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Rational, Irrational and Real Numbers

🎓 Class 9 Mathematics CBSE Theory Ch 3 — The World of Numbers ⏱ ~45 min
🌐 Language: [gtranslate]

This MCQ module is based on: Rational, Irrational and Real Numbers

This mathematics assessment will be based on: Rational, Irrational and Real Numbers
Targeting Class 9 level in Number Theory, with Intermediate difficulty.

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3.5 The Need for Rational Numbers

The integers \(\mathbb{Z}\) gave us debts and fortunes. But as civilisations grew, a new problem emerged: division. If three farmers wished to share two sacks of grain equally, what number describes each share? It is not 0, not 1, and not 2 — it is something between. The integers were not enough.

To solve this, mathematicians extended the number system to include all ratios of integers. These are the rational numbers?, denoted by \(\mathbb{Q}\) (from Quotient).

Definition: Rational Number
A number is called rational if it can be written in the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\). The set of all rational numbers is: \[\mathbb{Q} = \left\{\frac{p}{q} \,:\, p, q \in \mathbb{Z},\, q \neq 0 \right\}\]

Examples of rational numbers include:

\(\dfrac{2}{3},\ \dfrac{-7}{4},\ \dfrac{5}{1} = 5,\ \dfrac{0}{8} = 0,\ \dfrac{22}{7},\ -\dfrac{11}{3}\)

Notice that every integer is also a rational number: for any integer \(n\), we can write \(n = \frac{n}{1}\). Thus we have the chain of inclusions: \[ \mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \]

3.5.1 Equivalent Rationals — A Number Has Many Faces

A rational number can be written in infinitely many ways. For instance, \(\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{50}{100}\) — they all represent the same point on the number line. We say a rational number is in its simplest (lowest) form when \(\gcd(p, q) = 1\) and \(q > 0\).

3.5.2 Representing Rationals on the Number Line

Suppose we wish to mark \(\frac{5}{6}\) on the number line. Since \(0 < \frac{5}{6} < 1\), divide the unit segment between 0 and 1 into 6 equal parts and mark the 5th division point. Similarly, \(-\frac{3}{4}\) lies between 0 and \(-1\), three-fourths of the way to \(-1\).

-1012 5/6 -3/4 Subdivide each unit interval into q equal parts to plot p/q
Fig 3.3 — Plotting \(\frac{5}{6}\) and \(-\frac{3}{4}\) on the rational number line.

3.5.3 Density of Rationals — Infinitely Many Between Any Two

Between any two rational numbers, however close, there always lie infinitely many rational numbers. This property is called density.

Example 5. Find a rational number between \(\frac{3}{5}\) and \(\frac{4}{5}\).
Mean method: The midpoint of two rationals is also a rational. Mid \(= \frac{1}{2}\left(\frac{3}{5} + \frac{4}{5}\right) = \frac{1}{2} \cdot \frac{7}{5} = \frac{7}{10}\). Verify: \(\frac{3}{5} = 0.6,\ \frac{7}{10} = 0.7,\ \frac{4}{5} = 0.8\). ✓
Example 6. Find five rational numbers between \(\frac{1}{2}\) and \(\frac{3}{4}\).
Convert to common denominators large enough to give "room": \(\frac{1}{2} = \frac{12}{24}\) and \(\frac{3}{4} = \frac{18}{24}\). Five numbers between them: \(\frac{13}{24}, \frac{14}{24}, \frac{15}{24}, \frac{16}{24}, \frac{17}{24}\). In simplest forms: \(\frac{13}{24}, \frac{7}{12}, \frac{5}{8}, \frac{2}{3}, \frac{17}{24}\).

3.6 Decimal Expansions of Rational Numbers

Every rational number can be written as a decimal. To convert \(\frac{p}{q}\) to decimal form, perform long division. Two surprising things can happen:

Theorem 3.1 (NCERT)
The decimal expansion of any rational number \(\frac{p}{q}\) is either:
  1. Terminating — eventually ends (e.g. \(\frac{3}{4} = 0.75\)), or
  2. Non-terminating recurring — repeats a block of digits forever (e.g. \(\frac{1}{3} = 0.\overline{3}\)).
The converse is also true: every terminating or recurring decimal represents a rational number.

When does a decimal terminate?

A rational \(\frac{p}{q}\) (in lowest terms) gives a terminating decimal if and only if the prime factorisation of \(q\) contains only the primes 2 and 5 (i.e. \(q = 2^m \cdot 5^n\) for some \(m,n \geq 0\)).

FractionDenom qPrime factors of qDecimalType
\(\frac{3}{8}\)8\(2^3\)0.375Terminates
\(\frac{7}{20}\)20\(2^2 \cdot 5\)0.35Terminates
\(\frac{2}{3}\)33\(0.\overline{6}\)Recurs (3)
\(\frac{1}{7}\)77\(0.\overline{142857}\)Recurs (period 6)
\(\frac{1}{6}\)6\(2 \cdot 3\)\(0.1\overline{6}\)Recurs

3.6.1 Converting a Recurring Decimal Back to \(\frac{p}{q}\)

Example 7. Express \(0.\overline{6}\) in the form \(\frac{p}{q}\).
Let \(x = 0.\overline{6} = 0.6666\ldots\). Then \(10x = 6.6666\ldots\). Subtract: \(10x - x = 6.6666 - 0.6666 = 6\). Hence \(9x = 6\) and \(x = \frac{6}{9} = \frac{2}{3}\).
Example 8. Express \(2.\overline{357}\) as a fraction.
Let \(x = 2.\overline{357}\). The repeating block has length 3, so multiply by \(10^3 = 1000\): \(1000x = 2357.\overline{357}\). Subtract: \(1000x - x = 2357.\overline{357} - 2.\overline{357} = 2355\). So \(999x = 2355\), giving \(x = \frac{2355}{999} = \frac{785}{333}\).
Example 9. Express \(0.1\overline{6}\) as a fraction.
Mixed pattern (1 non-repeating digit + 1-digit cycle). Let \(x = 0.16666\ldots\). Then \(10x = 1.6666\ldots\) and \(100x = 16.6666\ldots\). Subtract: \(100x - 10x = 16.\overline{6} - 1.\overline{6} = 15\). So \(90x = 15\) and \(x = \frac{15}{90} = \frac{1}{6}\). ✓

3.7 Irrational Numbers — Beyond the Rationals

Around 500 BCE, the Pythagoreans of ancient Greece believed everything in the universe could be expressed as a ratio of whole numbers. According to legend, when their student Hippasus proved that \(\sqrt{2}\) cannot be written as \(\frac{p}{q}\), it caused such a crisis that he was reportedly thrown overboard at sea! He had discovered the first irrational number?.

Definition: Irrational Number
A number is called irrational if it cannot be expressed as \(\frac{p}{q}\) where \(p, q \in \mathbb{Z}\) and \(q \neq 0\). Equivalently, its decimal expansion is non-terminating and non-recurring.

Famous irrational numbers include:

  • \(\sqrt{2} = 1.41421356237\ldots\) — the diagonal of a unit square.
  • \(\sqrt{3} = 1.73205080757\ldots\), \(\sqrt{5}, \sqrt{7}, \ldots\) (in fact \(\sqrt{n}\) is irrational unless \(n\) is a perfect square).
  • \(\pi = 3.14159265358\ldots\) — the ratio of a circle's circumference to its diameter.
  • \(e = 2.71828182846\ldots\) — the base of natural logarithms.

3.7.1 Proof that \(\sqrt{2}\) is Irrational (Reductio ad Absurdum)

Theorem 3.2
\(\sqrt{2}\) is irrational.

Proof (by contradiction):
  1. Assume the opposite: \(\sqrt{2} = \frac{p}{q}\) where \(p, q\) are integers, \(q \neq 0\), with \(\gcd(p,q) = 1\) (lowest terms).
  2. Squaring: \(2 = \frac{p^2}{q^2}\), so \(p^2 = 2q^2\). Hence \(p^2\) is even, which forces \(p\) itself to be even.
  3. Write \(p = 2k\). Substitute: \((2k)^2 = 2q^2 \Rightarrow 4k^2 = 2q^2 \Rightarrow q^2 = 2k^2\). So \(q^2\) is even, hence \(q\) is even.
  4. But now both \(p\) and \(q\) are even, contradicting our assumption that \(\gcd(p, q) = 1\).
  5. The contradiction means our assumption was false. Therefore \(\sqrt{2}\) cannot be expressed as \(\frac{p}{q}\) — it is irrational. ∎

3.7.2 Locating \(\sqrt{2}\) on the Number Line — A Geometric Construction

How can a number that has no exact decimal still occupy an exact location? The answer comes from Pythagoras' theorem.

On the number line, mark O at 0 and A at 1. At point A, draw a perpendicular AB of length 1 unit. Then OB is the hypotenuse of a right triangle with legs 1 and 1, so: \[ OB = \sqrt{1^2 + 1^2} = \sqrt{2}. \] With centre O and radius OB, draw an arc cutting the number line at point P. Then OP = \(\sqrt{2}\), and P is the precise location of \(\sqrt{2}\).

0123 O A B 1 1 √2 P = √2 arc, radius OB
Fig 3.4 — Geometric construction of \(\sqrt{2}\) on the number line using Pythagoras' theorem.
Activity: The Square Root Spiral of Theodorus
L4 Analyse
Materials: A4 sheet, ruler, set-square, compass, pencil.
Predict: Can you locate \(\sqrt{3}, \sqrt{4}, \sqrt{5}, \ldots\) precisely on a number line using only a compass and a straight edge?
  1. Draw a horizontal segment OA = 1 unit. At A, draw AB ⟂ OA with AB = 1. Hypotenuse OB = \(\sqrt{1+1} = \sqrt{2}\).
  2. At B, draw BC ⟂ OB with BC = 1. Hypotenuse OC = \(\sqrt{2 + 1} = \sqrt{3}\).
  3. At C, draw CD ⟂ OC with CD = 1. Hypotenuse OD = \(\sqrt{4} = 2\).
  4. Continue. After \(n\) steps, the diagonal from O has length \(\sqrt{n+1}\).
  5. Now transfer each diagonal length onto a horizontal number line using a compass — you will have geometrically constructed \(\sqrt{2}, \sqrt{3}, \sqrt{4}, \ldots\)

Insight: This famous spiral of Theodorus proves that every \(\sqrt{n}\) (for any natural \(n\)) corresponds to a precise, locatable point on the number line — even though most of these are irrational and cannot be written as exact fractions.

O A=1 √2 √3 √4=2 √5 √6 Each new right triangle has one leg = previous hypotenuse, other leg = 1.
Fig 3.5 — Spiral of Theodorus: successive hypotenuses generate \(\sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5}, \sqrt{6}, \ldots\)

3.8 Real Numbers — The Complete Number Line

The union of rational and irrational numbers gives the real numbers?, denoted \(\mathbb{R}\). \[ \mathbb{R} = \mathbb{Q} \cup (\text{Irrationals}) \] A truly remarkable fact: every point on the number line corresponds to a unique real number, and every real number corresponds to a unique point. This one-to-one correspondence is sometimes called the completeness of the real line — there are no "gaps".

The Number System Hierarchy
\(\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\)
The set of irrational numbers is not a subset of \(\mathbb{Q}\) — it sits alongside it inside \(\mathbb{R}\).
ℝ — Real Numbers ℚ — Rationals ℤ — Integers 𝕎 — Whole ℕ — Naturals Irrationals √2, √3, √5 π, e 0.10110011… non-terminating,
non-recurring
Fig 3.6 — The hierarchy of number systems. Rationals and irrationals together fill the real line.

3.8.1 Constructing \(\sqrt{n}\) for any natural \(n\)

To construct \(\sqrt{n}\) on the number line for any positive integer \(n\):

  1. Mark a segment AB of length \(n\) units on a line.
  2. Extend by 1 unit to point C, so AC = \(n + 1\).
  3. Find midpoint O of AC and draw a semicircle with diameter AC.
  4. At B, erect a perpendicular meeting the semicircle at D. Then \(BD = \sqrt{n}\).

This works because of the geometric mean property: if AB and BC are two segments of the diameter, then the perpendicular from B to the semicircle has length \(\sqrt{AB \cdot BC} = \sqrt{n \cdot 1} = \sqrt{n}\).

🔵 Quick Check: Is \(\frac{22}{7}\) equal to π? Answer: No! \(\frac{22}{7} \approx 3.142857\overline{142857}\) — a recurring decimal — while \(\pi = 3.14159265\ldots\) is non-recurring. \(\frac{22}{7}\) is only an approximation of π.

Competency-Based Questions

Scenario: Aisha is a graphic designer building a poster. She lays out a square photograph of side 10 cm and wants to draw its diagonal as a guide line. She also marks decimal subdivisions on a 12 cm ruler attached to the canvas: \(\frac{1}{4}, \frac{1}{3}, \frac{2}{5}, \frac{3}{8}\) and the diagonal length itself.
Q1. Compute the diagonal of the 10 cm × 10 cm square photograph and classify it as rational or irrational.
L3 Apply
  • (a) 14.14 cm — rational
  • (b) \(10\sqrt{2}\) cm — irrational
  • (c) 20 cm — rational
  • (d) \(\sqrt{20}\) cm — irrational
Answer (b). Diagonal \(d = \sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2}\) cm. Since \(\sqrt{2}\) is irrational, so is \(10\sqrt{2}\) — option (a) is just a decimal approximation, not exact.
Q2. Among the four marked fractions \(\frac{1}{4}, \frac{1}{3}, \frac{2}{5}, \frac{3}{8}\), determine which give terminating decimal expansions and justify your reasoning.
L4 Analyse
Terminating: \(\frac{1}{4} = 0.25\) (denom \(2^2\)); \(\frac{2}{5} = 0.4\) (denom 5); \(\frac{3}{8} = 0.375\) (denom \(2^3\)). Non-terminating: \(\frac{1}{3} = 0.\overline{3}\) — denom 3 contains a prime other than 2 or 5.
Q3. Aisha wants to label the diagonal length precisely on her poster. She considers writing it as 14.14 cm, \(\sqrt{200}\) cm, or \(10\sqrt{2}\) cm. Evaluate which label is mathematically most accurate, and explain why.
L5 Evaluate
\(10\sqrt{2}\) is the most accurate. \(\sqrt{200}\) is also exact but unsimplified. 14.14 is rounded — losing information beyond the second decimal. In mathematical writing, the simplified surd form \(10\sqrt{2}\) is preferred for both exactness and brevity.
Q4. Design a square photo whose diagonal will be a rational number. State the side length(s) you would use and justify why this works.
L6 Create
Trick: If the side itself is irrational of the form \(s = k\sqrt{2}\), then the diagonal becomes \(s\sqrt{2} = k\sqrt{2}\cdot\sqrt{2} = 2k\), which is rational! Example: side \(= 5\sqrt{2}\) cm → diagonal = \(2 \times 5 = 10\) cm, rational. (Note: it is impossible for both side and diagonal to be rational integers — the irrationality of \(\sqrt{2}\) prevents it.)

Assertion–Reason Questions

Assertion (A): \(0.\overline{37}\) is a rational number.
Reason (R): Every non-terminating recurring decimal can be written as \(\frac{p}{q}\) with integer \(p, q\) and \(q \neq 0\).
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
(a) — \(0.\overline{37} = \frac{37}{99}\) (rational). R explains why all such decimals are rational.
Assertion (A): The sum of two irrational numbers is always irrational.
Reason (R): Irrational numbers cannot be expressed as \(\frac{p}{q}\).
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
(d) — A is FALSE: \((\sqrt{2}) + (-\sqrt{2}) = 0\) is rational; \(2 + \sqrt{3}\) and \(2 - \sqrt{3}\) are both irrational but their sum 4 is rational. R is TRUE — that is the correct definition of irrationals.
Assertion (A): The decimal expansion of \(\frac{7}{40}\) terminates.
Reason (R): The denominator 40 = \(2^3 \cdot 5\), having only primes 2 and 5.
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
(a) — \(\frac{7}{40} = 0.175\) terminates. R correctly identifies the rule: a fraction in lowest terms terminates iff its denominator factors into only 2's and 5's.

Frequently Asked Questions

What is a rational number with an example for Class 9?

A rational number is any number that can be written as p/q where p and q are integers and q is not zero. Examples include 3/4, -7, 0.5 (= 1/2) and 0.333... (= 1/3). Every integer n is rational because n = n/1.

What is an irrational number and how is it different from a rational number?

An irrational number cannot be expressed as p/q for any integers p, q with q != 0. Its decimal expansion is non-terminating and non-repeating. Examples are sqrt(2) = 1.41421356..., pi = 3.14159265... and e. Rational numbers, in contrast, always terminate or repeat.

How do you prove that sqrt(2) is irrational in Class 9?

Assume, for contradiction, that sqrt(2) = p/q in lowest terms. Squaring gives p^2 = 2q^2, so p^2 is even and hence p is even. Writing p = 2k gives 4k^2 = 2q^2, so q^2 = 2k^2, meaning q is also even. But then p and q share a factor 2, contradicting 'lowest terms'. So sqrt(2) cannot be rational.

What is the decimal expansion test for rational vs irrational numbers?

If the decimal expansion terminates (e.g., 0.75) or eventually repeats a fixed block (e.g., 0.142857142857...), the number is rational. If the digits never end and never settle into a repeating block (e.g., 0.10100100010000...), the number is irrational.

How do you locate sqrt(2) on the number line?

Draw a unit square OABC on the number line with side OA = 1. Its diagonal OB has length sqrt(1^2 + 1^2) = sqrt(2) by Pythagoras. With centre O and radius OB, draw an arc to cut the number line at point P. Then OP = sqrt(2).

What are real numbers and what does the real number line represent?

Real numbers R are the union of all rational and irrational numbers. Every point on the number line corresponds to exactly one real number, and every real number corresponds to exactly one point on the line. There are no 'gaps' on the real number line.

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