sumdifference

Sumdifference is a term used to describe the relationship between the sum and the difference of two numbers. If a and b are real numbers, the sum is s = a + b and the difference is d = a - b. The pair (s, d) uniquely determines the original numbers, through a = (s + d)/2 and b = (s - d)/2. Conversely, given a and b, one can compute s and d.

Key relationships arise from these definitions. The identities (a + b)^2 = a^2 + 2ab + b^2 and (a - b)^2 = a^2 - 2ab + b^2 show how the sum and difference relate to squares. The difference of squares identity, a^2 - b^2 = (a + b)(a - b), can be written as a^2 - b^2 = s d. Additionally, s + d = 2a and s - d = 2b, which provides a direct link between the pair (s, d) and the original numbers.

Example: take a = 7 and b = 3. Then s = 10 and d = 4. Using the inverse relations, a = (s + d)/2 = (10 + 4)/2 = 7 and b = (s - d)/2 = (10 - 4)/2 = 3.

Applications of the sumdifference framework include solving linear equations in two variables, changing variables to simplify expressions, and factoring expressions such as the difference of squares. It also provides a concise way to express products and powers in terms of the sum and difference, which can simplify algebraic manipulation. Related concepts include the sum and difference of squares and various algebraic factorization techniques.