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Number - Rational and Irrational Numbers

Grade 12A Level

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

🔑Concepts

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Definition of Rational Numbers (Q): Any number that can be expressed in the form p/qp/q, where pp and qq are integers and q≠0q \neq 0.

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Terminating and Recurring Decimals: All rational numbers, when written as decimals, either terminate (e.g., 0.25) or repeat in a pattern (e.g., 0.333...).

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Definition of Irrational Numbers: Numbers that cannot be expressed as a simple fraction. Their decimal expansions are non-terminating and non-recurring.

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Common Irrational Numbers: Square roots of non-perfect squares (surds) like 2\sqrt{2} or 3\sqrt{3}, and mathematical constants like π\pi and ee.

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Surds: Irrational roots of rational numbers. Rules for simplification include ab=ab\sqrt{ab} = \sqrt{a}\sqrt{b}.

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Rationalizing the Denominator: The process of removing surds from the bottom of a fraction using a conjugate.

📐Formulae

Rational Form: x=pq, where p,q∈Z,q≠0\text{Rational Form: } x = \frac{p}{q}, \text{ where } p, q \in \mathbb{Z}, q \neq 0

Surd Multiplication: a×b=ab\text{Surd Multiplication: } \sqrt{a} \times \sqrt{b} = \sqrt{ab}

Surd Division: ab=ab\text{Surd Division: } \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}

Rationalizing (Simple): 1a=aa\text{Rationalizing (Simple): } \frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}

Rationalizing (Conjugate): 1a+b=a−ba2−b\text{Rationalizing (Conjugate): } \frac{1}{a + \sqrt{b}} = \frac{a - \sqrt{b}}{a^2 - b}

💡Examples

Problem 1:

Convert the recurring decimal 0.47˙0.4\dot{7} (or 0.4777...0.4777...) into a fraction in its simplest form.

Solution:

Let x=0.4777...x = 0.4777... Multiply by 10: 10x=4.777...10x = 4.777... Multiply by 100: 100x=47.777...100x = 47.777... Subtract the two equations: 100x−10x=47.777−4.777100x - 10x = 47.777 - 4.777 90x=4390x = 43 x=4390x = \frac{43}{90}

Explanation:

To convert a recurring decimal, we create two equations with the same recurring tail and subtract them to eliminate the infinite decimal part.

Problem 2:

Simplify the expression by rationalizing the denominator: 43−5\frac{4}{3 - \sqrt{5}}

Solution:

Multiply the numerator and denominator by the conjugate (3+5)(3 + \sqrt{5}): 4(3+5)(3−5)(3+5)\frac{4(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})} =12+4532−(5)2= \frac{12 + 4\sqrt{5}}{3^2 - (\sqrt{5})^2} =12+459−5= \frac{12 + 4\sqrt{5}}{9 - 5} =12+454= \frac{12 + 4\sqrt{5}}{4} =3+5= 3 + \sqrt{5}

Explanation:

We use the identity (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2 to eliminate the surd in the denominator.

Problem 3:

Determine if the product of 8\sqrt{8} and 2\sqrt{2} is rational or irrational.

Solution:

8×2=8×2=16\sqrt{8} \times \sqrt{2} = \sqrt{8 \times 2} = \sqrt{16} 16=4\sqrt{16} = 4

Explanation:

Since 4 is an integer, it can be written as 4/14/1, making the result a rational number even though the individual factors were irrational.