8i

8i denotes eight times the imaginary unit i, where i is defined by i^2 = -1. As a complex number, 8i lies on the positive imaginary axis of the complex plane and has Cartesian coordinates (0, 8).

Its real part is 0 and its imaginary part is 8. The modulus (absolute value) of 8i is 8, and its argument (angle) is pi/2 (90 degrees). In polar form, 8i can be written as 8(cos pi/2 + i sin pi/2) or 8 e^(i pi/2).

The complex conjugate of 8i is -8i. In arithmetic, adding or multiplying 8i with other complex numbers follows standard rules: for a complex number a + bi, (8i) + (a + bi) = a + (8 + b)i, and (8i)(a + bi) = -8b + 8ai. Since i^2 = -1, (8i)^2 = -64.

8i is a purely imaginary number, meaning its value has no real component. It often appears in solving equations or in representations of complex numbers in the imaginary axis. For example, z^2 = -64 has solutions z = ±8i, and the equation z^2 + 64 = 0 likewise yields z = ±8i.

In a geometric sense, multiplying by i corresponds to a 90-degree rotation in the complex plane, so scaling by 8 and then multiplying by i places the result at eight units along the positive imaginary axis. This makes 8i a common example when illustrating basic complex arithmetic and polar form concepts.