Rational and Irrational Numbers

Overview

This lecture introduces rational and irrational numbers, explains how to identify them, and provides examples of each type.

Rational numbers can be written as fractions of two integers (numerator and denominator).
Integers include positive numbers, negative numbers, and zero.
The denominator of a rational number cannot be zero.
Terminating decimals (those that end) are rational.
Repeating decimals (those with a repeating pattern) are rational.
All whole numbers and their negatives are rational since they can be written as fractions (e.g., 6/1, -6/1).
Zero is rational (e.g., 0/1), but division by zero is undefined and not rational.
Any fraction with integer numerator and denominator (e.g., 1/4) is rational.
Examples: 6, -6, 0.7, 2.75, -2.75, 0, 0.3 (repeating), 0.18 (repeating), 1/4, √25.

Irrational numbers cannot be written as fractions of two integers.
Their decimal forms do not terminate or repeat.
The square root of a non-perfect square is irrational (e.g., √3, √2, √24, √50).
Famous irrational numbers include π (pi) and e.
Examples: √3, √2, √24, π (≈3.14159...), e (≈2.71828...), decimals like 3.1723... that do not terminate or repeat.

12, -12, 9/10, 0.2 (repeating) are all rational.
√49 = 7 is rational as 7 is a whole number.
Decimals with a repeating bar (e.g., 8.71425 repeating) are rational.
Non-terminating, non-repeating decimals, or roots of non-perfect squares, are irrational.

Rational number — A number that can be expressed as a fraction of two integers.
Irrational number — A number that cannot be written as a fraction of two integers; its decimal never terminates or repeats.
Terminating decimal — A decimal that ends.
Repeating decimal — A decimal with digits that repeat in a pattern forever.
Integer — Whole numbers, their negatives, and zero.
Undefined — The result of division by zero; not a rational number.