Decimal Expansions

The Key to Classifying Rational and Irrational Numbers

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Welcome to decimal expansions! Discover how decimal patterns reveal whether numbers are rational or irrational.

Introduction: Decimal Expansions

Every real number has a decimal expansion. The pattern of this expansion tells us whether the number is rational or irrational. This is the key to classification!

Decimal Pattern β†’ Number Type

Three Types of Decimal Expansions:

  • Terminating: 7/8 = 0.875 (stops)
  • Non-terminating Recurring: 1/3 = 0.333... (repeats)
  • Non-terminating Non-recurring: √2 = 1.414... (no pattern)

Terminating Decimals

A terminating decimal comes to an end after a finite number of digits. These are always rational numbers.

Example: 7/8

8 ) 7.000 6.4 --- 0.60 0.56 ---- 0.040 0.040 ----- 0.000 ← Remainder becomes 0!

Result: 7/8 = 0.875 (terminates)

When remainder = 0 β†’ Decimal terminates

Non-Terminating Recurring Decimals

A recurring decimal has a pattern that repeats forever. These are also rational numbers!

Example 1: 10/3

3 ) 10.000 9 --- 1.0 ← Remainder repeats! 0.9 ---- 0.10 0.09 ----- 0.01 ← Same remainder again!

Result: 10/3 = 3.333... = 3.3Μ„

Example 2: 1/7 (Longer cycle)

1/7 = 0.142857142857... = 0.1Μ„4Μ„2Μ„8Μ„5Μ„7Μ„

The block "142857" repeats forever!

Repeating remainder β†’ Repeating decimal digits

Non-Terminating Non-Recurring Decimals

These decimals go on forever with NO repeating pattern. These are irrational numbers!

Examples:

  • Ο€ = 3.14159265358979... (no pattern)
  • √2 = 1.41421356237309... (no pattern)
  • 0.10110111011110... (deliberate non-pattern)
No pattern, never stops β†’ Irrational

Division Examples - Finding Patterns

Key Observation About Remainders:

  • When dividing by n, remainders must be: 0, 1, 2, ..., n-1
  • Only n possible remainders!
  • Either remainder becomes 0 (terminates)
  • Or remainder repeats (recurring decimal)
  • Repeating block length < divisor

Complete Division: 1/7

Step 1: 10 Γ· 7 = 1 remainder 3 Step 2: 30 Γ· 7 = 4 remainder 2 Step 3: 20 Γ· 7 = 2 remainder 6 Step 4: 60 Γ· 7 = 8 remainder 4 Step 5: 40 Γ· 7 = 5 remainder 5 Step 6: 50 Γ· 7 = 7 remainder 1 Step 7: 10 Γ· 7 = 1 remainder 3 ← Back to remainder 3! Result: 0.142857142857...

Converting Decimals to Fractions

Example 1: Terminating Decimal

Convert 3.142678 to p/q

  • Count decimal places: 6 places
  • Multiply by 10⁢ = 1,000,000
  • Result: 3,142,678/1,000,000

Example 2: Simple Recurring (0.333...)

Let x = 0.333...

  • 10x = 3.333...
  • 10x - x = 3.333... - 0.333...
  • 9x = 3
  • x = 3/9 = 1/3

Example 3: Block Recurring (1.272727...)

Let x = 1.272727...

  • Repeating block has 2 digits, so multiply by 10Β² = 100
  • 100x = 127.272727...
  • 100x - x = 127.2727... - 1.2727...
  • 99x = 126
  • x = 126/99 = 14/11
Key: Multiply by 10ⁿ where n = length of repeating block

Classification Rules Summary

RATIONAL ⟷ Terminating OR Recurring
IRRATIONAL ⟷ Non-terminating Non-recurring

Complete Classification Chart:

Decimal Type Example Number Type
Terminating 0.875 Rational
Recurring 0.333... Rational
Non-recurring 1.414... Irrational

Quick Test - Classify These:

  • 0.75 β†’ Terminating β†’ Rational
  • 0.121212... β†’ Recurring β†’ Rational
  • Ο€ = 3.1415... β†’ Non-recurring β†’ Irrational
  • 2.8Μ„ = 2.888... β†’ Recurring β†’ Rational