Understanding Real Number Properties

• Real numbers encompass all rational and irrational numbers
• Real numbers can be positive, negative, or zero
• Real numbers are used to measure continuous quantities

Properties of Real Numbers

• Closure:
  • For real numbers a and b, a + b and a × b are also real numbers
• Commutativity:
  • For real numbers a and b, a + b = b + a and a × b = b × a
• Associativity:
  • For real numbers a, b, and c, (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)
• Identity:
  • There exists a real number 0 such that for any real number a, a + 0 = a
  • There exists a real number 1 such that for any real number a, a × 1 = a
• Inverse:
  • For every real number a, there exists a real number -a such that a + (-a) = 0
  • For every non-zero real number a, there exists a real number 1/a such that a × (1/a) = 1
• Distributivity:
  • For real numbers a, b, and c, a × (b + c) = (a × b) + (a × c)

Rational Numbers

• Rational numbers can be expressed as a fraction p/q
• p and q are integers and q ≠ 0
• All integers are rational numbers, since any integer n can be written as n/1
• Decimal representation either terminates (e.g., 0.25) or repeats (e.g., 0.333...)

Irrational Numbers

• Cannot be expressed as a fraction of two integers
• Decimal representation is non-terminating and non-repeating
• √2, π, and e, are examples of irrational numbers

Number Line

• Real numbers are represented on a number line
• Each point on the number line corresponds to a unique real number
• Numbers increase from left to right on the number line

Ordering of Real Numbers

• Real numbers can be compared using inequalities:
• a < b means a is less than b
• a > b means a is greater than b
• a ≤ b means a is less than or equal to b
• a ≥ b means a is greater than or equal to b

Absolute Value

• The absolute value of a real number a, denoted |a|, is its distance from 0 on the number line
• |a| = a if a ≥ 0
• |a| = -a if a < 0
• The absolute value is always non-negative

Operations on Real Numbers

• Addition, subtraction, multiplication, and division are defined for all real numbers, except division by zero
• These operations follow the closure, commutativity, associativity, identity, inverse, and distributivity properties

Intervals

• An interval is a set of real numbers between two given numbers
• Open interval (a, b): All real numbers between a and b, not including a and b
• Closed interval [a, b]: All real numbers between a and b, including a and b
• Half-open intervals (a, b] and [a, b): Include one endpoint but not the other

Density Property

• Between any two distinct real numbers, there exists another real number
• There are infinitely many real numbers between any two real numbers

Completeness Property

• Every non-empty set of real numbers that is bounded above has a least upper bound (supremum) in the real numbers
• Every non-empty set of real numbers that is bounded below has a greatest lower bound (infimum) in the real numbers
• Distinguishes real numbers from rational numbers

Decimal Representation

• Every real number can be represented as a decimal
• A decimal representation can be terminating, repeating, or non-terminating and non-repeating

Examples of Real Numbers

• Integers: -3, -2, -1, 0, 1, 2, 3
• Rational numbers: 1/2, -3/4, 0.75, -0.333...
• Irrational numbers: √2 ≈ 1.414, π ≈ 3.14159, e ≈ 2.71828
• Decimal numbers: 3.14, -2.5, 0.001

Applications of Real Numbers

• Measurement of length, area, volume, and time
• Financial calculations
• Scientific and engineering computations
• Modeling physical phenomena
• Computer graphics and simulations

Set of Real Numbers

• Denoted by the symbol ℝ
• It is an uncountable set; larger than the set of natural numbers