Rational vs. Irrational Numbers

Overview

This lecture explains the difference between rational and irrational numbers, focusing on their definitions, properties, and examples like pi.

A rational number can be written as a ratio of two integers (a fraction).
Whole numbers and finite or repeating decimal numbers are rational.
Rational numbers have decimal representations that either end or repeat in a pattern.
Any number whose decimal digits terminate or repeat is rational.

An irrational number cannot be written as a ratio of two integers (not as a fraction).
The decimal representation of an irrational number does not terminate or repeat in a pattern.
Pi (π) is a famous example of an irrational number.
Decimal approximations of irrational numbers (like 3.14 for pi) are never exact.
No fraction, even 22/7 or 355/113, gives the exact value of pi—these are only approximations.

For rational numbers, knowing the repeating pattern allows you to predict any decimal digit.
Irrational numbers have unpredictable decimal digits; each new digit must be calculated.

On the number line, an irrational number like pi never aligns exactly with a marked point, no matter how much you zoom in.
Rational numbers can be "found" precisely on the number line using subdivision; irrational numbers cannot.

There are infinitely many irrational numbers, and they outnumber rational numbers.

Rational Number — a number that can be expressed as a fraction (ratio) of two integers.
Irrational Number — a number that cannot be expressed as a fraction of two integers; its decimal never terminates or repeats.
Terminating Decimal — a decimal number that ends.
Repeating Decimal — a decimal with a digit or group of digits that repeats endlessly.
Pi (π) — an irrational number commonly approximated as 3.14, 22/7, or 355/113, but never exactly represented by a fraction.