Irrational Numbers
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Lesson Objectives
- Understand the definition of irrational numbers
- Distinguish between rational and irrational numbers
- Define surds as irrational square roots
- Simplify surds by identifying square factors
- Add, subtract, multiply, and divide surds
- Rationalise denominators containing surds
- Apply surd operations to solve problems
Introduction to Irrational Numbers
Irrational numbers are numbers that cannot be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$. Their decimal expansions are non-terminating and non-repeating. Famous examples include $\pi$, $e$, and $\sqrt{2}$. Understanding irrational numbers is essential for advanced mathematics, geometry, and science.
Irrational Numbers
Numbers that cannot be written as $\frac{p}{q}$ where $p, q \in \mathbb{Z}$, $q \neq 0$.
Examples: $\sqrt{2}$, $\pi$, $e$, $\sqrt{3}$, $\sqrt{5}$, $\phi$ (golden ratio)
Numbers that cannot be written as $\frac{p}{q}$ where $p, q \in \mathbb{Z}$, $q \neq 0$.
Examples: $\sqrt{2}$, $\pi$, $e$, $\sqrt{3}$, $\sqrt{5}$, $\phi$ (golden ratio)
Key Definitions:
• Irrational Number: A real number that cannot be expressed as a ratio of two integers.
• Rational Number: A number that can be expressed as $\frac{p}{q}$ where $p$ and $q$ are integers, $q \neq 0$.
• Surd: A root (square root, cube root, etc.) that is irrational (e.g., $\sqrt{2}$, $\sqrt{3}$, $\sqrt[3]{5}$).
• Simplest Surd Form: A surd written with the smallest possible number under the root sign.
• Irrational Number: A real number that cannot be expressed as a ratio of two integers.
• Rational Number: A number that can be expressed as $\frac{p}{q}$ where $p$ and $q$ are integers, $q \neq 0$.
• Surd: A root (square root, cube root, etc.) that is irrational (e.g., $\sqrt{2}$, $\sqrt{3}$, $\sqrt[3]{5}$).
• Simplest Surd Form: A surd written with the smallest possible number under the root sign.
Rational vs Irrational Numbers
All real numbers are either rational or irrational. Rational numbers include integers, fractions, terminating decimals, and repeating decimals. Irrational numbers include surds, $\pi$, $e$, and other non-repeating, non-terminating decimals.
Identifying Rational and Irrational Numbers:
1. Check if the number can be written as $\frac{p}{q}$ with integers $p$, $q$.
2. Terminating decimals (e.g., 0.5, 0.125) are rational.
3. Repeating decimals (e.g., $0.\dot{3}$, $0.\dot{1}4\dot{2}$) are rational.
4. Non-terminating, non-repeating decimals are irrational.
5. Square roots of non-perfect squares are irrational.
1. Check if the number can be written as $\frac{p}{q}$ with integers $p$, $q$.
2. Terminating decimals (e.g., 0.5, 0.125) are rational.
3. Repeating decimals (e.g., $0.\dot{3}$, $0.\dot{1}4\dot{2}$) are rational.
4. Non-terminating, non-repeating decimals are irrational.
5. Square roots of non-perfect squares are irrational.
Example 1: Identifying Rational Numbers
Problem: Which of these are rational? $0.5$, $\frac{2}{3}$, $\sqrt{4}$, $0.\dot{3}$, $\pi$
Solution:
$0.5 = \frac{1}{2}$ → rational
$\frac{2}{3}$ → rational
$\sqrt{4} = 2$ → rational
$0.\dot{3} = \frac{1}{3}$ → rational
$\pi$ → irrational
Answer: $0.5$, $\frac{2}{3}$, $\sqrt{4}$, $0.\dot{3}$ are rational.
Problem: Which of these are rational? $0.5$, $\frac{2}{3}$, $\sqrt{4}$, $0.\dot{3}$, $\pi$
Solution:
$0.5 = \frac{1}{2}$ → rational
$\frac{2}{3}$ → rational
$\sqrt{4} = 2$ → rational
$0.\dot{3} = \frac{1}{3}$ → rational
$\pi$ → irrational
Answer: $0.5$, $\frac{2}{3}$, $\sqrt{4}$, $0.\dot{3}$ are rational.
Example 2: Identifying Irrational Numbers
Problem: Which of these are irrational? $\sqrt{2}$, $\sqrt{9}$, $3.14159...$, $0.1010010001...$
Solution:
$\sqrt{2} \approx 1.414213...$ non-terminating, non-repeating → irrational
$\sqrt{9} = 3$ → rational
$3.14159...$ (π) → irrational
$0.1010010001...$ (pattern but not repeating) → irrational
Answer: $\sqrt{2}$, $\pi$, $0.1010010001...$ are irrational.
Problem: Which of these are irrational? $\sqrt{2}$, $\sqrt{9}$, $3.14159...$, $0.1010010001...$
Solution:
$\sqrt{2} \approx 1.414213...$ non-terminating, non-repeating → irrational
$\sqrt{9} = 3$ → rational
$3.14159...$ (π) → irrational
$0.1010010001...$ (pattern but not repeating) → irrational
Answer: $\sqrt{2}$, $\pi$, $0.1010010001...$ are irrational.
Number System Hierarchy
| Number Type | Examples | Decimal Form |
|---|---|---|
| Natural Numbers | $1, 2, 3, 4, ...$ | Terminating |
| Whole Numbers | $0, 1, 2, 3, ...$ | Terminating |
| Integers | $..., -2, -1, 0, 1, 2, ...$ | Terminating |
| Rational Numbers | $\frac{1}{2}, \frac{2}{3}, 0.75, 0.\dot{3}$ | Terminating or Repeating |
| Irrational Numbers | $\sqrt{2}, \pi, e, \sqrt{3}$ | Non-terminating, Non-repeating |
Watch Out!
Not all square roots are irrational! $\sqrt{4} = 2$ (rational), $\sqrt{9} = 3$ (rational), $\sqrt{16} = 4$ (rational). Only square roots of non-perfect squares are irrational.
Not all square roots are irrational! $\sqrt{4} = 2$ (rational), $\sqrt{9} = 3$ (rational), $\sqrt{16} = 4$ (rational). Only square roots of non-perfect squares are irrational.
Practice for Concept 1 (Rational vs Irrational)
- Is $0.75$ rational or irrational?
- Is $\sqrt{25}$ rational or irrational?
- Is $\sqrt{7}$ rational or irrational?
- Is $0.\dot{1}2\dot{3}$ rational or irrational?
- Classify: $\frac{22}{7}$, $\sqrt{10}$, $3.14$, $\pi$, $\sqrt{49}$
Introduction to Surds
A surd is a root (square root, cube root, etc.) that is irrational. The most common surds are square roots of non-perfect squares, such as $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{6}$, $\sqrt{7}$, $\sqrt{8}$, $\sqrt{10}$, etc. Surds are exact values and are often left in surd form for precision.
Properties of Surds:
1. $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ for $a, b \geq 0$
2. $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ for $a \geq 0, b > 0$
3. $\sqrt{a^2} = a$ for $a \geq 0$
4. $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ (useful for simplification)
1. $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ for $a, b \geq 0$
2. $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ for $a \geq 0, b > 0$
3. $\sqrt{a^2} = a$ for $a \geq 0$
4. $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ (useful for simplification)
Example 1: Identifying Surds
Problem: Which of these are surds? $\sqrt{2}$, $\sqrt{9}$, $\sqrt{12}$, $\sqrt{16}$, $\sqrt{20}$
Solution:
$\sqrt{2}$ → surd (non-perfect square)
$\sqrt{9} = 3$ → not a surd (perfect square)
$\sqrt{12}$ → surd (non-perfect square)
$\sqrt{16} = 4$ → not a surd
$\sqrt{20}$ → surd
Answer: $\sqrt{2}$, $\sqrt{12}$, $\sqrt{20}$ are surds.
Problem: Which of these are surds? $\sqrt{2}$, $\sqrt{9}$, $\sqrt{12}$, $\sqrt{16}$, $\sqrt{20}$
Solution:
$\sqrt{2}$ → surd (non-perfect square)
$\sqrt{9} = 3$ → not a surd (perfect square)
$\sqrt{12}$ → surd (non-perfect square)
$\sqrt{16} = 4$ → not a surd
$\sqrt{20}$ → surd
Answer: $\sqrt{2}$, $\sqrt{12}$, $\sqrt{20}$ are surds.
Example 2: Simplifying Basic Surds
Problem: Simplify $\sqrt{8}$.
Solution:
$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$
Answer: $2\sqrt{2}$
Problem: Simplify $\sqrt{8}$.
Solution:
$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$
Answer: $2\sqrt{2}$
Practice for Concept 2 (Identifying Surds)
- Which of these are surds? $\sqrt{18}$, $\sqrt{36}$, $\sqrt{50}$, $\sqrt{64}$
- Simplify $\sqrt{12}$.
- Simplify $\sqrt{45}$.
- Simplify $\sqrt{72}$.
- Simplify $\sqrt{32}$.
Simplifying Surds
To simplify a surd, find the largest square factor of the number under the root sign. Write the surd as the product of the square root of that square factor and the remaining surd.
Step-by-Step Method for Simplifying Surds:
1. Find the largest perfect square that divides the number under the root.
2. Write the number as a product of that perfect square and another factor.
3. Take the square root of the perfect square outside the root.
4. Leave the remaining factor under the root sign.
1. Find the largest perfect square that divides the number under the root.
2. Write the number as a product of that perfect square and another factor.
3. Take the square root of the perfect square outside the root.
4. Leave the remaining factor under the root sign.
Example 1: Simplifying $\sqrt{18}$
Problem: Simplify $\sqrt{18}$.
Solution:
Largest square factor of 18 is 9.
$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$
Answer: $3\sqrt{2}$
Problem: Simplify $\sqrt{18}$.
Solution:
Largest square factor of 18 is 9.
$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$
Answer: $3\sqrt{2}$
Example 2: Simplifying $\sqrt{50}$
Problem: Simplify $\sqrt{50}$.
Solution:
Largest square factor of 50 is 25.
$\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$
Answer: $5\sqrt{2}$
Problem: Simplify $\sqrt{50}$.
Solution:
Largest square factor of 50 is 25.
$\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$
Answer: $5\sqrt{2}$
Example 3: Simplifying $\sqrt{72}$
Problem: Simplify $\sqrt{72}$.
Solution:
Largest square factor of 72 is 36.
$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$
Answer: $6\sqrt{2}$
Problem: Simplify $\sqrt{72}$.
Solution:
Largest square factor of 72 is 36.
$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$
Answer: $6\sqrt{2}$
Example 4: Simplifying $\sqrt{48}$
Problem: Simplify $\sqrt{48}$.
Solution:
Largest square factor of 48 is 16.
$\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}$
Answer: $4\sqrt{3}$
Problem: Simplify $\sqrt{48}$.
Solution:
Largest square factor of 48 is 16.
$\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}$
Answer: $4\sqrt{3}$
Practice for Concept 3 (Simplifying Surds)
- Simplify $\sqrt{20}$.
- Simplify $\sqrt{28}$.
- Simplify $\sqrt{54}$.
- Simplify $\sqrt{75}$.
- Simplify $\sqrt{98}$.
Operations with Surds
Surds can be added, subtracted, multiplied, and divided like algebraic terms. Like surds (same number under the root) can be combined. Multiplication and division follow the properties of radicals.
Rules for Operations with Surds:
1. Addition/Subtraction: Only combine like surds: $a\sqrt{c} + b\sqrt{c} = (a+b)\sqrt{c}$
2. Multiplication: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$
3. Division: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
4. Rationalising the Denominator: Multiply numerator and denominator by the surd to eliminate the root from the denominator.
1. Addition/Subtraction: Only combine like surds: $a\sqrt{c} + b\sqrt{c} = (a+b)\sqrt{c}$
2. Multiplication: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$
3. Division: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
4. Rationalising the Denominator: Multiply numerator and denominator by the surd to eliminate the root from the denominator.
Example 1: Adding Like Surds
Problem: Simplify $3\sqrt{2} + 5\sqrt{2}$.
Solution:
$3\sqrt{2} + 5\sqrt{2} = (3+5)\sqrt{2} = 8\sqrt{2}$
Answer: $8\sqrt{2}$
Problem: Simplify $3\sqrt{2} + 5\sqrt{2}$.
Solution:
$3\sqrt{2} + 5\sqrt{2} = (3+5)\sqrt{2} = 8\sqrt{2}$
Answer: $8\sqrt{2}$
Example 2: Subtracting Like Surds
Problem: Simplify $7\sqrt{3} - 2\sqrt{3}$.
Solution:
$7\sqrt{3} - 2\sqrt{3} = (7-2)\sqrt{3} = 5\sqrt{3}$
Answer: $5\sqrt{3}$
Problem: Simplify $7\sqrt{3} - 2\sqrt{3}$.
Solution:
$7\sqrt{3} - 2\sqrt{3} = (7-2)\sqrt{3} = 5\sqrt{3}$
Answer: $5\sqrt{3}$
Example 3: Adding Unlike Surds (Simplify First)
Problem: Simplify $\sqrt{8} + \sqrt{18}$.
Solution:
$\sqrt{8} = 2\sqrt{2}$, $\sqrt{18} = 3\sqrt{2}$
$2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}$
Answer: $5\sqrt{2}$
Problem: Simplify $\sqrt{8} + \sqrt{18}$.
Solution:
$\sqrt{8} = 2\sqrt{2}$, $\sqrt{18} = 3\sqrt{2}$
$2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}$
Answer: $5\sqrt{2}$
Example 4: Multiplying Surds
Problem: Simplify $\sqrt{3} \times \sqrt{5}$.
Solution:
$\sqrt{3} \times \sqrt{5} = \sqrt{15}$
Answer: $\sqrt{15}$
Problem: Simplify $\sqrt{3} \times \sqrt{5}$.
Solution:
$\sqrt{3} \times \sqrt{5} = \sqrt{15}$
Answer: $\sqrt{15}$
Example 5: Multiplying Surds with Coefficients
Problem: Simplify $2\sqrt{3} \times 4\sqrt{5}$.
Solution:
$2 \times 4 \times \sqrt{3} \times \sqrt{5} = 8\sqrt{15}$
Answer: $8\sqrt{15}$
Problem: Simplify $2\sqrt{3} \times 4\sqrt{5}$.
Solution:
$2 \times 4 \times \sqrt{3} \times \sqrt{5} = 8\sqrt{15}$
Answer: $8\sqrt{15}$
Example 6: Dividing Surds
Problem: Simplify $\frac{\sqrt{12}}{\sqrt{3}}$.
Solution:
$\frac{\sqrt{12}}{\sqrt{3}} = \sqrt{\frac{12}{3}} = \sqrt{4} = 2$
Answer: $2$
Problem: Simplify $\frac{\sqrt{12}}{\sqrt{3}}$.
Solution:
$\frac{\sqrt{12}}{\sqrt{3}} = \sqrt{\frac{12}{3}} = \sqrt{4} = 2$
Answer: $2$
Example 7: Rationalising the Denominator
Problem: Rationalise $\frac{3}{\sqrt{2}}$.
Solution:
Multiply numerator and denominator by $\sqrt{2}$:
$\frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2}$
Answer: $\frac{3\sqrt{2}}{2}$
Problem: Rationalise $\frac{3}{\sqrt{2}}$.
Solution:
Multiply numerator and denominator by $\sqrt{2}$:
$\frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2}$
Answer: $\frac{3\sqrt{2}}{2}$
Example 8: Rationalising with Conjugate
Problem: Rationalise $\frac{1}{2+\sqrt{3}}$.
Solution:
Multiply numerator and denominator by the conjugate $2-\sqrt{3}$:
$\frac{1}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}} = \frac{2-\sqrt{3}}{4 - 3} = 2 - \sqrt{3}$
Answer: $2 - \sqrt{3}$
Problem: Rationalise $\frac{1}{2+\sqrt{3}}$.
Solution:
Multiply numerator and denominator by the conjugate $2-\sqrt{3}$:
$\frac{1}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}} = \frac{2-\sqrt{3}}{4 - 3} = 2 - \sqrt{3}$
Answer: $2 - \sqrt{3}$
Watch Out!
You cannot add unlike surds directly (e.g., $\sqrt{2} + \sqrt{3}$ cannot be simplified further). Always simplify surds first before attempting to combine them.
You cannot add unlike surds directly (e.g., $\sqrt{2} + \sqrt{3}$ cannot be simplified further). Always simplify surds first before attempting to combine them.
Practice for Concept 4 (Operations with Surds)
- Simplify $4\sqrt{5} + 2\sqrt{5}$.
- Simplify $5\sqrt{7} - 3\sqrt{7}$.
- Simplify $\sqrt{12} + \sqrt{27}$.
- Simplify $\sqrt{3} \times \sqrt{6}$.
- Simplify $2\sqrt{5} \times 3\sqrt{10}$.
- Simplify $\frac{\sqrt{20}}{\sqrt{5}}$.
- Rationalise $\frac{5}{\sqrt{3}}$.
- Rationalise $\frac{2}{3-\sqrt{2}}$.
Methods & Techniques
Mastering surds requires practice with simplification and operations. Use these strategies to avoid errors.
Verification / Checking Strategy:
1. For simplification: Square your answer to see if you get the original number.
2. For addition/subtraction: Check that surds are simplified before combining.
3. For rationalising: Multiply back to verify the denominator no longer contains a surd.
4. Use decimal approximations: Approximate both the original and simplified forms to check consistency.
1. For simplification: Square your answer to see if you get the original number.
2. For addition/subtraction: Check that surds are simplified before combining.
3. For rationalising: Multiply back to verify the denominator no longer contains a surd.
4. Use decimal approximations: Approximate both the original and simplified forms to check consistency.
Example: Checking Simplification
Original: $\sqrt{50}$ simplified to $5\sqrt{2}$.
Check:
$(5\sqrt{2})^2 = 25 \times 2 = 50$ ✓
Decimal check: $\sqrt{50} \approx 7.071$, $5\sqrt{2} \approx 5 \times 1.414 = 7.07$ ✓
Original: $\sqrt{50}$ simplified to $5\sqrt{2}$.
Check:
$(5\sqrt{2})^2 = 25 \times 2 = 50$ ✓
Decimal check: $\sqrt{50} \approx 7.071$, $5\sqrt{2} \approx 5 \times 1.414 = 7.07$ ✓
Common Pitfalls & How to Avoid Them:
• Pitfall 1: Adding unlike surds ($\sqrt{2} + \sqrt{3} \neq \sqrt{5}$) → Solution: Only combine like surds.
• Pitfall 2: Forgetting to simplify surds before adding → Solution: Always simplify each surd first.
• Pitfall 3: Incorrect rationalisation → Solution: Use the conjugate for binomial denominators.
• Pitfall 4: Confusing $\sqrt{a+b}$ with $\sqrt{a} + \sqrt{b}$ → Solution: Remember $\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}$ (e.g., $\sqrt{9+16}=5$, but $\sqrt{9}+\sqrt{16}=3+4=7$).
• Pitfall 5: Leaving surds unsimplified → Solution: Always look for square factors.
• Pitfall 1: Adding unlike surds ($\sqrt{2} + \sqrt{3} \neq \sqrt{5}$) → Solution: Only combine like surds.
• Pitfall 2: Forgetting to simplify surds before adding → Solution: Always simplify each surd first.
• Pitfall 3: Incorrect rationalisation → Solution: Use the conjugate for binomial denominators.
• Pitfall 4: Confusing $\sqrt{a+b}$ with $\sqrt{a} + \sqrt{b}$ → Solution: Remember $\sqrt{a+b} \neq \sqrt{a} + \sqrt{b}$ (e.g., $\sqrt{9+16}=5$, but $\sqrt{9}+\sqrt{16}=3+4=7$).
• Pitfall 5: Leaving surds unsimplified → Solution: Always look for square factors.
Common Surds and Their Simplified Forms
| Unsimplified Surd | Simplified Form |
|---|---|
| $\sqrt{8}$ | $2\sqrt{2}$ |
| $\sqrt{12}$ | $2\sqrt{3}$ |
| $\sqrt{18}$ | $3\sqrt{2}$ |
| $\sqrt{20}$ | $2\sqrt{5}$ |
| $\sqrt{27}$ | $3\sqrt{3}$ |
| $\sqrt{28}$ | $2\sqrt{7}$ |
| $\sqrt{32}$ | $4\sqrt{2}$ |
| $\sqrt{45}$ | $3\sqrt{5}$ |
| $\sqrt{48}$ | $4\sqrt{3}$ |
| $\sqrt{50}$ | $5\sqrt{2}$ |
Technique Practice
- Verify that $\sqrt{72} = 6\sqrt{2}$ by squaring both sides.
- Check if $\sqrt{2} + \sqrt{8} = 3\sqrt{2}$ is correct by simplifying $\sqrt{8}$ first.
- Identify the error: A student wrote $\frac{5}{\sqrt{5}} = \sqrt{5}$. Is this correct? Verify.
- For $\sqrt{50} + \sqrt{18}$, a student got $\sqrt{68}$. Is this correct? Explain.
Real-World Applications
Irrational numbers and surds appear in geometry, physics, engineering, and many other fields where exact values are required.
Application 1: Pythagoras' Theorem
Scenario: A right-angled triangle has legs of length 3 cm and 4 cm. The hypotenuse is $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$ cm (rational).
Problem: If legs are 3 cm and 5 cm, find the hypotenuse.
Solution:
$c = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}$ cm
$\sqrt{34}$ is a surd (cannot be simplified further).
Scenario: A right-angled triangle has legs of length 3 cm and 4 cm. The hypotenuse is $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$ cm (rational).
Problem: If legs are 3 cm and 5 cm, find the hypotenuse.
Solution:
$c = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}$ cm
$\sqrt{34}$ is a surd (cannot be simplified further).
Application 2: Diagonal of a Square
Scenario: A square has side length $s$. The diagonal is $s\sqrt{2}$.
Problem: If a square has side length 5 cm, find the diagonal length exactly.
Solution:
Diagonal = $5\sqrt{2}$ cm. This is exact and uses a surd.
Scenario: A square has side length $s$. The diagonal is $s\sqrt{2}$.
Problem: If a square has side length 5 cm, find the diagonal length exactly.
Solution:
Diagonal = $5\sqrt{2}$ cm. This is exact and uses a surd.
Application 3: Quadratic Formula
Scenario: Solving $x^2 - 4x + 1 = 0$ using the quadratic formula.
Problem: Find the exact solutions.
Solution:
$x = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3}$
The solutions $2 + \sqrt{3}$ and $2 - \sqrt{3}$ involve surds.
Scenario: Solving $x^2 - 4x + 1 = 0$ using the quadratic formula.
Problem: Find the exact solutions.
Solution:
$x = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3}$
The solutions $2 + \sqrt{3}$ and $2 - \sqrt{3}$ involve surds.
Application 4: Golden Ratio
Scenario: The golden ratio $\phi = \frac{1+\sqrt{5}}{2}$ appears in art, architecture, and nature.
Problem: Simplify $\frac{1+\sqrt{5}}{2}$.
Solution:
This is already in simplest surd form. $\phi \approx 1.618$ is irrational.
Scenario: The golden ratio $\phi = \frac{1+\sqrt{5}}{2}$ appears in art, architecture, and nature.
Problem: Simplify $\frac{1+\sqrt{5}}{2}$.
Solution:
This is already in simplest surd form. $\phi \approx 1.618$ is irrational.
Cross-Curricular Connections
- Geometry: Diagonal of a square ($s\sqrt{2}$), diagonal of a cube ($s\sqrt{3}$)
- Physics: Root-mean-square speed, period of a pendulum ($T = 2\pi\sqrt{\frac{L}{g}}$)
- Architecture: Golden ratio ($\phi = \frac{1+\sqrt{5}}{2}$) in design
- Trigonometry: Exact values like $\sin 45^\circ = \frac{\sqrt{2}}{2}$, $\sin 60^\circ = \frac{\sqrt{3}}{2}$
Cumulative Practice Exercises
Try these problems on your own. Show all working steps. Use the verification strategies to check your answers.
- Is $\sqrt{49}$ rational or irrational?
- Is $0.1010010001...$ rational or irrational?
- Simplify $\sqrt{28}$.
- Simplify $\sqrt{72}$.
- Simplify $3\sqrt{5} + 2\sqrt{5}$.
- Simplify $\sqrt{20} + \sqrt{45}$.
- Simplify $\sqrt{6} \times \sqrt{10}$.
- Simplify $3\sqrt{2} \times 2\sqrt{8}$.
- Simplify $\frac{\sqrt{50}}{\sqrt{2}}$.
- Rationalise $\frac{4}{\sqrt{2}}$.
- Rationalise $\frac{5}{2\sqrt{3}}$.
- Rationalise $\frac{3}{1+\sqrt{2}}$.
- Find the exact length of the diagonal of a square with side 7 cm.
- Error analysis: A student simplified $\sqrt{18} + \sqrt{8}$ as $\sqrt{26}$. Is this correct? If not, what is the correct answer?
- Prove that $\sqrt{2}$ is irrational. (Hint: Assume $\sqrt{2} = \frac{p}{q}$ in lowest terms, then show contradiction.)
Answers to Cumulative Exercises
- Problem: $\sqrt{49}$ rational or irrational?
Answer: $\sqrt{49} = 7$, rational - Problem: $0.1010010001...$ rational or irrational?
Answer: Non-terminating, non-repeating → irrational - Problem: Simplify $\sqrt{28}$.
Answer: $\sqrt{4 \times 7} = 2\sqrt{7}$ - Problem: Simplify $\sqrt{72}$.
Answer: $\sqrt{36 \times 2} = 6\sqrt{2}$ - Problem: $3\sqrt{5} + 2\sqrt{5}$.
Answer: $5\sqrt{5}$ - Problem: $\sqrt{20} + \sqrt{45}$.
Answer: $2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}$ - Problem: $\sqrt{6} \times \sqrt{10}$.
Answer: $\sqrt{60} = \sqrt{4 \times 15} = 2\sqrt{15}$ - Problem: $3\sqrt{2} \times 2\sqrt{8}$.
Answer: $6\sqrt{16} = 6 \times 4 = 24$ - Problem: $\frac{\sqrt{50}}{\sqrt{2}}$.
Answer: $\sqrt{25} = 5$ - Problem: Rationalise $\frac{4}{\sqrt{2}}$.
Answer: $\frac{4\sqrt{2}}{2} = 2\sqrt{2}$ - Problem: Rationalise $\frac{5}{2\sqrt{3}}$.
Answer: $\frac{5\sqrt{3}}{2 \times 3} = \frac{5\sqrt{3}}{6}$ - Problem: Rationalise $\frac{3}{1+\sqrt{2}}$.
Answer: $\frac{3(1-\sqrt{2})}{1-2} = \frac{3(1-\sqrt{2})}{-1} = -3 + 3\sqrt{2}$ or $3\sqrt{2} - 3$ - Problem: Diagonal of square side 7 cm.
Answer: $7\sqrt{2}$ cm - Problem: Error analysis: $\sqrt{18} + \sqrt{8} = \sqrt{26}$?
Answer: Incorrect. $\sqrt{18} = 3\sqrt{2}$, $\sqrt{8} = 2\sqrt{2}$, sum = $5\sqrt{2}$ - Problem: Prove $\sqrt{2}$ is irrational.
Answer: Assume $\sqrt{2} = p/q$ in lowest terms. Then $2 = p^2/q^2$ → $p^2 = 2q^2$, so $p^2$ is even → $p$ is even. Let $p = 2k$, then $4k^2 = 2q^2$ → $2k^2 = q^2$, so $q^2$ is even → $q$ is even. This contradicts $p/q$ being in lowest terms. Therefore $\sqrt{2}$ is irrational.
Conclusion & Summary
Irrational numbers are real numbers that cannot be expressed as fractions. Surds are a special type of irrational number that involve roots. Simplifying surds and performing operations with them are essential skills for algebra, geometry, and higher mathematics.
Key Takeaways:
1. Irrational numbers: Non-terminating, non-repeating decimals; cannot be written as $\frac{p}{q}$.
2. Surds: Roots of non-perfect squares (e.g., $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$).
3. Simplifying surds: Find the largest square factor and take its root outside.
4. Adding/Subtracting surds: Only combine like surds; simplify first.
5. Multiplying/Dividing surds: Use $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ and $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$.
6. Rationalising: Eliminate surds from denominators by multiplying by the surd or its conjugate.
7. Applications: Pythagoras' theorem, quadratic formula, golden ratio.
Keep practicing with surds. They appear frequently in geometry, trigonometry, and calculus!
Video Resource
Watch this video for more examples of simplifying surds and operations with surds.
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