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Year 8AC9M8N01

Rational, Irrational and Recurring

Recognise irrational numbers in applied contexts, including square roots and pi. Recognise terminating and recurring decimals, using digital tools as appropriate.

Every number is either a fraction in disguise or it is not. That single split explains why some decimals stop, some repeat, and some do neither.

Method

Rational means it can be a fraction
ab where b≠0\tfrac{a}{b} \text{ where } b \neq 0
Every terminating and every recurring decimal is rational.
Terminating decimals stop
38=0,375\tfrac{3}{8} = 0{,}375
This happens when the denominator has only 2s and 5s as factors.
Recurring decimals repeat forever
13=0,3‾\tfrac{1}{3} = 0{,}\overline{3}
The bar shows which digits repeat.
Irrational numbers never stop and never repeat
π, 2, 3\pi,\ \sqrt{2},\ \sqrt{3}
They cannot be written as a fraction at all.
A surd is an irrational root
9=3 is not a surd2 is\sqrt{9} = 3 \ \text{is not a surd} \quad \sqrt{2} \ \text{is}
Only roots that do not come out exactly.

Worked example

e.g. Decide whether 58, 23 and 7 are terminating, recurring or irrational.\text{Decide whether } \tfrac{5}{8},\ \tfrac{2}{3} \text{ and } \sqrt{7} \text{ are terminating, recurring or irrational.}
  1. Divide the first one out.5÷8=0,6255 \div 8 = 0{,}625
  2. It stops, so it terminates. Check why: 8 is 232^3.8=23only 2s8 = 2^3 \quad \text{only 2s}
  3. Divide the second out.2÷3=0,666…=0,6‾2 \div 3 = 0{,}666\ldots = 0{,}\overline{6}
  4. It repeats forever, so it is recurring. 3 is not a 2 or a 5.3 is neither3 \text{ is neither}
  5. Now the root. Is 7 a perfect square?4<7<9so no4 < 7 < 9 \quad \text{so no}
  6. It is not, so the root never stops or repeats.7≈2,6457… irrational\sqrt{7} \approx 2{,}6457\ldots \text{ irrational}

Practice

Divide it out. Then decide which of the three it is.

1
Is 720 terminating or recurring?\text{Is } \tfrac{7}{20} \text{ terminating or recurring?}
Answer0,35 — terminating, since 20=22×50{,}35 \text{ — terminating, since } 20 = 2^2 \times 5
2
Is 49 terminating or recurring?\text{Is } \tfrac{4}{9} \text{ terminating or recurring?}
Answer0,4‾ — recurring0{,}\overline{4} \text{ — recurring}
3
Is 25 a surd?\text{Is } \sqrt{25} \text{ a surd?}
AnswerNo. 25=5, a whole number.\text{No. } \sqrt{25} = 5, \text{ a whole number.}
4
Is 10 rational?\text{Is } \sqrt{10} \text{ rational?}
AnswerNo. 10 is not a perfect square.\text{No. 10 is not a perfect square.}
5
Why is π irrational?\text{Why is } \pi \text{ irrational?}
AnswerIts decimal never stops and never repeats.\text{Its decimal never stops and never repeats.}
6
Is 0,25 rational?\text{Is } 0{,}25 \text{ rational?}
AnswerYes. It is 14.\text{Yes. It is } \tfrac{1}{4}.
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