9i

9i denotes the complex number obtained by multiplying the imaginary unit i by 9. The imaginary unit i is defined by i^2 = -1. Consequently, 9i has real part 0 and imaginary part 9, making it a purely imaginary number and not a real number.

Algebraically, adding 9i to a real number a yields a + 9i. In general, for a complex number a + bi, 9i = 0 + 9i, so (a + bi) + 9i = a + (b+9)i. Scalar multiplication by a real number r gives r(9i) = 9ri. Its square is (9i)^2 = -81, and its reciprocal is 1/(9i) = -i/9. The complex conjugate of 9i is -9i, and its modulus is |9i| = 9.

Geometrically, in the complex plane 9i corresponds to the point (0, 9), on the positive imaginary axis. Its polar form is 9e^{iπ/2} or 9(cos(π/2) + i sin(π/2)).

Uses: 9i is a basic example of a purely imaginary number and is used in various algebraic and engineering contexts as a simple imaginary multiplier in complex arithmetic, phasor representations, and as a component in more complex numbers.