Practice question · Fill in the blanks
Complete the characterization of irrational numbers.
A real number is irrational exactly when its decimal expansion is ______.
Word bank: non-terminating · non-terminating and non-repeating · non-repeating only · longer than twenty digits
Hints
- 1/3 = 0.333... is non-terminating, yet it is rational, so one condition alone is not enough.
- Both conditions must fail for a rational, so both must hold for an irrational.
Show the answer
A real number is irrational exactly when its decimal expansion is non-terminating and non-repeating.
Why
Irrationality needs BOTH failures: never ending and never settling into a repeating block. Dropping either half misclassifies 1/3 and 1/7, which run forever but repeat. This is the lesson's headline pitfall.
Practise Rational and Irrational Numbers
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