Rational Numbers

Instructions

A rational number is defined as the quotient of two integers a and b, called the numerator and denominator, respectively, where b != 0.

The absolute value |r| of the rational number r = a/b is equal to |a|/|b|.

The sum of two rational numbers r₁ = a₁/b₁ and r₂ = a₂/b₂ is r₁ + r₂ = a₁/b₁ + a₂/b₂ = (a₁ * b₂ + a₂ * b₁) / (b₁ * b₂).

The difference of two rational numbers r₁ = a₁/b₁ and r₂ = a₂/b₂ is r₁ - r₂ = a₁/b₁ - a₂/b₂ = (a₁ * b₂ - a₂ * b₁) / (b₁ * b₂).

The product (multiplication) of two rational numbers r₁ = a₁/b₁ and r₂ = a₂/b₂ is r₁ * r₂ = (a₁ * a₂) / (b₁ * b₂).

Dividing a rational number r₁ = a₁/b₁ by another r₂ = a₂/b₂ is r₁ / r₂ = (a₁ * b₂) / (a₂ * b₁) if a₂ is not zero.

Exponentiation of a rational number r = a/b to a non-negative integer power n is r^n = (a^n)/(b^n).

Exponentiation of a rational number r = a/b to a negative integer power n is r^n = (b^m)/(a^m), where m = |n|.

Exponentiation of a rational number r = a/b to a real (floating-point) number x is the quotient (a^x)/(b^x), which is a real number.

Exponentiation of a real number x to a rational number r = a/b is x^(a/b) = root(x^a, b), where root(p, q) is the qth root of p.

Implement the following operations:

• addition, subtraction, multiplication and division of two rational numbers,
• absolute value, exponentiation of a given rational number to an integer power, exponentiation of a given rational number to a real (floating-point) power, exponentiation of a real number to a rational number.

Your implementation of rational numbers should always be reduced to lowest terms. For example, 4/4 should reduce to 1/1, 30/60 should reduce to 1/2, 12/8 should reduce to 3/2, etc. To reduce a rational number r = a/b, divide a and b by the greatest common divisor (gcd) of a and b. So, for example, gcd(12, 8) = 4, so r = 12/8 can be reduced to (12/4)/(8/4) = 3/2.

Assume that the programming language you are using does not have an implementation of rational numbers.

• Operator overloading is being introduced in this exercise. The Embarcadero docwiki on the subject will be very helpful to you in understanding how overriding class operators is possible along with Implicit and Explicit casting.

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Source

Wikipedia