isqrt2

Isqrt2 refers to the integer square root of 2, defined as the greatest integer less than or equal to the square root of 2. In formal terms, isqrt2 = floor(sqrt(2)). Since sqrt(2) is approximately 1.41421356, the value is 1. This reflects that 1^2 ≤ 2 and 2^2 > 2, so the integer square root of 2 is 1.

The concept extends to the general function isqrt(n), which returns floor(sqrt(n)) for nonnegative integers n. The value 1 for isqrt2 illustrates the common situation where the input is not a perfect square; the result is the largest integer whose square does not exceed the input.

In computing, isqrt functions provide exact integer results without using floating-point arithmetic, reducing rounding errors. Algorithms to compute isqrt(n) include binary search over integer values and Newton’s method adapted for integers. Some programming languages expose a built-in isqrt function, such as math.isqrt in Python, which returns 1 for the input 2. Other languages may implement similar functionality under names like integer_sqrt or floor_sqrt.

Notes on usage: isqrt is defined for nonnegative integers; negative inputs are outside its standard domain and may require error handling. The concept is widely used in number theory, combinatorics, and algorithm design, particularly in contexts requiring precise integer ranges or in the study of square roots and Diophantine problems.