Irreducible fraction
An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a
An equivalent definition is sometimes useful: if a and b are integers, then the fraction a/b is irreducible if and only if there is no other equal fraction c/d such that |c| < |a| or |d| < |b|, where |a| means the absolute value of a.[4] (Two fractions a/b and c/d are equal or equivalent if and only if ad = bc.)
For example, 1/4, 5/6, and −101/100 are all irreducible fractions. On the other hand, 2/4 is reducible since it is equal in value to 1/2, and the numerator of 1/2 is less than the numerator of 2/4.
A fraction that is reducible can be reduced by dividing both the numerator and denominator by a common factor. It can be fully reduced to lowest terms if both are divided by their
Examples
In the first step both numbers were divided by 10, which is a factor common to both 120 and 90. In the second step, they were divided by 3. The final result, 4/3, is an irreducible fraction because 4 and 3 have no common factors other than 1.
The original fraction could have also been reduced in a single step by using the greatest common divisor of 90 and 120, which is 30. As 120 ÷ 30 = 4, and 90 ÷ 30 = 3, one gets
Which method is faster "by hand" depends on the fraction and the ease with which common factors are spotted. In case a denominator and numerator remain that are too large to ensure they are coprime by inspection, a greatest common divisor computation is needed anyway to ensure the fraction is actually irreducible.
Uniqueness
Every rational number has a unique representation as an irreducible fraction with a positive denominator[3] (however 2/3 = −2/−3 although both are irreducible). Uniqueness is a consequence of the unique prime factorization of integers, since a/b = c/d implies ad = bc, and so both sides of the latter must share the same prime factorization, yet a and b share no prime factors so the set of prime factors of a (with multiplicity) is a subset of those of c and vice versa, meaning a = c and by the same argument b = d.
Applications
The fact that any rational number has a unique representation as an irreducible fraction is utilized in various proofs of the irrationality of the square root of 2 and of other irrational numbers. For example, one proof notes that if √2 could be represented as a ratio of integers, then it would have in particular the fully reduced representation a/b where a and b are the smallest possible; but given that a/b equals √2, so does 2b − a/a − b (since cross-multiplying this with a/b shows that they are equal). Since a > b (because √2 is greater than 1), the latter is a ratio of two smaller integers. This is a contradiction, so the premise that the square root of two has a representation as the ratio of two integers is false.
Generalization
The notion of irreducible fraction generalizes to the
See also
- Anomalous cancellation, an erroneous arithmetic procedure that produces the correct irreducible fraction by cancelling digits of the original unreduced form.
- Diophantine approximation, the approximation of real numbers by rational numbers.
References
- ^ Stepanov, S. A. (2001) [1994], "Fraction", Encyclopedia of Mathematics, EMS Press
- ISBN 9783540438267
- ^ a b Scott, William (1844), Elements of Arithmetic and Algebra: For the Use of the Royal Military College, College text books, Sandhurst. Royal Military College, vol. 1, Longman, Brown, Green, and Longmans, p. 75.
- ^ Scott (1844), p. 74.
- ISBN 9780821887981.
- ISBN 9781939512017.
- ISBN 9781584886907.
- ISBN 9780387715681.