The complex number a + bi. Exact when a and b are. Complex values take the ordinary operators + - * / and a whole-number power ^ (z * w, z^2, 1/z), == and !=, and abs(z) for the modulus; the name i, when bound to nothing, is the imaginary unit in an expression of numbers: (3 + 4*i)*(1 - 2*i) is 11 - 2i.
a (number) -- real partb (number) -- imaginary partThe complex number with modulus r and argument theta (radians).
r (number) -- theta (number) -- radiansThe real part of a complex number (also z.re).
z (complex) -- The imaginary part of a complex number (also z.im).
z (complex) -- The complex conjugate a - bi. Also the conjugate of a quaternion.
z (complex) -- |z| = sqrt(a^2 + b^2), exact (a surd when irrational).
z (complex) -- |z|^2 = a^2 + b^2, exact.
z (complex) -- The argument (angle) of z in radians, in (-pi, pi].
z (complex) -- The modulus r and argument theta of z, as a record (z.r, z.theta).
z (complex) -- True if the imaginary part is zero.
z (complex) -- True if the real part is zero.
z (complex) -- The product of two complex numbers (a plain number counts as complex). Also the product of two quaternions (not commutative: ij = k, ji = -k).
z (complex) -- w (complex) -- The quotient z / w of two complex numbers.
z (complex) -- w (complex) -- e^z for a complex z.
z (complex) -- The principal natural logarithm of a complex z.
z (complex) -- The principal square root of a complex z (also of a negative number), exact where its parts allow: csqrt(-3) is sqrt(3)i as NCERT writes it, csqrt(-4) is 2i, csqrt(complex(3, 4)) is 2 + i; otherwise a decimal.
z (complex) -- The complex conjugate: conj(3 + 4*i) is 3 - 4i. sqrt of a complex value gives its principal root, as csqrt.
z (complex) -- The complex numberz^n for a whole-number n, exact when z is.
z (complex) -- n (integer) -- All n n-th roots of z, as a list.
z (complex) -- n (integer) -- All roots of a polynomial of any degree with real or complex coefficients, given as a list from the highest power down; a root of multiplicity m appears m times. Roots whose parts are fractions are exact, the others are decimals; sorted by real part, then a + bi before a - bi. (For rational coefficients, solve(..., "complex") also gives radicals exactly.)
coefficients (list of complex) -- highest power firstcos(theta) + i sin(theta), theta in radians; exact at multiples of 30 and 45 degrees (cis(pi/3) is 1/2 + sqrt(3)/2 i).
theta (number) -- radiansThe complex number z turned counter-clockwise by theta radians about the point c: c + (z - c) cis(theta). Exact at multiples of 30 and 45 degrees. (rotate is for points, shapes and vectors.)
z (complex) -- c (complex) -- the centretheta (number) -- radiansTrue when three complex numbers (points of the Argand plane) lie on one line: im(conj(z2 - z1)(z3 - z1)) = 0. (collinear is for points.)
z1 (complex) -- z2 (complex) -- z3 (complex) -- The area of the triangle whose corners are three complex numbers: |im(conj(z2 - z1)(z3 - z1))| / 2.
z1 (complex) -- z2 (complex) -- z3 (complex) -- The distance from the complex number z to the line through a and b: |im(conj(b - a)(z - a))| / |b - a|.
z (complex) -- a (complex) -- b (complex) -- The sum of cis(2 pi j k / n) for j = 0, ..., n - 1, exact: n when n divides k, otherwise 0 (the sum of the n-th roots of unity, each raised to the k).
k (integer) -- n (integer) -- at least 1