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The World of Numbers - Irrational Numbers

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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An irrational number is a number that cannot be written in the form pq\frac{p}{q}, where pp and qq are integers and q≠0q \neq 0.

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The decimal expansion of an irrational number is non-terminating and non-recurring (non-repeating). For example, 0.10110111011110...0.10110111011110... is irrational.

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The collection of all rational numbers and irrational numbers together forms the set of Real Numbers, denoted by R\mathbb{R}.

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Every real number is represented by a unique point on the number line, and every point on the number line represents a unique real number.

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If nn is a positive integer that is not a perfect square (like 2,3,5,62, 3, 5, 6), then n\sqrt{n} is an irrational number.

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To locate n\sqrt{n} on a number line, we use the Pythagoras theorem. For instance, to locate 2\sqrt{2}, we construct a right-angled triangle with base 11 unit and height 11 unit, such that the hypotenuse is 12+12=2\sqrt{1^2 + 1^2} = \sqrt{2}.

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The sum or difference of a rational number and an irrational number is always irrational. For example, 2+32 + \sqrt{3} is irrational.

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The product or quotient of a non-zero rational number and an irrational number is irrational. For example, 252\sqrt{5} is irrational.

📐Formulae

s≠pq, where p,q∈Z,q≠0s \neq \frac{p}{q}, \text{ where } p, q \in \mathbb{Z}, q \neq 0

H2=P2+B2  ⟹  H=P2+B2H^2 = P^2 + B^2 \implies H = \sqrt{P^2 + B^2}

R=Rational Numbers∪Irrational Numbers\mathbb{R} = \text{Rational Numbers} \cup \text{Irrational Numbers}

Rational±Irrational=Irrational\text{Rational} \pm \text{Irrational} = \text{Irrational}

Rational (non-zero)×Irrational=Irrational\text{Rational (non-zero)} \times \text{Irrational} = \text{Irrational}

💡Examples

Problem 1:

State whether the following statement is true or false: Every real number is an irrational number.

Solution:

False

Explanation:

The set of Real Numbers consists of both rational and irrational numbers. Therefore, while every irrational number is a real number, every real number is not necessarily irrational (it could be rational, like 22 or 35\frac{3}{5}).

Problem 2:

Classify the number (3+23)−23(3 + \sqrt{23}) - \sqrt{23} as rational or irrational.

Solution:

(3+23)−23=3+23−23=3(3 + \sqrt{23}) - \sqrt{23} = 3 + \sqrt{23} - \sqrt{23} = 3

Explanation:

Since 33 can be written as 31\frac{3}{1}, it is a rational number. The irrational terms 23\sqrt{23} cancel each other out.

Problem 3:

Find an irrational number between 17\frac{1}{7} and 27\frac{2}{7}.

Solution:

17=0.142857‾ and 27=0.285714‾\frac{1}{7} = 0.\overline{142857} \text{ and } \frac{2}{7} = 0.\overline{285714} An irrational number between them is 0.150150015000...0.150150015000...

Explanation:

To find an irrational number between two rational numbers, we first find their decimal representations. Then, we pick a number that lies between them and has a non-terminating, non-recurring decimal pattern.

Problem 4:

Show how 5\sqrt{5} can be represented on the number line.

Solution:

5=22+12\sqrt{5} = \sqrt{2^2 + 1^2}

Explanation:

  1. Take a units line on the number line. 2. Mark point OO at 00 and point AA at 22 units. 3. Draw a perpendicular ABAB of length 11 unit at AA. 4. Join OBOB. By Pythagoras theorem, OB=22+12=5OB = \sqrt{2^2 + 1^2} = \sqrt{5}. 5. Using a compass with center OO and radius OBOB, draw an arc intersecting the number line at point PP. Point PP represents 5\sqrt{5}.