Real Numbers, Exponential Notation, and Properties of Real Numbers

Classification and Structure of the Real Number System

  • Real Numbers (R\mathbb{R}): The overarching set containing all rational and irrational numbers. Every real number is categorized as either rational or irrational.

  • Natural Numbers:

    • Definition: The set of positive counting numbers.
    • Set Representation: {1,2,3,… }\{1, 2, 3, \dots\}

  • Whole Numbers:

    • Definition: The set containing zero plus all natural numbers.
    • Set Representation: {0,1,2,3,4,5,… }\{0, 1, 2, 3, 4, 5, \dots\}

  • Integers:

    • Definition: The set containing all whole numbers and their negative opposites.
    • Set Representation: {…,−3,−2,−1,0,1,2,3,… }\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}

  • Rational Numbers:

    • Definition: Any number that can be expressed as a ratio of two integers in the form pq\frac{p}{q}, where pp and qq are integers and q≠0q \neq 0.
    • Decimal Representations:
      • Terminating Decimals: Decimals that contain a finite number of digits after the decimal point.
        • Examples: 36.836.8, 0.1250.125, 4.54.5
      • Repeating Decimals: Decimals that contain an infinite number of digits that repeat in a specific pattern.
        • Examples: 0.333…0.333\dots, 1.999…1.999\dots, 7.689689…7.689689\dots

  • Irrational Numbers:

    • Definition: Any real number that cannot be expressed in the form pq\frac{p}{q}. They are non-terminating and non-repeating decimals that extend infinitely without a repeating block or pattern.
    • Examples: π\pi, 2\sqrt{2}, 7\sqrt{7}, −10-\sqrt{10}, 0.34334333433334…0.34334333433334\dots, 45.86745893…45.86745893\dots, 0.070070007…0.070070007\dots

  • Hierarchical Nesting of Real Numbers:

    • Natural numbers are a subset of whole numbers.
    • Whole numbers are a subset of integers.
    • Integers are a subset of rational numbers.
    • Rational numbers and irrational numbers are mutually exclusive sets that together comprise the real number system.

The Real Number System Diagram

Graphical Representation on the Number Line

  • The Number Line: A visual, graphical representation of real numbers on a continuous horizontal scale.

  • Representation of Integers:

    • Positions discrete integer values spaced evenly across positive, negative, and zero coordinates.

Integers on the Number Line

  • Representation of Rational Numbers:
    • Includes fractional values and improper fractions situated precisely between integers.
    • Examples plotted on the number line:
      • −92-\frac{9}{2} (or −4.5-4.5)
      • −53-\frac{5}{3} (or −1.666…-1.666\dots)
      • −18-\frac{1}{8} (or −0.125-0.125)
      • 00
      • 22
      • 2142\frac{1}{4} (or 2.252.25)

Rational Numbers on the Number Line

Number Classification Analysis

  • Classification Breakdown for Specific Numbers:
    • Number 00: Belongs to Whole numbers, Integers, Rational numbers, and Real numbers.
    • Number −3-3: Belongs to Integers, Rational numbers, and Real numbers.
    • Number 4.54.5: Belongs to Rational numbers (terminating decimal) and Real numbers.
    • Number 0.333…0.333\dots: Belongs to Rational numbers (repeating decimal, equivalent to 13\frac{1}{3}) and Real numbers.
    • Number 2\sqrt{2}: Belongs to Irrational numbers (non-repeating, non-terminating radical) and Real numbers.
    • Number π\pi: Belongs to Irrational numbers (non-repeating infinite decimal) and Real numbers.
    • Number 7.689689…7.689689\dots: Belongs to Rational numbers (repeating pattern 689689) and Real numbers.
    • Number −10-\sqrt{10}: Belongs to Irrational numbers (negative radical of a non-perfect square) and Real numbers.

Exponential Notation Principles

  • Structure of Exponential Expression (bnb^n):

    • Base (bb): The number or quantity being multiplied repeatedly.
    • Exponent (nn): The number indicating how many times the base is used as a factor.
    • Power: The computed numerical value of the entire exponential expression.

  • Worked Example of Structure:

    • In the expression 43=644^3 = 64:
      • Base = 44
      • Exponent = 33
      • Power = 6464 (calculated as 4×4×4=644 \times 4 \times 4 = 64)

  • Evaluated Exponential Expressions:

    • Expression 1: 23+(−3)22^3 + (-3)^2
      • Evaluate 23=82^3 = 8
      • Evaluate (−3)2=9(-3)^2 = 9
      • Sum: 8+9=178 + 9 = 17
    • Expression 2: −52+73-5^2 + 7^3
      • Evaluate −52=−(5×5)=−25-5^2 = -(5 \times 5) = -25
      • Evaluate 73=3437^3 = 343
      • Sum: −25+343=318-25 + 343 = 318
    • Expression 3: (−1)97+(1)97(-1)^{97} + (1)^{97}
      • Evaluate (−1)97=−1(-1)^{97} = -1 (odd power of −1-1 is −1-1)
      • Evaluate (1)97=1(1)^{97} = 1
      • Sum: −1+1=0-1 + 1 = 0
    • Expression 4: 63−436^3 - 4^3
      • Evaluate 63=2166^3 = 216
      • Evaluate 43=644^3 = 64
      • Difference: 216−64=152216 - 64 = 152
    • Expression 5: (3)3(10)2(3)^3(10)^2
      • Evaluate (3)3=27(3)^3 = 27
      • Evaluate (10)2=100(10)^2 = 100
      • Product: 27×100=270027 \times 100 = 2700

Fundamental Properties of Real Numbers

  • Commutative Properties: Order of terms or factors does not affect the outcome.

    • Addition: a+b=b+aa + b = b + a
      • Terms can be added in any order without changing the sum.
    • Multiplication: a⋅b=b⋅aa \cdot b = b \cdot a
      • Factors can be multiplied in any order without changing the product.

  • Associative Properties: Grouping of terms or factors does not affect the outcome.

    • Addition: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c)
      • Terms can be regrouped in addition without altering the sum.
    • Multiplication: (a⋅b)⋅c=a⋅(b⋅c)(a \cdot b) \cdot c = a \cdot (b \cdot c)
      • Factors can be regrouped in multiplication without altering the product.

  • Identity Properties: Operations that leave a value unchanged.

    • Additive Identity: x+0=xx + 0 = x
      • Zero is the additive identity element.
    • Multiplicative Identity: 1⋅x=x1 \cdot x = x
      • One is the multiplicative identity element.

  • Inverse Properties: Operations that yield identity elements.

    • Additive Inverse: x+(−x)=0x + (-x) = 0
      • The sum of any real number xx and its opposite −x-x equals zero.
    • Multiplicative Inverse: (pq)⋅(qp)=1\left(\frac{p}{q}\right) \cdot \left(\frac{q}{p}\right) = 1
      • The product of a fraction pq\frac{p}{q} and its reciprocal qp\frac{q}{p} (where p,q≠0p, q \neq 0) equals one.

  • Distributive Property of Multiplication over Addition:

    • Distributing a Factor: a(b+c)=ab+aca(b + c) = ab + ac
      • A factor outside a sum multiplies each term inside the sum.
    • Undistributing a Common Factor: ab+ac=a(b+c)ab + ac = a(b + c)
      • A factor common to terms in a sum can be factored out and written outside parentheses.

Summary of Real Number System Principles

  • Dichotomy of Real Numbers: Every real number is either rational or irrational.
  • Rational Form: Rational numbers take the form pq\frac{p}{q} (q≠0q \neq 0) and convert to terminating or repeating decimals.
  • Irrational Form: Irrational numbers cannot be written as pq\frac{p}{q} and form non-terminating, non-repeating decimals such as π\pi, 2\sqrt{2}, 7\sqrt{7}, and −10-\sqrt{10}.
  • Exponential Definition: In bnb^n, bb is the base, nn is the exponent, and the resulting calculation is the power.
  • Five Core Properties: Real-number arithmetic is dictated by Commutative, Associative, Identity, Inverse, and Distributive properties.