12i

12i is a complex number that is purely imaginary, equal to 0 + 12i in rectangular form. Here i denotes the imaginary unit, which satisfies i^2 = -1. The modulus of 12i is |12i| = 12, and its principal argument is π/2 (90 degrees), placing it on the positive imaginary axis of the complex plane. In exponential form, 12i can be written as 12 e^{iπ/2}, or equivalently 12 (cos(π/2) + i sin(π/2)).

Algebraically, powers of i repeat in a cycle: i^0 = 1, i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. Therefore (12i)^n = 12^n i^n for any integer n. For example, (12i)^2 = -144 and (12i)^3 = -1728 i.

The reciprocal of 12i is 1/(12i) = -i/12. Adding 12i to a real number yields a complex number with the real part unchanged and an imaginary part increased by 12.

Geometrically, 12i is located on the imaginary axis at a distance of 12 from the origin in the complex plane.

Note: In engineering contexts, the letter i is sometimes replaced by j to denote the imaginary unit, so 12j would correspond to 12i in those settings.

See also: imaginary unit, complex number, complex plane, polar form of a complex number.