L3 · Classifying and ordering real numbers

Learning goal

“I can classify a number as rational or irrational from its decimal expansion, and order a mixed list of real numbers - fractions, decimals and square roots - from least to greatest.”

Terminating decimal — a decimal that ends: 0.625, 3.5. Always rational.

Repeating decimal — a decimal whose digits repeat in a block forever: 0.454545.... Always rational.

Decimal expansion — the decimal form of a number, used to classify it: terminating or repeating -> rational; non-terminating and non-repeating -> irrational.

Perfect square — a whole number that is the square of a whole number (1, 4, 9, 16, ...). Square roots of perfect squares are rational; square roots of other whole numbers are irrational.

Worked example

Classify 5/11 and 0.101001000100001... . Step 1: 5/11 = 0.454545... - the block "45" repeats forever. Repeating -> rational (consistent: 5/11 is already in a/b form). Step 2: 0.101001000100001... - the gaps between the 1s keep growing (one 0, then two, then three...). The pattern never settles into a repeating block and never ends -> irrational. Say it plainly: "never ends" is NOT the test. The test is "never ends AND never repeats."

Check yourself — explain it

Why is 0.333... rational even though its decimal never terminates?

Paired problem — example still visible

Classify 0.202002000200002... (the block of zeros grows by one each time) -> Irrational - non-terminating and non-repeating.

Faded completion — last step is yours

"7/9 = 0.777..., so 7/9 is ______ because its decimal ______." -> rational; repeats (the digit 7 repeats forever)

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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