Basics of Rational Numbers

Overview

Rational number: any number that can be written as A/B where A and B are integers and B ≠ 0.
Integers include negative numbers, zero, and positive numbers (no fractional part).
The denominator cannot be zero because division by zero is undefined.

Same number, many forms: one rational number can be written in many equivalent fractions (e.g., 6 = 6/1 = 12/2).
Decimal connection:
- Terminating decimals are rational (example: 0.71 = 71/100).
- Repeating decimals are rational (example: 0.‾3 = 1/3).
Whole numbers and integers are rational because they can be written with denominator 1 (e.g., 4 = 4/1).
Zero is rational (e.g., 0/1 = 0), but 1/0 is undefined.
Irrational numbers have decimals that neither terminate nor repeat and cannot be written as integer fractions (example: √3).

Whole numbers:
- 6 → 6/1 or 12/2; rational.
- −6 → −6/1; rational.
Terminating decimals:
- 0.71 → 71/100; rational.
- 2.75 → 275/100 = 11/4; rational.
Zero:
- 0 → 0/1 or 0/100; rational.
Repeating decimals:
- 0.‾3 → 1/3; rational.
- 0.‾18 → 18/99 = 2/11; rational.
Fractions:
- 1/4 → decimal 0.25 (terminating); rational.
Roots:
- √25 = 5 (an integer) → rational.
- √3 ≈ 1.73205… (non-terminating, non-repeating) → irrational.

Rational Number: number expressible as A/B with integers A, B and B ≠ 0.
Integer: whole number that can be negative, zero, or positive.
Terminating Decimal: decimal that ends; always rational.
Repeating Decimal: decimal with a repeating pattern; always rational.
Irrational Number: number not expressible as an integer fraction; decimal neither terminates nor repeats.

Can the number be written as A/B with integers and B ≠ 0? Then it is rational.
For a decimal, if it terminates or repeats, it is rational; if it neither terminates nor repeats, it is irrational.
For roots, if the root is a whole number (from a perfect square or cube), it is rational; otherwise it is usually irrational.