Arithmetic

• Addition
• Subtraction
• Multiplication
• Division

Addition

\[ \frac{n_1}{d_1} + \frac{n_2}{d_2} = \frac{n_1d_2 + n_2d_1}{d_1d_2} \]

add : Rational -> Rational -> Rational
add (Rational n1 d1) (Rational n2 d2) =
    makeRational (n1 * d2 + n2 * d1) (d1 * d2)

Subtraction

\[ \frac{n_1}{d_1} - \frac{n_2}{d_2} = \frac{n_1d_2 - n_2d_1}{d_1d_2} \]

sub : Rational -> Rational -> Rational
sub (Rational n1 d1) (Rational n2 d2) =
    makeRational (n1 * d2 - n2 * d1) (d1 * d2)

Multiplication

\[ \frac{n_1}{d_1} \times \frac{n_2}{d_2} = \frac{n_1n_2}{d_1d_2} \]

mul : Rational -> Rational -> Rational
mul (Rational n1 d1) (Rational n2 d2) =
    makeRational (n1 * n2) (d1 * d2)

Division

\[ \frac{n_1}{d_1} \div \frac{n_2}{d_2} = \frac{n_1}{d_1} \times \frac{d_2}{n_2} = \frac{n_1d_2}{d_1n_2} \text{, } n_2 \neq 0 \]

div : Rational -> Rational -> Rational
div (Rational n1 d1) (Rational n2 d2) =
    if n2 == 0 then
        zero

    else
        makeRational (n1 * d2) (d1 * n2)

For division by zero, I chose to return the zero rational number rather than to signal an error.

Exercise

Implement div : Rational -> Rational -> Maybe Rational, such that div r zero == Nothing, and solve the downstream consequences of this change.