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Complex

structure Complex {
    re: Real
    im: Real
}

Complex numbers consist of a real part and an imaginary part. They extend the real numbers and satisfy the equation i² = -1.

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abs_squared​

define abs_squared(self) -> Real {
    self.re * self.re + self.im * self.im
}

Computes the squared magnitude |z|² = re² + im².

add​

define add(self, other: Complex) -> Complex {
    Complex.new(self.re + other.re, self.im + other.im)
}

Adds two complex numbers component-wise.

conj​

define conj(self) -> Complex {
    Complex.new(self.re, -self.im)
}

Yields the complex conjugate (negates the imaginary part).

div​

define div(self, other: Complex) -> Complex {
    self * other.reciprocal
}

Divides this complex number by another. Division by zero returns zero (making this a total function).

from_real​

let from_real: Real -> Complex = function(r: Real) {
    Complex.new(r, Real.0)
}

Converts a real number to a complex number (with zero imaginary part).

i​

let i: Complex = Complex.new(Real.0, Real.1)

The imaginary unit, satisfying i² = -1.

im​

Complex.im: Complex -> Real

The imaginary part of the complex number.

is_imaginary​

define is_imaginary(self) -> Bool {
    self.re = Real.0 and self.im != Real.0
}

True if this complex number is purely imaginary (no real component).

is_real​

define is_real(self) -> Bool {
    self.im = Real.0
}

True if this complex number has no imaginary component.

mul​

define mul(self, other: Complex) -> Complex {
    Complex.new(self.re * other.re - self.im * other.im, self.re * other.im + self.im * other.re)
}

Multiplies two complex numbers using the formula (a+bi)(c+di) = (ac-bd)+(ad+bc)i.

neg​

define neg(self) -> Complex {
    Complex.new(-self.re, -self.im)
}

Yields the additive inverse of this complex number.

re​

Complex.re: Complex -> Real

The real part of the complex number.

reciprocal​

define reciprocal(self) -> Complex {
    self.conj * Complex.from_real(self.abs_squared.reciprocal)
}

Computes the multiplicative inverse (1/z) for non-zero complex numbers. Yields 0 when applied to 0 (division by zero yields zero).