Understanding Rational Numbers

Overview

This lecture explains what rational numbers are, how to identify them, and provides examples and non-examples to clarify the concept.

Rational numbers can be written as a fraction of two integers (numerator and denominator are both integers).
Denominator cannot be zero because division by zero is undefined.
Rational numbers include positive numbers, negative numbers, and zero.

Any whole number is rational (e.g., 6 can be written as 6/1).
Negative numbers are rational if they can be written as a fraction of integers (e.g., -6 = -6/1).
Terminating decimals (decimals that end) are rational (e.g., 0.7 = 7/10).
Repeating decimals (decimals with a repeating pattern) are rational (e.g., 0.333... = 1/3).
Zero is rational because it can be written as 0 over any nonzero integer (e.g., 0/1).

6 (as 6/1, 12/2, 36/6, etc.)
-6 (as -6/1, -12/2)
0.7 (as 7/10, 21/30)
2.75 (as 275/100, simplified to 11/4)
0 (as 0/1, 0/25)
0.3 repeating (as 1/3, 3/9)
0.18 repeating (as 18/99, 2/11)
1/4 (already a fraction; decimal form 0.25)
Square root of 25 (equals 5, which is rational).

Square root of 3 is not rational because it cannot be written as a fraction of two integers (its decimal neither terminates nor repeats).
Expressions like √3/1 are not rational if the numerator is not an integer.
Roots of numbers that are not perfect squares or cubes are generally irrational.

Rational Number — A number that can be written as a fraction of two integers (denominator not zero).
Integer — Whole numbers and their negatives, including zero.
Terminating Decimal — A decimal that ends after a finite number of digits.
Repeating Decimal — A decimal with a digit or group of digits that repeats infinitely.
Irrational Number — A number that cannot be written as a fraction of two integers; its decimal does not terminate or repeat.

Review additional examples of irrational numbers in the next video as suggested.
Practice writing different numbers as fractions to check if they are rational.