32i

32i is a complex number formed by multiplying the imaginary unit i by 32. The imaginary unit i is defined by i^2 = -1, so 32i has no real part and can be written in rectangular form as 0 + 32i.

In terms of magnitude and direction, 32i has a modulus of 32, since sqrt(0^2 + 32^2) = 32. Its argument (angle) is pi/2 radians (90 degrees), placing it on the positive imaginary axis. The polar form is 32 e^{i pi/2}, or equivalently 32(cos pi/2 + i sin pi/2).

As an algebraic object, 32i follows the standard rules for complex numbers. Its square is (32i)^2 = -1024, and more generally i^n cycles every four powers (i, -1, -i, 1). For arithmetic, the sum of complex numbers (a + bi) and (c + di) is (a + c) + (b + d)i, and the product (a + bi)(c + di) = (ac - bd) + (ad + bc)i. Specifically, multiplying 32i by a + bi yields (−32b) + (32a)i.

32i is purely imaginary, so its real part is zero. It serves as a simple example in teaching and applying complex-number concepts, such as converting between rectangular and polar forms or performing basic operations. In some programming contexts, the imaginary unit is represented by i in math notation but by j in code, so expressions like 32i may appear in mathematical discussions or be rendered as 32j in certain languages.