The square root of negative one has no solution within the real numbers. Within the complex numbers, it has two solutions: and .
An imaginary number is a complex number such that it is the product of a real number and the imaginary unit, the principal square root of negative one within the complex numbers, which is often written as i (j is often used in electrical engineering since I is used for current). The sum of a real number and an imaginary number is a complex number and the product of two imaginary numbers is a negative number.
The term "imaginary" for these numbers was coined as a derogatory term for these numbers as they were regarded as fictional or useless[1], but many applications for imaginary and complex numbers have since been found.
The complex line can be viewed as an object isomorphic to a Euclidean plane. The imaginary line is a line orthogonal to the real line that intersects it at 0 and complex numbers can be viewed as points on the complex plane or two dimensional vectors. Multiplying a complex number by a positive imaginary number would scale the vector by the absolute value of the imaginary number and rotate it 90 degrees counterclockwise and when multiplying any two complex numbers, the absolute values multiply and the arguments add.
More generally, an "imaginary number" could refer to the product of a real number and any element of a basis of hypercomplex numbers that is not the identity element one. Hypercomplex numbers are distributive unital algebras over the field of real numbers. For instance, the quaternions have three principal imaginary units i, j, and k that satisfy i2 = j2 = k2= ijk = -1 and are analogous to the imaginary unit in the complex numbers. The multiplicative group of the elements of the basis of quaternions and their additive inverses is nonabelian and therefore multiplication of quaternions is noncommutative.