Fractions

Fractions are a way to represent parts of a whole. A fraction is written as a/b, where the numerator a indicates how many parts are taken, and the denominator b indicates into how many equal parts the whole is divided, with b not equal to zero. Fractions express rational numbers and can be proper (a < b), improper (a ≥ b), or mixed numbers (an integer plus a proper fraction).

Equivalent fractions name the same value. They are obtained by multiplying numerator and denominator by the same nonzero integer. Fractions can be reduced to simplest form by dividing both parts by their greatest common divisor.

Operations with fractions follow specific rules. Addition and subtraction require a common denominator; for a/b and c/d, the result is (ad ± bc) / bd after converting to a common denominator. Multiplication is straightforward: (a/b) × (c/d) = (ac)/(bd). Division is handled by multiplying by the reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c), provided c ≠ 0.

Fractions can be converted to decimals. Finite decimals occur when the denominator factors into powers of 2 and/or 5 only; other denominators yield repeating decimals. Fractions also correspond to points on the number line and are used to express ratios, probabilities, measurements, and portions in daily life.

Special notes: the numerator zero yields zero, provided the denominator is nonzero. A fraction with a zero denominator is undefined. Understanding fractions involves recognizing equivalence, simplifying, and applying the rules for arithmetic operations.