HP 50g manual Chapter Calculations with complex numbers, Definitions

Chapter 4

Calculations with complex numbers

This chapter shows examples of calculations and application of functions to complex numbers.

Definitions

A complex number z is a number z = x + iy, where x and y are real numbers, and i is the imaginary unit defined by i² = –1. The complex number x + iy has a real part, x = Re(z), and an imaginary part, y = Im(z). The complex number z = zx + iy is often used to represent a point P(x,y) in the x–yplane, with the x-axis referred to as the real axis, and the y-axis referred to as the imaginary axis.

A complex number in the form x + iy is said to be in a rectangular representation. An alternative representation is the ordered pair z = (x,y). A complex number can also be represented in polar coordinates (polar

representation) as z = reiθ = r·cosθ + i r·sinθ, where r = z = x 2 + y 2

is the magnitude of the complex number z, and θ = Arg(z) = arctan(y/x) is the argument of the complex number z.

The relationship between the Cartesian and polar representation of complex numbers is given by the Euler formula: ei iθ = cosθ + i sinθ. The complex conjugate of a complex number (z = x + iy = re iθ) is z = x – iy = re –iθ . The complex conjugate of i can be thought of as the reflection of

zabout the real (x) axis. Similarly, the negative of z, –z= –x–iy= –reiθ, can be thought of as the reflection of z about the origin.

Setting the calculator to COMPLEX mode

To work with complex numbers select the CAS complex mode:

H)@@CAS@˜˜™

The COMPLEX mode will be selected if the CAS MODES screen shows the option _Complex checked, i.e.,

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