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
Root mean square in the frequency domain: If formula_25 is the arithmetic mean and formula_26 is the standard deviation of a population or a waveform then: From this it is clear that the RMS value is always greater than or equal to the average, in that the RMS includes the "error" / square deviation as well. Physical scientists often use the term "root mean square" as a synonym for standard deviation when it can be assumed the input signal has zero mean, i.e., referring to the square root of the mean squared deviation of a signal from a given baseline or fit. This