Complex numbers

In Imaginary mode $i$ is the imaginary unit and results may be complex, written as $a + bi$. Otherwise inputs are read as in Algebra: letters are variables, there are no units.

Calculate

Input	Result
(2 + 3i)(1 - i)	$5 + i$
\frac{1}{1+i}	$0{,}5 - 0{,}5i$
i^{23}	$-i$
\sqrt{-9}	$3i$
e^{i\pi}	$-1$
\ln(-1)	$3{,}14159\ldots i$

Sums, products, quotients and integer powers are exact. e^{z}, z^w, \ln and the trigonometric functions of a complex number use the principal branch.

Parts of a complex number

Input	Meaning	Result for $3 + 4i$
|3+4i|	modulus	$5$
\Re(3+4i)	real part	$3$
\Im(3+4i)	imaginary part	$4$
\arg(3+4i)	argument in $(-\pi, \pi]$	$0{,}927\ldots$
\overline{3+4i}	conjugate	$3 - 4i$

Complex variables

A variable can hold a complex number:

z_1 = 3+4i
z_1^2

$$-7 + 24i$$

Equations

Quadratic equations with a negative discriminant get their complex roots:

Input	Result
x^2 + 1 = 0	$x = -i \lor x = i$
x^2 + 2x + 5 = 0	$x = -1 - 2i \lor x = -1 + 2i$

In Algebra mode the same equations report no real solution.

See also

• Known limits: what is not covered