|- |align="right" | 7 || |- |align="right" | 8 || |- |align="right" | 9 || 3 |- |align="right" | 10 || |- |} As expansions in other numeral systems As with before, the square roots of the perfect squares (e.g., 0, 1, 4, 9, 16) are integers. In all other cases, the square roots of positive integers are irrational numbers, and therefore have non-repeating digits in any standard positional notation system. The square roots of small integers are used in both the SHA-1 and SHA-2 hash function designs to provide nothing up my sleeve numbers. As periodic continued fractions One of the most intriguing results from the study of irrational numbers as continued fractions was obtained by Joseph Louis Lagrange 1780. Lagrange found that the representation of the square root of any non-square positive integer as a continued fraction is periodic. That is, a certain pattern of partial denominators repeats indefinitely in the continued fraction. In a sense these square roots are the very simplest irrational numbers, because they can be represented with a simple repeating pattern of integers. {| |- |align="right"||| = [1; 2, 2, ...] |- |align="right"||| = [1; 1, 2, 1, 2, ...] |- |align="right"||| = [2] |- |align="right"||| = [2; 4, 4, ...] |- |align="right"||| = [2; 2, 4, 2, 4, ...] |- |align="right"||| = [2; 1, 1, 1, 4, 1, 1, 1, 4, ...] |- |align="right"|||= [2; 1, 4, 1, 4, ...] |- |align="right"||| = [3] |- |align="right"||| = [3; 6, 6, ...] |- |align="right"||| = [3; 3, 6, 3, 6, ...] |- |align="right"||| = [3; 2, 6, 2, 6, ...] |- |align="right"||| = [3; 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, ...] |- |align="right"||| = [3; 1, 2, 1, 6, 1, 2, 1, 6, ...] |- |align="right"||| = [3; 1, 6, 1, 6, ...] |- |align="right"||| = [4] |- |align="right"||| = [4; 8, 8, ...] |- |align="right"||| = [4; 4, 8, 4, 8, ...] |- |align="right"||| = [4; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, ...] |- |align="right"||| = [4; 2, 8, 2, 8, ...] |} The square bracket notation used above is a short form for a continued fraction. Written in the more suggestive algebraic form, the simple continued fraction for the square root of 11, [3; 3, 6, 3, 6, ...], looks like this: where the two-digit pattern {3, 6} repeats over and over