Complex numbers

19.1Turn on complex numbers#

A complex number is $a+bi$ with $i^2=-1$. Type i with 2ndi.. In REAL mode square roots of negative numbers give an error, so first choose a+bi (or re^θi) in MODE, sixth row.

Example 19.1

The square root of −4

$$\sqrt{-4}=2i$$

1. In MODE move down 5 rows, right to a+bi, enter, home.

   mode▼▼▼▼▼▶enter2ndQUITmode

2. Type √(−4).

   2nd√x²(−)4)enter

Example 19.2

i squared

1. In a+bi mode type i².

   mode▼▼▼▼▼▶enter2ndQUITmode2ndi.x²enter

19.2Add, multiply and divide complex numbers#

Example 19.3

Multiply

$$(1+2i)(3-i)=5+5i$$

1. Choose a+bi mode and type (1+2i)(3−i).

   mode▼▼▼▼▼▶enter2ndQUITmode(1+22ndi.)(3−2ndi.)enter

Example 19.4

Divide

$$\frac{1+i}{1-i}=i$$

1. Type (1+i)/(1−i).

   mode▼▼▼▼▼▶enter2ndQUITmode(1+2ndi.)÷(1−2ndi.)enter

19.3The CMPLX menu#

Press math then ▶ twice for the CMPLX tab.

Example 19.5

Conjugate

1. a+bi mode. Press math, right, right, 1 and type 3+4i.

   mode▼▼▼▼▼▶enter2ndQUITmodemath▶▶13+42ndi.)enter

Example 19.6

Real part and imaginary part

1. real( is item 2.

   mode▼▼▼▼▼▶enter2ndQUITmodemath▶▶23+42ndi.)enter

2. imag( is item 3.

   math▶▶33+42ndi.)enter

Example 19.7

Modulus

$$|3+4i|=5$$

1. abs( is item 5.

   mode▼▼▼▼▼▶enter2ndQUITmodemath▶▶53+42ndi.)enter

Example 19.8

Angle

1. angle( is item 4 (radians here).

   mode▼▼▼▼▼▶enter2ndQUITmodemath▶▶43+42ndi.)enter

Example 19.9

Polar form

$$3+4i=5e^{0.927i}$$

1. Calculate 3+4i.

   mode▼▼▼▼▼▶enter2ndQUITmode3+42ndi.enter

2. Press math, right, right, 7 (►Polar) and enter.

   math▶▶7enter

19.4Always show polar form#

Choose re^θi on the MODE screen and every complex answer is shown in polar form. See the modes chapter.

See also Modes and settings: Real or complex results