Square root any nonnegative number, then the principal square root of −"x" is The right side (as well as its negative) is indeed a square root of −"x", since For every non-zero complex number "z" there exist precisely two numbers "w" such that : the principal square root of "z" (defined below), and its negative. There are two complex numbers that square to a given arbitrary non-zero imaginary number formula_11 with real formula_12 where the two-digit pattern {3, 6} repeats over and over again in the partial denominators. Since , the above is also identical to the following generalized continued fractions: The
Square root of 2 The number can also be expressed by taking the Taylor series of a trigonometric function. For example, the series for gives The Taylor series of with and using the double factorial gives The convergence of this series can be accelerated with an Euler transform, producing It is not known whether can be represented with a BBP-type formula. BBP-type formulas are known for and , however. The number can be represented by an infinite series of Egyptian fractions, with denominators defined by 2th terms of a Fibonacci-like recurrence relation a(n)=34a(n-1)-a(n-2), a(0)=0, a(1)=6. The square root of two has the following continued