Working with Irrational Numbers Revision Notes for Grade 12 NSC Matric Mathematics

Working with Irrational Numbers

Understanding irrational numbers and surds

Irrational numbers are real numbers that cannot be expressed as simple fractions. These numbers have decimal expansions that neither terminate nor repeat in a pattern. When we work with square roots, cube roots, and other radicals that don't simplify to whole numbers, we often encounter irrational numbers.

Surds are a special type of irrational number. They are expressions involving square roots, cube roots, or higher-order roots that cannot be simplified to rational numbers. Common examples include $\sqrt{3}$ , $\sqrt{5}$ , $\sqrt{6}$ , $\sqrt{7}$ , and $\sqrt{8}$ .

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It's important to distinguish surds from rational square roots. Not all square roots are surds. For instance:

$\sqrt{1} = 1$ (rational)
$\sqrt{4} = 2$ (rational)
$\sqrt{9} = 3$ (rational)
$\sqrt{16} = 4$ (rational)

These are not surds because they simplify to whole numbers.

Locating surds on a number line

To estimate where a surd falls on a number line, you need to identify two consecutive whole numbers between which the surd lies. This technique helps you understand the approximate value of irrational numbers.

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Method: Find perfect squares close to your number and determine which whole numbers they correspond to.

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Worked Example 1: Locating √5

To locate $\sqrt{5}$ on a number line:

Consider that $2^2 = 4$ and $3^2 = 9$
Since $4 < 5 < 9$ , we know that $2 < \sqrt{5} < 3$
$\sqrt{5}$ is closer to 2 than to 3

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Worked Example 2: Locating √17

To locate $\sqrt{17}$ on a number line:

Consider that $4^2 = 16$ and $5^2 = 25$
Since $16 < 17 < 25$ , we know that $4 < \sqrt{17} < 5$
$\sqrt{17}$ is very close to 4

Simplifying surds through operations

Multiplication of surds

When multiplying surds, you can combine them under a single radical sign using the rule:

$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$

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Worked Example 3: Multiplying Surds

$\sqrt{5} \times \sqrt{3} = \sqrt{15}$

This rule makes calculations more efficient and helps simplify complex expressions.

Division of surds

Division of surds follows a similar pattern:

$\sqrt{a} \div \sqrt{b} = \sqrt{a \div b}$

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Worked Example 4: Dividing Surds

$\sqrt{18} \div \sqrt{2} = \sqrt{9} = 3$

Simplifying square roots by factoring

The key to simplifying surds is to identify and extract perfect square factors. Look for factors of the number under the radical that are perfect squares, then separate them out.

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Method:

Factor the number under the square root
Identify perfect square factors
Take the square root of perfect squares and move them outside the radical

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Worked Example 5: Simplifying by Factoring

$\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}$

The 9 is a perfect square ( $3^2$ ), so its square root (3) comes out of the radical, leaving $\sqrt{3}$ inside.

Addition and subtraction of surds

You can only add or subtract surds that have the same radical part - these are called like terms.

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Rule: Surds with identical radical parts can be combined by adding or subtracting their coefficients.

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Worked Example 6: Adding Like Terms

$4\sqrt{2} + 3\sqrt{2} = 7\sqrt{2}$

The radical parts ( $\sqrt{2}$ ) are identical, so we add the coefficients: $4 + 3 = 7$ .

Rationalising the denominator

Rationalising means removing surds from the denominator of a fraction. This makes fractions easier to work with and is often required in final answers.

Basic rationalisation

To rationalise a simple surd in the denominator, multiply both numerator and denominator by the same surd.

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Worked Example 7: Basic Rationalisation

To rationalise $\frac{\sqrt{3}}{\sqrt{2}}$ :

Multiply top and bottom by $\sqrt{2}$
$\frac{\sqrt{3} \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{\sqrt{6}}{2}$

The denominator becomes 2 (a rational number) because $\sqrt{2} \times \sqrt{2} = 2$ .

Using the difference of two squares

For more complex denominators involving sums or differences of surds, use the conjugate method.

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Worked Example 8: Using Conjugates

To rationalise $\frac{3}{\sqrt{3} - 1}$ :

Multiply by the conjugate $\frac{\sqrt{3} + 1}{\sqrt{3} + 1}$
$\frac{3(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)}$
$\frac{3(\sqrt{3} + 1)}{3 - 1} = \frac{3(\sqrt{3} + 1)}{2} = \frac{3\sqrt{3} + 3}{2}$

Additional worked examples

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Worked Example 9: Simplify √8 × √2

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

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Worked Example 10: Simplify √50/√2

$\frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}} = \sqrt{25} = 5$

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Worked Example 11: Rationalise 4/√5

To rationalise $\frac{4}{\sqrt{5}}$ :

Multiply top and bottom by $\sqrt{5}$
$\frac{4 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{4\sqrt{5}}{5}$

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Key Points to Remember:

Surds are irrational square roots that cannot be simplified to whole numbers
Multiplication and division of surds can be combined under one radical sign
Addition and subtraction only works with like terms (same radical part)
Simplification involves factoring out perfect squares from under the radical
Rationalisation removes surds from denominators by strategic multiplication

Working with Irrational Numbers (Grade 12 NSC Matric Mathematics): Revision Notes