Write 0.2777... (only the 7 recurring) in the form p/q.
- Let x = 0.2777... Then 10x = 2.777... and 100x = 27.777...
- Subtract: 100x − 10x = 27.777... − 2.777..., so 90x = 25.
- x = 25/90 = 5/18.
Answer: 0.2777... = 5/18
Real numbers
Maths · Number systems. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
Every real number is either rational or irrational. A rational number can be written as p/q, where p and q are integers and q is not zero. Its decimal form either stops (terminating) or repeats a block of digits for ever (recurring).
An irrational number cannot be written as p/q. Its decimal never stops and never repeats, for example √2, √7 and π. Note that 22/7 is only an approximation of π.
The real numbers obey the field properties for addition and multiplication (closure, commutative, associative, identity, inverse, distributive) and the order properties for inequalities.
Rules to remember
- Rational number: p/q with p, q integers and q ≠ 0; its decimal is terminating or recurring.
- Irrational number: non-terminating, non-recurring decimal, e.g. √2, √3, π, e.
- Additive identity is 0 and multiplicative identity is 1; the additive inverse of a is −a; the multiplicative inverse of a is 1/a (a ≠ 0).
- Commutative: a + b = b + a, ab = ba. Associative: (a + b) + c = a + (b + c), (ab)c = a(bc). Distributive: a(b + c) = ab + ac.
- Order: if a > b then a + c > b + c; if a > b and c > 0 then ac > bc; if a > b and c < 0 then ac < bc.
- Recurring decimal to p/q: let x be the decimal, multiply by powers of 10 so the recurring parts line up, then subtract.
- Closure: N is not closed under subtraction (3 − 5 is not in N); Z, Q and R are.
Common mistake
Students call every decimal that goes on for ever irrational. A recurring decimal such as 0.333... is rational (it equals 1/3); only a decimal that neither stops nor repeats is irrational.
Worked example
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