An
imaginary number is a
complex number that can be written as a
real number multiplied by the
imaginary unit i,
[note 1] which is defined by its property
i2 = −1.
[1] The
square of an imaginary number
bi is −
b2. For example, 5
i is an imaginary number, and its square is −25. Except for 0 (which is both real and imaginary
[2]), imaginary numbers produce negative real numbers when squared.
An imaginary number
bi can be added to a real number
a to form a complex number of the form
a +
bi, where the real numbers
a and
b are called, respectively, the
real part and the
imaginary part of the complex number.
[3][note 2] Imaginary numbers can therefore be thought of as complex numbers whose real part is zero. The name "imaginary number" was coined in the 17th century as a derogatory term, as such numbers were regarded by some as fictitious or useless. The term "imaginary number" now means simply a complex number with a
real part equal to 0, that is, a number of the form
bi.