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Math College Notes: Real Numbers, Fractions, and PEMDAS

Real Numbers and Subsets

  • Real numbers include all rational and irrational numbers:
    • \mathbb{R} = \mathbb{Q} \cup \mathbb{I}
  • Natural numbers: {1,2,3,…}
  • Whole numbers: {0,1,2,…}
  • Integers: {…,-2,-1,0,1,2,…}
  • Rational numbers: numbers expressible as a fraction \frac{p}{q}, with p \in \mathbb{Z}, q \in \mathbb{Z}\setminus{0\}
  • Irrational numbers: cannot be written as a fraction; examples: \sqrt{2}, \pi
  • Decimal forms: terminating or repeating decimals are rational; non-terminating non-repeating are irrational
  • Imaginary numbers: of the form \bi with b \in \mathbb{R}, i^2 = -1
  • Complex numbers: a + bi with a,b \in \mathbb{R}
  • Examples:
    • 0.5 is rational, \sqrt{2} is irrational
    • 1.2.3… (non-repeating) would be irrational (conceptual)

Fractions and Mixed Numbers

  • Fraction: \frac{n}{d}, with d \neq 0
  • Mixed number: a \frac{n}{d}, where 0 < n < d; convert to improper: a \frac{n}{d} = \frac{ad + n}{d}
  • Improper fraction: a fraction where the numerator >= denominator
  • Least Common Denominator (LCD): the least common denominator (often called LCD; same as the least common multiple of denominators) used to add/subtract fractions
  • To add \frac{a}{d1} + \frac{b}{d2}: set LCD = lcm(d1, d2); \frac{a}{d1} = \frac{a \cdot (LCD/d1)}{LCD}, \frac{b}{d2} = \frac{b \cdot (LCD/d2)}{LCD}; sum = \frac{a \cdot (LCD/d1) + b \cdot (LCD/d2)}{LCD}

Operations with Fractions

  • Multiplication: \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}
  • Division: \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}
  • Mixed numbers: convert to improper fractions for operations; e.g., 2 \frac{1}{3} = \frac{7}{3}
  • Sign handling: track positive/negative signs carefully

PEMDAS and Basic Properties

  • PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right)
  • Multiplication and Addition properties (commutative and associative):
    • Commutative: a + b = b + a; a \cdot b = b \cdot a
    • Associative: (a + b) + c = a + (b + c); (a \cdot b) \cdot c = a \cdot (b \cdot c)
  • Subtraction and division are not generally commutative or associative
  • Note: Within MD and AS steps, evaluate from left to right