Complex numbers

A complex number type: the `i` suffix on a number makes it imaginary, arithmetic mixes complex and real values, and every engine (the interpreter, C and wasm) computes the complex functions with the same formulas, so they print the same digits. Run a cell to see its output beside the expected one.

Imaginary literals and arithmetic

2.5i is an imaginary literal (a bare identifier i is still a variable — only digits take the suffix). +, -, *, / work between complex and real values; reals promote.

Viv [1]

z = 2.0 + 3.0i
w = complex(1.0, -1.0)
show z
show z * w
show z / w
show z + 1.0

re, im, conj, abs

abs of a complex number is its modulus; z times its conjugate is exactly |z| squared.

Viv [2]

z = 3.0 + 4.0i
show re(z)
show im(z)
show conj(z)
show abs(z)
show re(z * conj(z))

Euler's identity, computed

exp, log, sqrt, sin, cos, tan all work on complex arguments. e^{i pi} comes out as -1 with an imaginary part of 1.2e-16, which is sin of the double nearest pi. sqrt takes the half-angle formulas, so the square root of -1 is exactly i.

Viv [3]

pi = 3.141592653589793
show exp(pi * 1.0i)
show abs(exp(1.0i))
show sqrt(-1.0 + 0.0i)

A rotation is a complex multiply

Multiplying by e^{i theta} rotates the plane by theta. A function that takes and returns complex values needs no annotations; the compiler infers the types.

Viv [4]

def rotate(z, theta) {
    return z * exp(complex(0.0, theta))
}
q = rotate(1.0 + 0.0i, 1.5707963267948966)
show im(q)
test "quarter turn sends 1 to i" {
    assert_near(im(q), 1.0, 1.0e-12)
    assert_near(re(q), 0.0, 1.0e-12)
}

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