47i

47i is a complex number with zero real part and an imaginary part of 47. In standard form it is written as 0 + 47i, placing it on the positive imaginary axis of the complex plane. The imaginary unit i satisfies i^2 = -1, and any real number multiplied by i yields a purely imaginary number such as 47i.

The magnitude and argument of 47i are straightforward: its magnitude (or modulus) is |47i| = 47, since it lies 47 units away from the origin. Its argument (angle from the positive real axis) is π/2, corresponding to the positive imaginary direction. In polar form, 47i can be expressed as 47 e^{iπ/2}, or equivalently 47 (cos(π/2) + i sin(π/2)).

In terms of algebraic operations, multiplying 47i by i gives 47i^2 = -47, so i times 47i equals -47. Adding 47i to a real number a yields a + 47i, shifting the point vertically in the complex plane without changing the real part. Conjugation changes the sign of the imaginary part, so the complex conjugate of 47i is -47i.

In engineering contexts, the imaginary unit is often denoted by j, so 47i is written as 47j in electrical engineering and related fields. 47i serves as a representative example of a purely imaginary complex number used in various mathematical and applied analyses, such as phasor representations and signal processing.