What you’ll learn
Understand irrational decimal expansions.
Key idea
Rational decimals terminate or eventually repeat; endlessness by itself is not enough to infer irrationality.
Finite displays do not prove nonrepetition; repeating nines may equal a terminating representation.
Worked example
Classify examples and non-examples for the concept 'Understand irrational decimal expansions'.
Notation 0.5(2) means prefix digit 5 followed by block 2 repeated forever. Classify with full-pattern evidence.
- In 0.5(2), digit 5 is a finite prefix and block 2 repeats forever, not merely in a displayed sample.
- Its value is 5/10+2/(10×9)=47/90; eventual repetition makes it rational.
- An irrational decimal is nonterminating and never eventually repeats a fixed finite block; a finite display cannot prove that condition.
Answer: Rational — block 2 eventually repeats forever after the finite prefix.
Quick Check
Classify examples and non-examples for the concept 'Understand irrational decimal expansions'.
Notation 0.0(5) means prefix digit 0 followed by block 5 repeated forever. Classify with full-pattern evidence.