# 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.

# How to get a fraction in Ruby[edit]

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^{[1]}:

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[edit]

## Numerator[edit]

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

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

The result is **not** 24, why?

## Denominator[edit]

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

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

## Value[edit]

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

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

# Operations[edit]

## Unary operations[edit]

### Negation[edit]

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[edit]

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

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

## Binary operations[edit]

### Addition[edit]

Ta add two fractions, one uses the "+" symbol, but the result will always be a fraction even if it is actually an integer:

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

### Subtraction[edit]

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

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

### Multiplication[edit]

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

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

### Division[edit]

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[edit]

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[edit]

## Farey mediant[edit]

*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[edit]

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:

- 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.

## Notes[edit]

- ↑ Well, it is true that but anyway...