Real Numbers and Fundamental Algebraic Properties

Real numbers

• Real numbers (denoted by R) include every decimal number and limits of decimal expansions.
• Examples:
  • 1 can be written as $1 = 1.000\dots$
  • $\pi \approx 3.14159265\dots$
• Visual idea: Think of R as forming a line called the real number line, with points corresponding to real numbers.
• Subsets of Real Numbers:
  • $\mathbb{N} = {1, 2, 3, 4, \dots}\; ext{(natural numbers)}$
  • $\mathbb{Z} = {\dots, -2, -1, 0, 1, 2, \dots}\; ext{(integers)}$
  • \mathbb{Q} = \left{ \frac{a}{b} \big|\ a, b \in \mathbb{Z}, \ b \neq 0 \right}
  • Rational numbers: fractions of integers.
  • Examples: $\frac{1}{2}, \frac{2}{2} = 1, 2 = \frac{2}{1}$
  • $\mathbb{I}$ (irrational numbers): real numbers that are not rational.
  • Examples: $\pi, \sqrt{2}, \sqrt{5}, e$
  • Characteristic: their decimal expansion goes on forever without repeating patterns.
  • Together, $\mathbb{Q} \cup \mathbb{I} = \mathbb{R}$ (the rationals and irrationals comprise all real numbers).
• Irrational numbers:
  • Decimal expansion neither terminates nor repeats.
  • Examples given: $\pi, \sqrt{2}, \sqrt{5}, e$
• Important note from the transcript:
  • Rational numbers have decimal expansions that either terminate or repeat.
  • Irrationals have non-terminating, non-repeating decimals.

Real number line visualization

• Origin is the point 0 on the line.
• Positive side contains numbers > 0; negative side contains numbers < 0.
• Distances correspond to absolute value, i.e., the distance between two numbers a and b is $|a - b|.$

Properties of addition

• Associative property:
  • $(a + b) + c = a + (b + c)$
• Commutative property:
  • $a + b = b + a$
• Additive identity:
  • There exists an element 0 such that $a + 0 = 0 + a = a$
• Additive inverse:
  • For every a, there exists an element $-a$ such that $a + (-a) = (-a) + a = 0$
• Examples:
  • $(2 + 1) + 3 = 3 + 3 = 6$
  • $2 + (1 + 3) = 2 + 4 = 6$
  • $3 + 0 = 3$

Properties of multiplication

• Associative property:
  • $(a \cdot b) \cdot c = a \cdot (b \cdot c)$
• Commutative property:
  • $a \cdot b = b \cdot a$
• Multiplicative identity:
  • There exists an element 1 such that $a \cdot 1 = 1 \cdot a = a$
• Multiplicative inverse (nonzero):
  • For every a ≠ 0, there exists an element $a^{-1}$ such that $a \cdot a^{-1} = a^{-1} \cdot a = 1$
• Example:
  • $2 \cdot 3 = 3 \cdot 2 = 6$

Distributive law (interaction of + and ×)

• Left distributive:
  • $a \cdot (b + c) = a b + a c$
• Right distributive:
  • $(a + b) \cdot c = a c + b c$

Subtraction and division in terms of addition and multiplication

• Subtraction defined via addition:
  • $a - b = a + (-b)$
  • Example: $3 - 2 = 3 + (-2) = 1$
• Division defined via multiplication (for nonzero divisor):
  • $\frac{a}{b} = a \cdot b^{-1}, \quad b \neq 0$

Zero properties and the zero product rule

• Additive identity again: $a + 0 = a$
• Zero-product property:
  • If $a \cdot b = 0$, then either $a = 0$ or $b = 0$ (or both).
• Example problem from transcript:
  • Solve $x(x - 1) = 0$
  • Solutions: $x = 0 \quad \text{or} \quad x = 1$