Decimal representation: does it stop, repeat, or neither?

Three kinds of decimals

Write a number as a decimal and watch the digits after the point. Your book sorts every decimal into one of three kinds.

A terminating decimal has a finite number of digits after the point, like 0.375 or 0.8. It is rational.

A non-terminating, recurring decimal never stops but repeats a pattern forever: 1/3 = 0.333…, 22/7 = 3.142857142857…. Your book says these are rational too.

A non-terminating, non-recurring decimal never stops and never repeats. These are the irrational numbers: π, e and √2.

Stops or repeats: rational. Neither stops nor repeats: irrational. The bar over digits marks the part that repeats.

Key terms

Terminating decimal

A decimal that stops.

Recurring decimal

Never stops, but repeats; rational.

Non-recurring decimal

Never stops, never repeats; irrational.

Euler's number (e)

A famous irrational constant.

Short questions with model answers

1. Q1. Example 1 (iii): is 3.5̄ rational or irrational?

   • 3.5̄ = 3.5555…
   • non-terminating, recurring

   Rational.

2. Q2. Example 1 (iv): is 3.36788542… rational or irrational?

   • 3.36788542…
   • no repeating block

   Irrational.

Common mistakes

• ✗ Calling 0.444… irrational because it never ends.

  ✓ It repeats, so it is rational (Example 1 ii).

• ✗ Saying π equals 22/7.

  ✓ 22/7 = 3.142857 repeating, so it is rational; π never repeats. 22/7 is only close to π.

• ✗ Reading 0.16̄ as 0.161616….

  ✓ The bar is over the 6 only: 0.1666….

MCQs

1. 1. 0.35 is:

   1. (a) terminating, so rational
   2. (b) recurring, so irrational
   3. (c) non-recurring, so irrational
   4. (d) not a real number
   Show answer

   (a) Two digits after the point, then it stops (Example 1 i).

2. 2. 1.709975947… is:

   1. (a) terminating
   2. (b) recurring
   3. (c) irrational
   4. (d) an integer
   Show answer

   (c) No repeating pattern and no end (Example 1 v).

3. 3. Which one is irrational?

   1. (a) 22/7
   2. (b) √2
   3. (c) 0.444…
   4. (d) 0.8
   Show answer

   (b) √2 = 1.41421356… never repeats (p. 4).

4. 4. 0.16̄ means:

   1. (a) 0.161616…
   2. (b) 0.16
   3. (c) 0.1666…
   4. (d) 0.666…
   Show answer

   (c) Only the 6 is under the bar, so only the 6 repeats.

5. 5. Non-terminating, recurring decimals are:

   1. (a) irrational
   2. (b) not real
   3. (c) only integers
   4. (d) rational
   Show answer

   (d) Your book: they are also rational numbers (p. 4).

Quick revision

• Terminating decimals stop; they are rational.
• Recurring decimals repeat forever; they are rational too.
• Non-terminating, non-recurring decimals are irrational: π, e, √2.