sqrtsumi

Sqrtsumi is a term used in recreational mathematics to refer to the sum of square roots across a finite multiset of nonnegative integers. Given a finite multiset A = {a1, a2, ..., an}, sqrtsumi(A) is defined as the sum of the principal square roots of its elements: sqrtsumi(A) = sum sqrt(ai). The expression is rational exactly when every ai is a perfect square; otherwise it is irrational. A common variant, sqrtsumi_rounded(A) = round(sqrtsumi(A)), is used in puzzle contexts to produce integer targets.

Examples illustrate the concept. For A = {4, 9, 16}, sqrtsumi(A) = sqrt(4) + sqrt(9) + sqrt(16) = 2 + 3 + 4 = 9. For A = {2, 2, 8}, sqrtsumi(A) = sqrt(2) + sqrt(2) + sqrt(8) = 2*sqrt(2) + 2*sqrt(2) = 4*sqrt(2) ≈ 5.65685.

Variants and usage. In puzzles and teaching, sqrtsumi is used to craft target numbers or to explore the properties of irrational sums and their approximations. The operation is additive over disjoint multisets: sqrtsumi(A ∪ B) = sqrtsumi(A) + sqrtsumi(B). This makes it useful for constructing composed sums and examining when sums of surds simplify or remain irrational.

See also: square root, irrational number, sum of surds, Diophantine approximation.

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