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:
“If the physicians had not their cassocks and their mules, if the doctors had not their square caps and their robes four times too wide, they would never had duped the world, which cannot resist so original an appearance.”
—Blaise Pascal (1623–1662)
“April is the cruellest month, breeding
Lilacs out of the dead land, mixing
Memory and desire, stirring
Dull roots with spring rain.”
—T.S. (Thomas Stearns)
“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)
“All of life and human relations have become so incomprehensibly complex that, when you think about it, it becomes terrifying and your heart stands still.”
—Anton Pavlovich Chekhov (1860–1904)
“I had but three chairs in my house; one for solitude, two for friendship; three for society. When visitors came in larger and unexpected numbers there was but the third chair for them all, but they generally economized the room by standing up.”
—Henry David Thoreau (1817–1862)