i

"The" imaginary number (also called the imaginary unit) is defined as the square root of , i.e., . Although there are two possible square roots of any number, the square roots of a negative number cannot be distinguished until one of the two is defined as the imaginary unit, at which point and can then be distinguished. Since either choice is possible, there is no ambiguity in defining as "the" square root of .

In the Wolfram Language, the imaginary number is implemented as I. For some reason, engineers and physicists prefer the symbol j to , probably because the symbol (or ) is commonly used to denote current.

In the novel The Da Vinci Code, the character Robert Langdon jokes that character Sophie Neveu "believes in the imaginary number because it helps her break code" (Brown 2003, p. 351). In the movie Proof (2005), the character Hal Dobbs (played by Jake Gyllenhaal) is a mathematics graduate student whose rock-and-roll band "plays" a song called "." The joke is that when the band "plays" the song, they remain silent and motionless for several minutes since the song is "imaginary." In Isaac Asimov's short story "The Imaginary" (1942), eccentric psychologist Tan Porus explains the behavior of a mysterious species of squid by using imaginary numbers in the equations which describe its psychology. The anthology Imaginary Numbers: An Anthology of Marvelous Mathematical Stories, Diversions, Poems, and Musings (Frucht 2000) includes many other works involving imaginary numbers.

Numbers of the form , where is a real number, are called imaginary numbers (or sometimes, for emphasis, purely imaginary numbers). Numbers of the form where and are real numbers are called complex numbers, and when is used to denote a complex number, it is sometimes (in older texts) called an "affix."

The square root of is

(1)

since

(2)

This can be immediately derived from the Euler formula with ,

(3)

(4)

Amazingly, the principal value of is a real number given by

(5)

(OEIS A049006; Uhler 1921; Wells 1986, p. 26; Derbyshire 2004, p. 205).

Interestingly, all higher-order power towers have complex values, but the infinite power tower converges to the value

			(6)
			(7)
			(8)

(OEIS A077589 and A077590; Macintyre 1966), where is the Lambert W-function, as illustrated above.

The following mathematical joke exhibits the strange way in which mathematicians think. "Rrrrrring. Operator: I'm sorry, the number you have dialed is imaginary. Please multiply by and dial again." A variant of this joke, actually left on one mathematician's phone by his son states, "I'm sorry, the number you have dialed is an imaginary number. Please rotate by and try again." Taking this joke one step further gives the "identity"

(9)

The double-struck capital letter I, , is a symbol sometimes used instead of for the ring of integers. In contrast, the lower case symbol is used to refer to the imaginary unit .