Real Numbers Properties and Operations

Study Notes

Properties of Real Numbers

Commutative Property: The order of real numbers does not change the result of addition or multiplication.
- a + b = b + a
- a × b = b × a
Associative Property: The order in which real numbers are added or multiplied does not change the result.
- (a + b) + c = a + (b + c)
- (a × b) × c = a × (b × c)
Distributive Property: Real numbers can be distributed across addition and multiplication.
- a × (b + c) = a × b + a × c
- a + (b × c) = a + b × a + c

Irrational Numbers

Definition: A real number that cannot be expressed as a finite decimal or fraction.
Examples: π, e, √2, √3
Properties:
- Irrational numbers cannot be written in simplest form.
- Irrational numbers have an infinite number of decimal places that never repeat.
- Irrational numbers can be represented on a number line, but not exactly.

Real Number Operations

Addition: Combining two real numbers to get a total or a sum.
- a + b = c
Subtraction: Finding the difference between two real numbers.
- a - b = c
Multiplication: Combining two real numbers to get a product.
- a × b = c
Division: Finding the quotient of two real numbers.
- a ÷ b = c

Real Number Sets

Natural Numbers (N): {1, 2, 3, ...}
Whole Numbers (W): {0, 1, 2, 3, ...}
Integers (Z): {..., -3, -2, -1, 0, 1, 2, 3, ...}
Rational Numbers (Q): {a/b, where a and b are integers and b ≠ 0}
Real Numbers (R): All rational and irrational numbers

Absolute Value

Definition: The distance of a number from zero on a number line.
Notation: |a|, where a is a real number.
Properties:
- |a| ≥ 0
- |a| = 0 if and only if a = 0
- |a × b| = |a| × |b|
- |a + b| ≤ |a| + |b|

Description

Test your knowledge of real numbers, including properties, operations, and sets. Learn about commutative, associative, and distributive properties, and how to work with irrational numbers, natural numbers, whole numbers, integers, rational numbers, and real numbers.