Overview
This lecture covers the differences between rational and irrational numbers, focusing on their definitions and how to identify them using examples.
Rational Numbers
- Rational numbers can be written as a ratio (fraction) of two integers.
- Examples: 3/4 and -7/6 are rational because both numerator and denominator are integers.
- All integers (e.g., 8, -5) are rational because they can be written as a fraction (e.g., 8 = 16/2, -5 = -10/2).
- Finite decimal numbers (that end), such as 0.25, are rational because they can be converted to fractions (0.25 = 1/4).
- Repeating decimals, like 0.666... (0.6 with a line over the 6), are rational numbers (0.6Ì… = 2/3).
- Any decimal that repeats or ends is rational (e.g., 0.17Ì… = 17/99).
Irrational Numbers
- Irrational numbers cannot be written as a fraction of two integers.
- Decimals that go on forever without repeating are irrational (e.g., the square root of 7 ≈ 2.6457513...).
- Square roots that can't be simplified to an integer are typically irrational.
- Numbers like π (pi ≈ 3.14159265...) and e (≈ 2.71828182...) are irrational, as their decimal parts never end or repeat.
Key Terms & Definitions
- Rational Number — A number that can be written as a fraction of two integers.
- Irrational Number — A number that cannot be written as a fraction of two integers; its decimal form never ends or repeats.
- Integer — A whole number, either positive, negative, or zero.
Action Items / Next Steps
- Practice identifying rational and irrational numbers from a list of examples.
- Review square roots and decimal expansions to determine their type.