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RPN Power 48 manual · Chapter 10 of 19

Complex numbers

Build, split and calculate with complex numbers in rectangular or polar form.

10.1Make a complex number#

A complex number is shown as (re,im). Type it directly with right shift 8 and 9, or push two reals and press left shift 1 (R→C). Right shift 1 (C→R) splits it again. Left shift 2 types the imaginary unit i.

Example 10.1

Make 3 + 4i from two reals

  1. Type 3, ENTER, 4 and press left shift 1.

    3ENTER4◀R→C1
Result of the example "Make 3 + 4i from two reals" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: (3,4).
You should see1: (3,4)
Example 10.2

Type a complex number directly

  1. Right shift 8 gives parentheses, right shift 9 a comma.

    ▶( )81▶,92ENTER
Result of the example "Type a complex number directly" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: (1,2).
You should see1: (1,2)
Example 10.3

Split a complex number

  1. Press right shift 1 (C→R).

    3ENTER4◀R→C1▶C→R1
Result of the example "Split a complex number" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: 3 / 1: 4.
You should see2: 3 1: 4
Example 10.4

Use the imaginary unit

  1. 2 times i plus 3.

    2◀i2×3+
Result of the example "Use the imaginary unit" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: (3,2).
You should see1: (3,2)

10.2Add, multiply and divide complex numbers#

(1+2i)(3+4i)=−5+10i(1+2i)(3+4i)=-5+10i
Example 10.5

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

  1. Make each number, then multiply.

    1ENTER2◀R→C13ENTER4◀R→C1×
Result of the example "Multiply (1+2i) by (3+4i)" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: (-5,10).
You should see(-5,10)
Example 10.6

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

  1. Make each number, then divide.

    1ENTER2◀R→C13ENTER4◀R→C1÷
Result of the example "Divide (1+2i) by (3+4i)" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: (.44,.08).
You should see(.44,.08)
Example 10.7

The square of i

  1. i times i is −1.

    ◀i2◀i2×
Result of the example "The square of i" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: (-1,0).
You should see(-1,0)
Example 10.8

Square root of a negative number

  1. A negative real has a complex square root.

    4+/−√x
Result of the example "Square root of a negative number" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: (0,2).
You should see(0,2)
Example 10.9

Euler: e to the power i

ei=cos⁡1+isin⁡1e^{i}=\cos1+i\sin1

  1. Put i on the stack and press EXP (left shift power).

    ◀i2◀eˣyˣ
Result of the example "Euler: e to the power i" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: (.540302305868,.841470984808).
You should see(.540302305868,.841470984808)

10.3The CPLX menu#

CPLX page 1: R→C, C→R, RE, IM, ABS, ARG. The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: / R→C C→R RE IM ABS ARG .
CPLX page 1: R→C, C→R, RE, IM, ABS, ARG.
CPLX page 2: CONJ, NEG, i. The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: / CONJ NEG i .
CPLX page 2: CONJ, NEG, i.
CommandWhat it does
R→C, C→RBuild or split.
RE, IMThe real part or imaginary part.
ABSThe magnitude.
ARGThe angle, in the current angle unit.
CONJThe complex conjugate.
NEGChange the sign.
iPush the imaginary unit.
∣z∣=x2+y2,arg⁡z=tan⁡−1yx|z|=\sqrt{x^{2}+y^{2}},\qquad \arg z=\tan^{-1}\frac{y}{x}
Example 10.10

Real part

  1. Make 3 + 4i and press RE (third key).

    3ENTER4◀R→C1▶CPLXAPPSF3
Result of the example "Real part" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: 3 / R→C C→R RE IM ABS ARG .
You should see1: 3
Example 10.11

Imaginary part

  1. IM is the fourth key.

    3ENTER4◀R→C1▶CPLXAPPSF4
Result of the example "Imaginary part" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: 4 / R→C C→R RE IM ABS ARG .
You should see1: 4
Example 10.12

Magnitude

∣3+4i∣=5|3+4i|=5

  1. ABS is the fifth key.

    3ENTER4◀R→C1▶CPLXAPPSF5
Result of the example "Magnitude" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: 5 / R→C C→R RE IM ABS ARG .
You should see1: 5
Example 10.13

Angle

  1. ARG is the sixth key (radians by default).

    3ENTER4◀R→C1▶CPLXAPPSF6
Result of the example "Angle" The display shows: RAD { HOME } / 6: / 5: / 4: / 3: / 2: / 1: .927295218002 / R→C C→R RE IM ABS ARG .
You should see1: .927295218002

10.4Show complex numbers in polar form#

In POLAR mode a complex number is shown as (r,∠angle) with the angle in the current unit, and the status line shows R∠Z. The object is the same; only the display changes.

Example 10.14

Polar display

  1. Make 3 + 4i and choose POLAR.

    3ENTER4◀R→C1MODENXTF3
Result of the example "Polar display" The display shows: RAD R∠Z { HOME } / 6: / 5: / 4: / 3: / 2: / 1: (5,∠.927295218002) / GRAD RECT POLA■ HEX■ DEC BIN .
You should see(5,∠.927295218002)

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