# 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 | edit source]

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 | edit source]

## Numerator[edit | edit source]

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[edit | edit source]

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

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

## Value[edit | edit source]

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

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

# Operations[edit | edit source]

## Unary operations[edit | edit source]

### Negation[edit | edit source]

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 | edit source]

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

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

## Binary operations[edit | edit source]

### Addition[edit | edit source]

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[edit | edit source]

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 | edit source]

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

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

### Division[edit | edit source]

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 | edit source]

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 | edit source]

## Farey mediant[edit | edit source]

*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 | edit source]

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 | edit source]

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