2i

2i denotes the complex number formed by multiplying the imaginary unit i by 2. It is purely imaginary, having real part 0 and imaginary part 2.

Geometrically, 2i lies on the positive imaginary axis of the complex plane at a distance of 2 from the origin. Its modulus is |2i| = 2, and the principal argument is π/2 (90 degrees).

(2i)^2 = -4, and in general 2^n i^n; because i cycles through i, -1, -i, 1, powers of 2i alternate between real and imaginary values. Even powers are real (for example, (2i)^4 = 16), and odd powers are imaginary (for example, (2i)^3 = -8i).

The reciprocal of 2i is 1/(2i) = -i/2, and the complex conjugate is -2i. The product with its conjugate is |2i|^2 = 4, so |1/(2i)| = 1/2.

2i is a root of the quadratic equation z^2 + 4 = 0, together with -2i as the other root.

In applications, 2i appears in complex arithmetic, signal processing, and the broader study of complex numbers, where it represents a purely imaginary quantity used in combining real and imaginary components.

See also: imaginary unit, complex number, and Euler's formula.