Complex numbers

13.1Type the imaginary unit#

Press ctrl𝑖π to type $i$. Complex arithmetic works in any mode. In Real mode, a result that needs $i$ from real inputs (such as $\sqrt{-4}$) gives "Non-real result"; switch to Rectangular in Settings to allow it.

Example 13.1

Multiply complex numbers

$$(2+i)(3-i)=7+i$$

1. Type (2+i)(3−i).

   (2+ctrl𝑖π)(3−ctrl𝑖π)enter

Example 13.2

Another product

1. Type (1+2i)(3−i).

   (1+2ctrl𝑖π)(3−ctrl𝑖π)enter

Example 13.3

i squared

1. Type i².

   ctrl𝑖π^2▶enter

Example 13.4

Divide complex numbers

1. Type (1+i)/(1−i).

   (1+ctrl𝑖π)÷(1−ctrl𝑖π)enter

13.2Real part, imaginary part, size and angle#

$$\lvert a+bi\rvert=\sqrt{a^{2}+b^{2}},\qquad \overline{a+bi}=a-bi,\qquad \arg(a+bi)=\arctan\frac{b}{a}$$

Example 13.5

Real part

1. Type real(3+4i).

   real(3+4ctrl𝑖π)enter

Example 13.6

Imaginary part

1. Type imag(3+4i).

   imag(3+4ctrl𝑖π)enter

Example 13.7

Conjugate

1. Type conj(3+4i).

   conj(3+4ctrl𝑖π)enter

Example 13.8

Absolute value

1. Type abs(3+4i).

   abs(3+4ctrl𝑖π)enter

Example 13.9

Angle

1. Type angle(1+i).

   angle(1+ctrl𝑖π)enter

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