Complex numbers

9.1Make a complex number#

Shift STO (COMPLEX) combines Y and X into one complex number: Y is the first part and X the second. A complex number takes one stack level. In rectangular form (REC) the parts are real and imaginary; in polar form (POL) they are radius and angle.

Example 9.1

Make 3 + 4i

1. Type 3, ENTER, 4, then shift STO.

   3ENTER4⇧COMPLEXSTO

Example 9.2

Split a complex number back into two reals

1. Press shift STO again on a complex X.

   3ENTER4⇧COMPLEXSTO⇧COMPLEXSTO

9.2Add, multiply and divide complex numbers#

Example 9.3

Add (1+2i) and (3+4i)

1. Make each number, then add.

   1ENTER2⇧COMPLEXSTO3ENTER4⇧COMPLEXSTO+

Example 9.4

Multiply (1+2i) by (3+4i)

1. Make each number, then multiply.

   1ENTER2⇧COMPLEXSTO3ENTER4⇧COMPLEXSTO×

Example 9.5

Divide (1+2i) by (3+4i)

1. Make each number, then divide.

   1ENTER2⇧COMPLEXSTO3ENTER4⇧COMPLEXSTO÷

Example 9.6

Reciprocal of 1+2i

1. Press 1/x.

   1ENTER2⇧COMPLEXSTO1/x

Example 9.7

Change the sign

1. Press +/−.

   1ENTER2⇧COMPLEXSTO+/−

9.3Functions of complex numbers#

Square root, square, power, logarithm, exponential and the trigonometric functions all accept complex values. Absolute value (ABS in CONVERT page 2) gives the magnitude.

Example 9.8

Square root of 1+2i

1. Make the number and press the square-root key.

   1ENTER2⇧COMPLEXSTO√x

Example 9.9

Square of i

1. Make 0 + 1i and square it.

   0ENTER1⇧COMPLEXSTO⇧x²√x

Example 9.10

Euler: e to the power i

$$e^{i}=\cos 1+i\sin 1$$

1. Make i and press shift LN (e to the power x).

   0ENTER1⇧COMPLEXSTO⇧eˣLN

Example 9.11

Magnitude of 3+4i

$$|3+4i|=\sqrt{3^2+4^2}=5$$

1. Make the number, open CONVERT page 2 and press ABS.

   3ENTER4⇧COMPLEXSTO⇧CONVERT5▼LOG

Example 9.12

Complex sine

1. Sine of 1+i.

   1ENTER1⇧COMPLEXSTOSIN

9.4Show complex numbers in polar form#

Choose POL in the MODES menu. Complex values then show as radius, an angle sign, and the angle in the current angle unit. The polar annunciator appears. In POL mode shift STO takes the radius from Y and the angle from X.

Example 9.13

Convert 3+4i to polar

1. Make 3 + 4i.

   3ENTER4⇧COMPLEXSTO

2. Open MODES and press POL (sixth key), then close.

   ⇧MODES+/−XEQEXIT

   

Example 9.14

Enter a number by radius and angle

1. Switch to POL, then make 2 at 30 degrees.

   ⇧MODES+/−XEQEXIT2ENTER30⇧COMPLEXSTO

9.5Complex or real results#

With RL (real results) a square root or logarithm of a negative real gives Invalid Data. With CPX they return complex answers. If the input is already complex, you always get complex answers.

Example 9.15

Complex input always works

1. Make -4 + 0i and take the square root, even in real-result mode.

   4+/−ENTER0⇧COMPLEXSTO√x