Real Numbers and Their Classification

Classification of Real Numbers

• Real numbers are divided into two main categories: rational numbers and irrational numbers.

Rational Numbers

• Definition: Any number that can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

• Fractions: Examples include $\frac{1}{2}$ and $-\frac{3}{7}$.

• Integers and Natural Numbers: Any integer can be written as a fraction over $1$ (e.g., $46 = \frac{46}{1}$).

• Terminating Decimals: Decimals that end after a finite number of digits (e.g., $0.17$, $0.6$).

• Repeating Decimals: Non-terminating decimals that repeat a pattern (e.g., $0.33333...$, $\frac{2}{3}$, $\frac{1}{9}$).

• Algebraic Proof ($0.33333... = \frac{1}{3}$):

  • Let $x = 0.33333...$

  • Multiply by $10$: $10x = 3.33333...$

  • Subtract $x$ from $10x$: $10x - x = 3.33333... - 0.33333...$

  • Simplify: $9x = 3$

  • Divide by $9$: $x = \frac{3}{9} = \frac{1}{3}$

Irrational Numbers

• Definition: Numbers whose decimal expansions are non-terminating and non-repeating.

• Examples:

  • $\pi$

  • $e$ (natural exponential base)

Non-Real Numbers and Undefined Quantities

• Imaginary Numbers: Non-real numbers such as $i = \sqrt{-1}$.

• Infinity: An unbounded concept, not a real number.

• Division by Zero: Expressions such as $\frac{5}{0}$ are undefined.

Questions & Discussion

• Question: What is a number written in the form $\frac{a}{b}$ called?

  • Response: A fraction.

• Question: What is a standard repeating decimal and its fractional equivalent?

  • Response: $0.33333...$, which equals $\frac{1}{3}$.

• Question: Is $\pi$ a real number?

  • Response: Yes, $\pi$ is a real, irrational number.

• Question: Is $e$ a real number?

  • Response: Yes, $e$ is a real, irrational number.

• Question: What are examples of numbers that are not real?

  • Response: Imaginary numbers ($i = \sqrt{-1}$), infinity, and division by zero (e.g., $\frac{5}{0}$).