2.1 Introduction to Real Numbers
Real numbers are all the numbers you can find on the cosmic number line of the universe. They include:
- 🌠 Counting numbers (1, 2, 3...)
- 🪐 Whole numbers (0, 1, 2...)
- ☄️ Integers (..., -2, -1, 0, 1, 2...)
- 🔮 Fractions (½, ¾, etc.)
- 💫 Decimals (0.5, 3.14, etc.)
- ⚡ Irrational numbers (√2, π, etc.)
🚀 Quick Cosmic Quiz!
Which of these are real numbers?
2.2 Rational Numbers
A rational number is any number that can be expressed as a fraction p/q where p and q are integers and q ≠ 0.
🌈 Examples: ½, 0.75 (which is ¾), -2 (which is -2/1), 0.333... (which is ⅓)
Rational Numbers Visualization
Rational numbers can be terminating decimals (like 0.5) or repeating decimals (like 0.333...).
Examples in Different Forms
- 🔢 Fraction: ⅔
- ⏹️ Terminating decimal: 0.4
- ♾️ Repeating decimal: 0.142857142857...
- 🔷 Integer: -7
Example: Is 0.125 a rational number?
Yes! Because 0.125 can be written as 125/1000, which simplifies to ⅛.
2.3 Irrational Numbers
An irrational number cannot be expressed as a simple fraction. Its decimal form is non-terminating and non-repeating.
🌌 Examples: √2, π (pi), e (Euler's number), φ (golden ratio)
Famous Irrational Numbers
- π ≈ 3.141592653589793...
- √2 ≈ 1.414213562373095...
- e ≈ 2.718281828459045...
- φ ≈ 1.618033988749894...
Proof that √2 is irrational
Assume √2 is rational. Then √2 = a/b where a and b have no common factors.
Squaring both sides: 2 = a²/b² → 2b² = a²
This means a² is even, so a must be even. Let a = 2k.
Then 2b² = (2k)² → 2b² = 4k² → b² = 2k²
Now b² is even, so b must be even. But this contradicts our assumption that a and b have no common factors.
Therefore, √2 cannot be rational.
🔍 Irrational Number Explorer
Calculate square roots to see which are irrational:
2.4 Real Numbers
Real numbers include all rational and irrational numbers. They can be represented on the cosmic number line.
Every point on the number line corresponds to a real number, and every real number corresponds to a point on the number line.
Real Number System
Classification of Real Numbers
- Rational Numbers
- Integers (..., -2, -1, 0, 1, 2...)
- Fractions (½, ¾, etc.)
- Terminating decimals (0.5)
- Repeating decimals (0.333...)
- Irrational Numbers
- √2, √3, π, e, etc.
- Non-terminating, non-repeating decimals
Example: Classify these numbers
Identify each as rational or irrational:
2.5 Radical Notation
Radical notation is used to represent roots of numbers. The most common is the square root (√).
General form: ⁿ√a where n is the index (degree of the root) and a is the radicand.
When n=2, we usually write √a instead of ²√a.
Parts of a Radical
- Radical symbol: √
- Index: n (small number outside)
- Radicand: a (number inside)
Examples
- √9 = 3 (since 3² = 9)
- ³√8 = 2 (since 2³ = 8)
- ⁴√16 = 2 (since 2⁴ = 16)
- √2 ≈ 1.414 (irrational)
🧮 Radical Calculator
2.6 Surds
A surd is an irrational number expressed as a root (especially a square root) that cannot be simplified to remove the root.
Examples: √2, √3, √5, ³√7
Non-examples: √4 (because it equals 2), √(9/4) (because it equals 3/2)
Types of Surds
- Simple surds: Single term like √2 or √5
- Compound surds: Sum or difference of two or more surds like √2 + √3
- Mixed surds: Product of a rational number and a surd like 2√5
- Pure surds: Surds with coefficient 1 like √7
Simplifying Surds
√50 = √(25 × 2) = √25 × √2 = 5√2
√72 = √(36 × 2) = √36 × √2 = 6√2
√18 + √8 = 3√2 + 2√2 = 5√2
Practice: Simplify these surds
√12 = √
√45 = √
2.7 Rationalisation of Surds
Rationalisation is the process of eliminating a surd from the denominator of a fraction.
For a denominator with a single surd (a√b), multiply numerator and denominator by √b.
For a denominator with (a + √b), multiply numerator and denominator by the conjugate (a - √b).
Simple Rationalisation
1/√2 = (1 × √2)/(√2 × √2) = √2/2
3/√5 = (3 × √5)/(√5 × √5) = 3√5/5
5/(2√3) = (5 × √3)/(2√3 × √3) = 5√3/6
Using Conjugates
1/(1 + √2) = [1 × (1 - √2)]/[(1 + √2)(1 - √2)] = (1 - √2)/(1 - 2) = (1 - √2)/(-1) = √2 - 1
3/(√5 - √2) = [3 × (√5 + √2)]/[(√5 - √2)(√5 + √2)] = (3√5 + 3√2)/(5 - 2) = (3√5 + 3√2)/3 = √5 + √2
Rationalisation Practice
Rationalise the denominator:
2.8 Scientific Notation
Scientific notation expresses numbers as a × 10ⁿ where 1 ≤ a < 10 and n is an integer.
Used for very large or very small numbers.
Examples: 3.2 × 10⁵ = 320,000; 4.5 × 10⁻³ = 0.0045
Large Numbers
Speed of light: 299,792,458 m/s = 2.99792458 × 10⁸ m/s
Earth's population: ~8,000,000,000 = 8 × 10⁹
Distance to Sun: 149,600,000 km = 1.496 × 10⁸ km
Small Numbers
Mass of electron: 0.000000000000000000000000000910938356 g = 9.10938356 × 10⁻²⁸ g
Planck's constant: 0.000000000000000000000000000000000662607015 J·s = 6.62607015 × 10⁻³⁴ J·s