Square Roots of Negative and Complex Numbers
Second leaf of the complex square root Using the Riemann surface of the square root, one can see how the two leaves fit together
The square of any positive or negative number is positive, and the square of 0 is 0. Therefore, no negative number can have a real square root. However, it is possible to work with a more inclusive set of numbers, called the complex numbers, that does contain solutions to the square root of a negative number. This is done by introducing a new number, denoted by i (sometimes j, especially in the context of electricity where "i" traditionally represents electric current) and called the imaginary unit, which is defined such that i2 = –1. Using this notation, we can think of i as the square root of –1, but notice that we also have (–i)2 = i2 = –1 and so –i is also a square root of –1. By convention, the principal square root of –1 is i, or more generally, if x is any positive 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 w2 = z: the principal square root of z (defined below), and its negative.
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