sumanddifference

Sum and difference refers to two quantities derived from two numbers a and b: the sum s = a + b and the difference d = a − b. The pair (s, d) uniquely determines a and b, since a = (s + d)/2 and b = (s − d)/2. Conversely, s and d are easy to compute from a and b.

Key identities involve these quantities. For example, (a + b)^2 = a^2 + 2ab + b^2 and (a − b)^2 = a^2 − 2ab + b^2. Subtracting the two equations yields 4ab = (a + b)^2 − (a − b)^2, and the product (a + b)(a − b) equals a^2 − b^2. These relationships are useful in simplifying expressions and solving problems.

Algebraic use includes solving simple linear systems, converting between variables, and factoring. Knowing the sum and difference can make certain tasks more efficient than working directly with a and b. For integer a and b, s and d must have the same parity; otherwise, (s ± d)/2 do not yield integers.

Examples help illustrate the method. If a = 7 and b = 3, then s = 10 and d = 4. Conversely, if s = 10 and d = 4, then a = (10 + 4)/2 = 7 and b = (10 − 4)/2 = 3.

Extensions of the idea appear in broader contexts, including vector and complex-number settings, where analogous sum and difference concepts apply. In trigonometry, separate sum and difference identities serve related but distinct purposes.