singlefraction

Single fraction is an algebraic expression that can be written as a single ratio of polynomials, with one numerator and one denominator. In other words, it is a fraction in which neither the numerator nor the denominator contains a nested fraction or a sum of fractions. This distinguishes it from complex fractions, where the top or bottom may include fractions or sums of fractions. A single fraction can be proper or improper depending on the relative degrees of the numerator and denominator.

To obtain a single fraction from a sum or difference of fractions, one typically finds the least common multiple of the denominators, rewrites each term over that common denominator, and then combines the numerators. The result is then simplified by factoring and canceling common factors where allowed. It is important to track domain restrictions: any values that make an original denominator zero must be excluded from the domain, and if factors are canceled during simplification, those excluded values must still be excluded even if they disappear from the simplified expression.

Example: Convert (1/x) + (2/(x+1)) to a single fraction. The common denominator is x(x+1). Rewriting gives (x+1)/[x(x+1)] + (2x)/[x(x+1)], which combines to (3x+1)/[x(x+1)]. The domain excludes x ≠ 0 and x ≠ -1. Another example: (1/x) − (1/(x−1)) becomes [(x−1) − x]/[x(x−1)] = −1/[x(x−1)], with the same domain exclusions.

Single fractions are fundamental in simplifying rational expressions and are widely used in algebra, calculus, and mathematical problem solving.