Mathematics with Python and Ruby/Fractions in Ruby

A fraction is nothing more than the exact quotient of an integer by another one. As the result is not necessarily a decimal number, fractions have been used way before the decimal numbers: They are known at least since the Egyptians and Babylonians. And they still have many uses nowadays, even when they are more or less hidden like in these examples:

1. If a man is said to be 5'7" high, this means that his height, in feet, is $5+{\frac {7}{12}}={\frac {67}{12}}$.
2. Hearing that it is 8 hours 13, one can infer that, past midday, $8+{\frac {13}{60}}={\frac {493}{60}}$ hours have passed.
3. If a price is announced as three quarters it means that, in dollars, it is ${\frac {3}{4}}$: Once again, a fraction!
4. Probabilities are often given as fraction (mostly of the egyptian type). Like in "There is one chance over 10 millions that a meteor fall on my head" or "Stallion is favorite at five against one".
5. Statistics too like fractions: "5 people over 7 think there are too many surveys".

The equality 0.2+0.5=0.7 can be written as ${\frac {2}{10}}+{\frac {5}{10}}={\frac {7}{10}}$ but conversely, ${\frac {1}{2}}+{\frac {1}{3}}={\frac {5}{6}}$ cannot be written as a decimal equality because such an equality would not be exact.

How to get a fraction in Ruby

To enter a fraction, the Rational object is used:

a=Rational(24,10)
puts(a)

The simplification is automatic. An other way is to use mathn, which changes the behavior of the slash operator:

require 'mathn'
a=24/10
puts(a)

It is also possible to get a fraction from a real number with its to_r method. Yet the fraction is ensured to be correct only if its denominator is a power of 2:

a=1.2
b=a.to_r
puts(b)

In this case, to_r from String is more exact:

puts "1.2".to_r   #=> (6/5)
puts "12/10".to_r #=> (6/5)

Properties of the fractions

Numerator

To get the numerator of a fraction f, one enters f.numerator:

a=Rational(24,10)
puts(a.numerator)

Notice, the result is not 24, because the fraction will be reduced to 12/5.

Denominator

To get the denominator of a fraction f, one enters f.denominator:

a=Rational(24,10)
puts(a.denominator)

Value

An approximate value of a fraction is obtained by a conversion to a float:

a=Rational(24,10)
puts(a.to_f)

Operations

Unary operations

Negation

Like any number, the negation of a fraction is obtained while preceding its name by the minus sign "-":

a=Rational(2,-3)
puts(-a)

inverse

To invert a fraction, one divides 1 by this fraction:

a=Rational(5,4)
puts(1/a)

Binary operations

Addition

To add two fractions, one uses the "+" symbol, and the result will be a fraction.

a=Rational(34,21)
b=Rational(21,13)
puts(a+b)

If a non-fraction is added to a fraction, the resulting type will vary. Adding a fraction and an integer will result in a fraction. But, adding a fraction and a float results in a float.

Subtraction

Likewise, to subtract two fractions, one writes the minus sign between them:

a=Rational(34,21)
b=Rational(21,13)
puts(a-b)

Multiplication

The product of two fractions will ever be a fraction either:

a=Rational(34,21)
b=Rational(21,13)
puts(a*b)

Division

The integer quotient and remainder are still defined for fractions:

a=Rational(34,21)
b=Rational(21,13)
puts(a/b)
puts(a%b)

Exponentiation

If the exponent is an integer, the power of a fraction will still be a fraction:

a=Rational(3,2)
puts(a**12)
puts(a**(-2))

But if the exponent is a float, even if the power is actually a fraction, Ruby will give it as a float:

a=Rational(9,4)
b=a**0.5
puts(b)
puts(b.to_r)

Algorithms

Farey mediant

Ruby has no method to compute the Farey mediant of two fractions, but it is easy to create it with a definition:

def Farey(a,b)
    n=a.numerator+b.numerator
    d=a.denominator+b.denominator
    return Rational(n,d)
end
 
 
a=Rational(3,4)
b=Rational(1,13)
puts(Farey(a,b))

Egyptian fractions

To write a fraction like the Egyptians did, one can use Fibonacci's algorithm:

def egypt(f)
    e=f.to_i
    f-=e
    list=[e]
    begin
        e=Rational(1,(1/f).to_i+1)
        f-=e
        list.push(e)
    end while f.numerator>1
    list.push(f)
    return list
end
 
require 'mathn'
 
a=21/13
puts(egypt(a))

The algorithm can be summed up like this:

1. One extracts the integer part of the fraction (with to_i) and stores it in a list;
2. One subtracts to f (the remaining fraction) the largest integer inverse possible;
3. And so on while the numerator of f is larger than one.
4. Finally one adds the last fraction to the list.