating or recurring38, 0.333...
IrrationalNon-terminating, non-recurring√2, π
RealBothEvery number-line point

Practice questions

Easy
  1. 1What is an irrational number?
  2. 2Decimal form of an irrational number?
  3. 3Is √16 irrational?
Medium
  1. 4Name a famous irrational constant.
  2. 5Is 2 + √7 rational?
  3. 6Is √5 × √5 rational?
Challenging
  1. 7Classify: 0.5, √10, 73, 0.2020020002....
  2. 8Why does the proof for √2 start by assuming it is rational?
Show answers (try them yourself first)
  1. 1A number that cannot be written as pq.
  2. 2Non-terminating, non-recurring.
  3. 3No, it equals 4.
  4. 4π.
  5. 5No, irrational.
  6. 6Yes, it is 5.
  7. 7Rational, irrational, rational, irrational.
  8. 8To reach a contradiction, which shows the assumption is false.

Surds, simplest form, like surds and comparison

From daily life

The number √50 looks complicated, but 50 = 25 × 2 lets us write it as 5√2, which is much easier to compare and add.

Simple definition

A surd (करणी) is an irrational root of a positive rational number, written n√a.

Rule 1: What is a surd — In n√a, √ is the radical sign, n the order and a the radicand. ∛7 and ∜8 are surds; ∛27 = 3 and ∛8 = 2 are not, as they are rational. Order 2 surds are quadratic surds. The square root of a negative real number is not real.

Root of a positive rational number that is irrational is a surd; roots like root 16 are notSurd or not?Surds√3, ∛5, √(2/3)never a whole numberNot surds√16 = 4, ∛27 = 3the root is rational

In the figure : In the figure a surd is an irrational root of a positive rational number.

  • Easy : Easy: √3 is a surd.
  • Medium : Medium: ∛27 = 3 is not a surd.
  • Exam-type : Exam: Which is a surd: √49, ∛5, ∛64? Only ∛5.

Rule 2: Simplest form and comparing — √72 = √36 × √2 = 6√2; √200 = √100 × √2 = 10√2; √75 = 5√3. Surds with equal order and equal radicand (in simplest form) are like surds (सजातीय करणी): √12 = 2√3 and √27 = 3√3. To compare, compare squares: (4√3)2 = 48 > 45 = (3√5)2, so 4√3 > 3√5; (5√2)2 = 50 > 49, so 5√2 > 7.

  • Easy : Easy: √18 = 3√2.
  • Medium : Medium: √12 and √27 are like surds.
  • Exam-type : Exam: Compare 6√2 and 5√3: 72 < 75, so 6√2 < 5√3.
Remember

Simplify first (take out squares). Compare by squaring.

Common mistake

Calling √4 a surd. It equals 2, which is rational.

Shortcut

Memory clue: "वर्ग बाहेर काढा, मग तुलना".

Spot the difference

SurdSimplest form
√726√2
√20010√2
√755√3

Practice questions

Easy
  1. 1What is a surd?
  2. 2What is the order in 3√5?
  3. 3Simplify √48.
Medium
  1. 4Simplify √98.
  2. 5Are 2√5 and 7√5 like surds?
  3. 6Which is greater: 3√5 or 4√3?
Challenging
  1. 7Simplify √128.
  2. 8Why do we compare squares to compare surds?
Show answers (try them yourself first)
  1. 1An irrational root of a positive rational number.
  2. 23.
  3. 34√3.
  4. 47√2.
  5. 5Yes.
  6. 64√3, as 48 > 45.
  7. 7√64 × √2 = 8√2.
  8. 8Squaring removes the root and keeps the order for positive numbers.

Operations on surds and rationalizing factors

From daily life

Like apples and apples add easily, but apples and mangoes don't. Like surds add like apples; unlike surds stay apart.

Simple definition

Only like surds can be added or subtracted; any surds can be multiplied or divided. A rationalizing factor (परिमेयकारक घटक) turns a surd into a rational number.

Rule 1: Add, subtract, multiply, divide — 5√2 + 3√2 = 8√2; 10√3 − 4√3 = 6√3. √18 + √50 − √8 = 3√2 + 5√2 − 2√2 = 6√2. √6 × √15 = √90 = 3√10. √72 ÷ √2 = √36 = 6. √12 × √27 = 2√3 × 3√3 = 18. The product or quotient of surds can be rational.

  • Easy : Easy: 4√5 + √5 = 5√5.
  • Medium : Medium: √8 + √32 = 2√2 + 4√2 = 6√2.
  • Exam-type : Exam: Simplify √45 + √80 − √20 = 3√5 + 4√5 − 2√5 = 5√5.

Rule 2: Rationalizing factor — If the product of two surds is rational, each is a rationalizing factor of the other: √3 × √12 = 6. It is not unique; the simplest for √12 = 2√3 is √3. To remove a root from a denominator: 1√7 = √77; 52√3 = 5√36 (multiply top and bottom by √3).

  • Easy : Easy: √5 × √5 = 5.
  • Medium : Medium: 3√2 = 3√22.
  • Exam-type : Exam: Rationalize 103√5 = 10√515 = 2√53.
Remember

Add only like surds. Multiply top and bottom by the rationalizing factor.

Common mistake

Adding √2 + √3 as √5. Unlike surds cannot be combined.

Shortcut

Memory clue: "सजातीय = मिळवा; छेदात करणी = घटकाने गुणा".

Spot the difference

OperationExample
Add like surds5√2 + 3√2 = 8√2
Multiply√6 × √15 = 3√10
Divide√72 ÷ √2 = 6
Rationalize1√7 = √77

Practice questions

Easy
  1. 1Can we add √2 and √3 to get one surd?
  2. 26√7 − 2√7 = ?
  3. 3√6 × √6 = ?
Medium
  1. 4√50 ÷ √2 = ?
  2. 5Rationalize 1√7.
  3. 6Simplest rationalizing factor of √12?
Challenging
  1. 7Simplify √12 × √27.
  2. 8Why is a rationalizing factor not unique?
Show answers (try them yourself first)
  1. 1No, they are unlike surds.
  2. 24√7.
  3. 36.
  4. 4√25 = 5.
  5. 5√77.
  6. 6√3.
  7. 72√3 × 3√3 = 18.
  8. 8Any multiple that still gives a rational product works.

Binomial surds, conjugates and absolute value

From daily life

A shop price tag of ₹ (√5 + √3) is odd, but multiplying by its "mirror" (√5 − √3) gives plain 2. Such mirrors are called conjugates.

Simple definition

Surds like √5 + √3 are binomial surds; (√a + √b) and (√a − √b) are conjugates (संयुग) with product a − b. ∣x∣ is the absolute value (निरपेक्ष मूल्य).

Rule 1: Conjugates — (√a + √b)(√a − √b) = a − b, which is rational. (√7 + √2)(√7 − √2) = 5; (√6 + √3)(√6 − √3) = 3. The conjugate of 7 + √3 is 7 − √3. Each is the rationalizing factor of the other.

To rationalize 1 over root 3 plus root 2, multiply by root 3 minus root 2 over root 3 minus root 2; the denominator becomes 1Rationalizing with the conjugate1/(√3 + √2)× (√3 − √2)/(√3 − √2)multiply by the conjugate(√3 − √2)/(3 − 2)denominator is rational√3 − √2

In the figure : In the figure multiplying by the conjugate removes the root from the denominator.

  • Easy : Easy: Conjugate of 3 + √2 is 3 − √2.
  • Medium : Medium: (√5 + 1)(√5 − 1) = 5 − 1 = 4.
  • Exam-type : Exam: Rationalize 1(√7 − √2) = (√7 + √2)5.

Rule 2: Rationalizing with conjugates — 6(√5 + √2): multiply top and bottom by (√5 − √2). The denominator becomes 5 − 2 = 3. So the value is 6(√5 − √2)3 = 2√5 − 2√2.

Both minus 4 and 4 are 4 steps from zero, so their absolute value is 4Absolute value = distance from 0−6−4−2024644|−4| = 4 |4| = 4 |0| = 0

In the figure : In the figure absolute value is a distance, so it is never negative.

  • Easy : Easy: ∣−7∣ = 7.
  • Medium : Medium: ∣4 − 9∣ = 5.
  • Exam-type : Exam: ∣x − 3∣ = 4 gives x = 7 or x = −1.

Rule 3: Absolute value — ∣x∣ is the distance of x from zero: ∣x∣ = x if x > 0, ∣0∣ = 0, ∣x∣ = −x if x < 0. It is never negative. ∣x + 2∣ = 5 gives x + 2 = 5 or x + 2 = −5, so x = 3 or x = −7.

  • Easy : Easy: ∣5∣ = 5.
  • Medium : Medium: ∣−2.5∣ = 2.5.
  • Exam-type : Exam: Solve ∣2x − 1∣ = 7: 2x − 1 = 7 or −7, so x = 4 or x = −3.
Remember

Conjugates multiply to a − b. ∣x∣ is a distance, never negative.

Common mistake

Writing ∣−3∣ = −3. It is 3.

Shortcut

Memory clue: "चिन्ह उलट = संयुग".

Spot the difference

ExpressionConjugateProduct
√5 + √3√5 − √32
7 + √37 − √346
√7 + √2√7 − √25

Practice questions

Easy
  1. 1What is the conjugate of √5 + √3?
  2. 2(√a + √b)(√a − √b) = ?
  3. 3Product of (√6 + √3)(√6 − √3)?
Medium
  1. 4∣−9∣ = ?
  2. 5Can ∣x∣ be negative?
  3. 6Solve ∣x − 5∣ = 2.
Challenging
  1. 7Rationalize 1(√5 − √3).
  2. 8Find (7 + √3)(7 − √3).
Show answers (try them yourself first)
  1. 1√5 − √3.
  2. 2a − b.
  3. 33.
  4. 49.
  5. 5No.
  6. 6x = 7 or x = 3.
  7. 7(√5 + √3)2.
  8. 849 − 3 = 46.

Quick revision

  • Terminating decimal: denominator has only 2 and 5 as prime factors. Irrational numbers have non-terminating, non-recurring decimals.
  • A surd is an irrational root of a positive rational number; like surds can be added and subtracted.
  • (√a + √b)(√a − √b) = a − b is rational; the conjugate is the rationalizing factor; ∣x∣ is never negative.