Rational numbers

Measuring needs parts of a whole. A field shared among three children gives each 1/3; half a cup of ghee is 1/2. Numbers for parts of a whole are fractions.

Just as 3 has the opposite –3, every positive fraction has a negative fraction: –3/4 for 3/4. The minus sign can go in three places: –1/5 = (–1)/5 = 1/(–5).

Integers and all fractions, positive and negative, together make the rational numbers, written ℚ (for quotient). A rational number is any number that can be written p/q, where p and q are integers and q ≠ 0.

"Why can't q be 0?" p/q is the number that gives p when you multiply it by q. But every number times 0 gives 0, so 'dividing by 0' cannot give one definite answer.

Every integer is rational: 5 = 5/1 and –10 = –10/1. So ℚ contains the natural numbers, the whole numbers and the integers.

One rational number, many names. –1/3 = –2/6 = –3/9 = –10/30 = –2026/6078: these are equivalent. Cancel common factors to simplify: 12/30 = 2/5 (divide both by 6). We usually write p/q in lowest terms, where p and q are co-prime (no common factor except 1).

Are two fractions equal? Cross-multiply. a/b = c/d exactly when ad = bc. For example 2/3 = 4/6 because 2 × 6 = 12 = 3 × 4.