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 instantaneous 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.
Read more about this topic: Sqrt
Famous quotes containing the words square, roots, negative, complex and/or numbers:
“I would say it was the coffin of a midget
Or a square baby
Were there not such a din in it.”
—Sylvia Plath (1932–1963)
“Though of erect nature, man is far above the plants. For man’s superior part, his head, is turned toward the superior part of the world, and his inferior part is turned toward the inferior world; and therefore he is perfectly disposed as to the general situation of his body. Plants have the superior part turned towards the lower world, since their roots correspond to the mouth, and their inferior parts towards the upper world.”
—Thomas Aquinas (c. 1225–1274)
“Our role is to support anything positive in black life and destroy anything negative that touches it. You have no other reason for being. I don’t understand art for art’s sake. Art is the guts of the people.”
—Elma Lewis (b. 1921)
“In ordinary speech the words perception and sensation tend to be used interchangeably, but the psychologist distinguishes. Sensations are the items of consciousness—a color, a weight, a texture—that we tend to think of as simple and single. Perceptions are complex affairs that embrace sensation together with other, associated or revived contents of the mind, including emotions.”
—Jacques Barzun (b. 1907)
“All experience teaches that, whenever there is a great national establishment, employing large numbers of officials, the public must be reconciled to support many incompetent men; for such is the favoritism and nepotism always prevailing in the purlieus of these establishments, that some incompetent persons are always admitted, to the exclusion of many of the worthy.”
—Herman Melville (1819–1891)