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:
- If a man is said to be 5'7" high, this means that his height, in feet, is .
- Hearing that it is 8 hours 13, one can infer that, past midday, hours have passed.
- If a price is announced as three quarters it means that, in dollars, it is : Once again, a fraction!
- 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".
- 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 but conversely, cannot be written as a decimal equality because such an equality would not be exact.
- 1 How to get a fraction in Ruby
- 2 Properties of the fractions
- 3 Operations
- 4 Algorithms
How to get a fraction in Ruby
To enter a fraction, the Rational object is used:
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
To get the numerator of a fraction f, one enters f.numerator:
Notice, the result is not 24, because the fraction will be reduced to 12/5.
To get the denominator of a fraction f, one enters f.denominator:
An approximate value of a fraction is obtained by a conversion to a float:
Like any number, the negation of a fraction is obtained while preceding its name by the minus sign "-":
To invert a fraction, one divides 1 by this fraction:
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.
Likewise, to subtract two fractions, one writes the minus sign between them:
a=Rational(34,21) b=Rational(21,13) puts(a-b)
The product of two fractions will ever be a fraction either:
a=Rational(34,21) b=Rational(21,13) puts(a*b)
The integer quotient and remainder are still defined for fractions:
a=Rational(34,21) b=Rational(21,13) puts(a/b) puts(a%b)
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)
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))
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:
- One extracts the integer part of the fraction (with to_i) and stores it in a list;
- One subtracts to f (the remaining fraction) the largest integer inverse possible;
- And so on while the numerator of f is larger than one.
- Finally one adds the last fraction to the list.
- Well, it is true that but anyway...