Numbers come in different flavors: rational and irrational. Rational numbers can be written as fractions, while irrational ones can't. This distinction is crucial for understanding the nature of numbers and their relationships.

Real numbers include both rational and irrational types. They form a continuous line, with rational numbers at specific points and irrational numbers filling the gaps. This concept helps us grasp the full spectrum of numbers we use in math.

Understanding Rational and Irrational Numbers

Rational vs irrational numbers

• Rational numbers express as a ratio of two integers ($\frac{a}{b}$, $b \neq 0$) include integers (3), fractions ($\frac{2}{5}$), and terminating (0.75) or repeating decimals ($0.\overline{3}$)
• Irrational numbers cannot express as a ratio of two integers have decimal expansions that neither terminate nor repeat ($\sqrt{2}$, $\pi$, $e$)

Decimal to fraction conversions

• Terminating decimals to fractions write the decimal as a fraction over a power of 10, then simplify ($0.25 = \frac{25}{100} = \frac{1}{4}$)
• Repeating decimals to fractions let $x$ equal the repeating decimal, multiply by a power of 10 to shift the decimal, and solve for $x$ ($0.\overline{3} = \frac{1}{3}$)
  1. Let $x = 0.\overline{3}$
  2. $10x = 3.\overline{3}$
  3. $10x - x = 3.\overline{3} - 0.\overline{3}$
  4. $9x = 3$
  5. $x = \frac{1}{3}$
• Fractions to decimals divide the numerator by the denominator ($\frac{3}{8} = 0.375$)

Categorization of real numbers

• Real numbers include all rational and irrational numbers
• Subsets of real numbers
  • Whole numbers non-negative integers ($0, 1, 2, 3, \ldots$)
  • Integers positive, negative, and zero ($\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$)
  • Rational numbers express as $\frac{a}{b}$, $a$ and $b$ are integers and $b \neq 0$
  • Irrational numbers real numbers that are not rational ($\sqrt{3}$, $\pi$)

Relationships between Number Sets

• Whole numbers $\subset$ Integers $\subset$ Rational numbers $\subset$ Real numbers
• Irrational numbers subset of real numbers but do not overlap with rational numbers
• All real numbers can be represented on a number line, with rational numbers at specific points and irrational numbers filling the gaps

Advanced Number Classifications

• Algebraic numbers: numbers that are roots of polynomial equations with integer coefficients (e.g., $\sqrt{2}$, $\sqrt[3]{5}$)
• Transcendental numbers: irrational numbers that are not algebraic (e.g., $\pi$, $e$)
• The set of real numbers exhibits the property of density, meaning between any two real numbers, there are infinitely many other real numbers