1.1 Introduction
- Numbers are used to count, measure, and label things.
- Types of numbers:
- Natural numbers (1, 2, 3…)
- Whole numbers (0, 1, 2, 3…)
- Integers (…, -2, -1, 0, 1, 2…)
- Rational numbers (p/q, q ≠ 0)
- Irrational numbers (cannot be expressed as fraction p/q)
Key Idea: Every number is either rational or irrational.
1.2 Irrational Numbers
- Cannot be expressed as p/q (fraction).
- Their decimal expansions are non-terminating and non-repeating.
- Examples: √2, √3, π, e
Properties
- Sum of a rational and irrational number → irrational
- Product of a non-zero rational number and irrational number → irrational
- Sum/product of two irrational numbers → may be rational or irrational
1.3 Real Numbers and Their Decimal Expansions
- Real numbers = Rational numbers + Irrational numbers
- Decimal expansions:
- Rational numbers: Terminating or repeating decimals
- Irrational numbers: Non-terminating, non-repeating decimals
Representation on Number Line
- Every real number corresponds to a unique point on the number line.
- Conversely, every point on the number line represents a unique real number.
1.4 Operations on Real Numbers
- Closure: Sum, difference, product, and quotient of real numbers → real number
- Commutative law: a + b = b + a, a × b = b × a
- Associative law: (a + b) + c = a + (b + c), (a × b) × c = a × (b × c)
- Distributive law: a × (b + c) = a × b + a × c
- Additive identity: a + 0 = a
- Multiplicative identity: a × 1 = a
- Additive inverse: a + (-a) = 0
- Multiplicative inverse: a × (1/a) = 1, a ≠ 0
1.5 Laws of Exponents for Real Numbers
For any real numbers a, b (a ≠ 0) and integers m, n:
- a^m × a^n = a^(m+n)
- a^m ÷ a^n = a^(m−n)
- (a^m)^n = a^(mn)
- (ab)^m = a^m × b^m
- (a/b)^m = a^m / b^m, b ≠ 0
- a^0 = 1, a ≠ 0
- a^−m = 1 / a^m, a ≠ 0
Quick Short Q&A (Most Possible)
| Question | Short Answer |
|---|---|
| What is a rational number? | Can be expressed as p/q, q ≠ 0 |
| What is an irrational number? | Cannot be expressed as fraction |
| Example of irrational numbers? | √2, π |
| Real numbers = ? | Rational + Irrational |
| Decimal of rational numbers? | Terminating or repeating |
| Decimal of irrational numbers? | Non-terminating, non-repeating |
| Law of exponents for a^m × a^n? | a^(m+n) |
| Multiplicative inverse of a? | 1/a |
| Commutative law of multiplication? | a × b = b × a |
| Distributive law? | a(b + c) = ab + ac |