Properties of Real Numbers

Study Notes

Commutative Property: The order of numbers does not change the result.
- a + b = b + a
- a × b = b × a
Associative Property: The order in which numbers are grouped does not change the result.
- (a + b) + c = a + (b + c)
- (a × b) × c = a × (b × c)
Distributive Property: Multiplication distributes over addition.
- a × (b + c) = a × b + a × c
Additive Identity: The number 0 does not change the result when added.
- a + 0 = a
Multiplicative Identity: The number 1 does not change the result when multiplied.
- a × 1 = a
Additive Inverse: Each number has an opposite that, when added, results in 0.
- a + (-a) = 0
Multiplicative Inverse: Each non-zero number has a reciprocal that, when multiplied, results in 1.
- a × (1/a) = 1

Rational Numbers: Can be expressed as a fraction (p/q) where p and q are integers and q ≠ 0.
- Examples: 3/4, 22/7, 1/2
Irrational Numbers: Cannot be expressed as a fraction.
- Examples: π, e, √2

Every positive integer can be expressed as a product of prime numbers in a unique way.
- Example: 60 = 2 × 2 × 3 × 5

Method of Contradiction: Assume the number is rational and show that it leads to a contradiction.
Example: Proving √2 is Irrational
- Assume √2 is rational, so √2 = p/q where p and q are integers.
- Square both sides: 2 = (p/q)²
- Simplify: 2 = p²/q²
- Multiply both sides by q²: 2q² = p²
- Since the left side is even, the right side must also be even, so p is even.
- Let p = 2k, then 2q² = (2k)²
- Simplify: 2q² = 4k²
- Divide both sides by 2: q² = 2k²
- Since the left side is odd, the right side must also be odd, so q is odd.
- This is a contradiction, since we assumed p and q are integers, and p is even while q is odd.
- Therefore, √2 is irrational.

The Commutative Property states that the order of numbers does not change the result, e.g. a + b = b + a and a × b = b × a.
The Associative Property states that the order in which numbers are grouped does not change the result, e.g. (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
The Distributive Property states that multiplication distributes over addition, e.g. a × (b + c) = a × b + a × c.
The Additive Identity is 0, which does not change the result when added, e.g. a + 0 = a.
The Multiplicative Identity is 1, which does not change the result when multiplied, e.g. a × 1 = a.
The Additive Inverse is the opposite of each number, which when added, results in 0, e.g. a + (-a) = 0.
The Multiplicative Inverse is the reciprocal of each non-zero number, which when multiplied, results in 1, e.g. a × (1/a) = 1.

Rational Numbers are numbers that can be expressed as a fraction (p/q) where p and q are integers and q ≠ 0, e.g. 3/4, 22/7, 1/2.
Irrational Numbers are numbers that cannot be expressed as a fraction, e.g. π, e, √2.

The Fundamental Theorem of Arithmetic states that every positive integer can be expressed as a product of prime numbers in a unique way, e.g. 60 = 2 × 2 × 3 × 5.

The Method of Contradiction is used to prove that a number is irrational, by assuming it is rational and showing that it leads to a contradiction.
The Method of Contradiction is used to prove that √2 is irrational, by assuming √2 is rational, and then showing that it leads to a contradiction, thus proving that √2 is irrational.