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Number - Surds and Irrational Numbers

Grade 10IGCSE

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

🔑Concepts

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Definition of Irrational Numbers: Numbers that cannot be expressed as a fraction p/q where p and q are integers (e.g., π\pi, 2\sqrt{2}).

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Surds: Irrational numbers left in root form to maintain exact precision.

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Simplifying Surds: Factoring out the largest perfect square from under the radical symbol.

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Like Surds: Surds with the same number under the root, which can be added or subtracted.

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Rationalising the Denominator: The process of removing a radical from the bottom of a fraction by multiplying the numerator and denominator by an appropriate factor.

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Conjugate Pairs: Using the identity (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2 to rationalise denominators containing two terms.

📐Formulae

ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b}

ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}

ac±bc=(a±b)ca\sqrt{c} \pm b\sqrt{c} = (a \pm b)\sqrt{c}

ka=kaa\frac{k}{\sqrt{a}} = \frac{k\sqrt{a}}{a}

ka+b=k(a−b)a2−b\frac{k}{a + \sqrt{b}} = \frac{k(a - \sqrt{b})}{a^2 - b}

💡Examples

Problem 1:

Simplify 72+50\sqrt{72} + \sqrt{50}.

Solution:

62+52=1126\sqrt{2} + 5\sqrt{2} = 11\sqrt{2}

Explanation:

First, find the largest square factor for each: 72=36×2=62\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2} and 50=25×2=52\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}. Since they are now 'like surds', they can be added.

Problem 2:

Expand and simplify (3+5)(3−2)(\sqrt{3} + 5)(\sqrt{3} - 2).

Solution:

3−23+53−10=33−73 - 2\sqrt{3} + 5\sqrt{3} - 10 = 3\sqrt{3} - 7

Explanation:

Use the FOIL method: 33=3\sqrt{3}\sqrt{3} = 3, 3×(−2)=−23\sqrt{3} \times (-2) = -2\sqrt{3}, 5×3=535 \times \sqrt{3} = 5\sqrt{3}, and 5×(−2)=−105 \times (-2) = -10. Combine the integers and the surd terms.

Problem 3:

Rationalise the denominator of 43−5\frac{4}{3 - \sqrt{5}}.

Solution:

4(3+5)(3−5)(3+5)=12+459−5=12+454=3+5\frac{4(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})} = \frac{12 + 4\sqrt{5}}{9 - 5} = \frac{12 + 4\sqrt{5}}{4} = 3 + \sqrt{5}

Explanation:

Multiply the numerator and denominator by the conjugate of the denominator (3+53 + \sqrt{5}). The denominator simplifies using the difference of two squares: 32−(5)2=9−5=43^2 - (\sqrt{5})^2 = 9 - 5 = 4. Finally, divide each term in the numerator by 4.