4.2 Rational & Irrational Numbers

What Is a Rational Number?

A rational number is any number that can be written as a fraction a/b, where a and b are integers and b ≠ 0. The name comes from "ratio."

Examples:

• 3/4 — a fraction of two integers
• −7 = −7/1 — every integer is rational (denominator 1)
• 0 = 0/1 — zero is rational
• 0.5 = 1/2 — terminating decimals are rational
• 0.333... = 1/3 — repeating decimals are rational
• 2.75 = 11/4 — terminating decimal
• −1.2 = −6/5 — terminating decimal

A decimal is rational if it terminates (ends) or repeats with a fixed, recognizable block.

What Is an Irrational Number?

An irrational number cannot be written as a ratio of two integers. Its decimal form is non-terminating and non-repeating — the digits go on forever with no repeating block.

Common examples:

• √2 ≈ 1.41421356...
• √3 ≈ 1.7320508...
• √5 ≈ 2.2360679...
• π ≈ 3.14159265...

A useful check: the square root of a perfect square is rational (√4 = 2, √9 = 3, √16 = 4). Only square roots of non-perfect squares are irrational. The same idea applies to cube roots: ∛8 = 2 is rational, but ∛2 is irrational.

Trap (π is irrational): Although 22/7 ≈ 3.142857... is a handy approximation, π is NOT equal to 22/7. Pi cannot be written as any fraction of integers.

Trap (0.333... is rational): A repeating decimal is rational even though it never "ends." 0.333... = 1/3. The repeating block (the digit 3) is the giveaway.

The Real Number Hierarchy

Real numbers split into two big buckets: rational and irrational. Within rational, integers form a subset; within integers, whole numbers; within whole numbers, natural numbers.

Every natural number is whole; every whole number is an integer; every integer is rational; every rational number is real. Irrational numbers are real but not rational.

              Real Numbers
             /            \
        Rational        Irrational
         /   \
     Integers   (fractions, terminating/repeating decimals)
       /   \
    Whole    (negative integers)
      |
   Natural (1, 2, 3, ...)

Classifying Numbers — Worked Examples

Example 1. Classify each number into every set it belongs to.

• 6 → natural, whole, integer, rational, real
• −4 → integer, rational, real (not natural, not whole)
• 0.25 → rational, real (terminating decimal = 1/4)
• √7 → irrational, real
• 0.121212... → rational, real (repeating block "12" = 12/99 = 4/33)
• π → irrational, real

Example 2. Is √(81/16) rational or irrational?

√(81/16) = √81 / √16 = 9/4. That is a ratio of integers (9 and 4), so it is rational. The square root of a perfect-square fraction is rational.

Example 3. Is 0.1010010001... (each time adding one more 0 before the next 1) rational or irrational?

This decimal never terminates and never repeats — the pattern itself keeps changing. It is irrational.

Decimal Expansion Types

Every real number has a decimal expansion that falls into one of three buckets:

1. Terminating — the decimal ends. Examples: 0.75, 2.4, 7.0. Always rational.
2. Repeating — a block of digits cycles forever. Examples: 0.333..., 0.142857142857... (= 1/7). Always rational.
3. Non-terminating, non-repeating — digits go on forever with no cycle. Always irrational.

A terminating decimal corresponds to a fraction whose denominator (in lowest terms) has only 2s and 5s as prime factors. For example, 3/8 = 0.375 terminates because 8 = 2³. But 1/3 = 0.333... does not terminate because 3 is not 2 or 5.

Converting Repeating Decimals to Fractions (Brief)

0.777... Let x = 0.777... Then 10x = 7.777... Subtract: 10x − x = 7.777... − 0.777... = 7 9x = 7 x = 7/9

0.4545... (two-digit repeat) Let x = 0.4545... Then 100x = 45.4545... Subtract: 100x − x = 45 99x = 45 x = 45/99 = 5/11 (divide top and bottom by 9)

Shortcut: one repeating digit → divide by 9. Two repeating digits → divide by 99. In general, n repeating digits → divide by (10ⁿ − 1), then simplify.

Recognizing Rational vs Irrational from Decimal Form

Given a decimal, decide rational or irrational:

• Ends after a few digits → rational (terminating)
• Has a repeating block (look for a bar or "...") → rational (repeating)
• Goes on forever with no repeating pattern → irrational

Examples:

• 0.625 → rational (terminates)
• 0.818181... → rational (repeats "81")
• 3.14159265358979... → irrational (π never repeats)
• 0.123456789101112... → irrational (no repeating block — it just counts up)

Common Traps Recap

• π is irrational. 22/7 and 3.14 are rational approximations, not π itself.
• 0.333... is rational (= 1/3). A non-terminating decimal can still be rational if it repeats.
• √(perfect square) is rational. √16 = 4 is rational; √(non-perfect square) is irrational.
• −5 is rational (it is −5/1). Negative does not mean irrational.
• 0 is rational (0/1). Zero is neither positive nor negative, but it is a ratio of integers.