# 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 $5+{\frac {7}{12}}={\frac {67}{12}}$.
* Hearing that it is 8 hours 13, one can infer that, past midday, $8+{\frac {13}{60}}={\frac {493}{60}}$ hours have passed.
* If a price is announced as three quarters it means that, in dollars, it is ${\frac {3}{4}}$: 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 ${\frac {2}{10}}+{\frac {5}{10}}={\frac {7}{10}}$ 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:
The simplification is automatic. An other way is to use mathn, which changes the behavior of the slash operator:
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:
In this case, to_r from String is more exact:

# Properties of the fractions
## Numerator

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.

## Denominator

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

## Value

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

# Operations
## Unary operations
### Negation

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

### inverse

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

## Binary operations
### Addition

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

### Multiplication

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

### Division

The integer quotient and remainder are still defined for fractions:

### Exponentiation

If the exponent is an integer, the power of a fraction will still be a fraction:
But if the exponent is a float, even if the power is actually a fraction, Ruby will give it as a float:

# 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:

## Egyptian fractions

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

* ↑ Well, it is true that ${\frac {5404319552844595}{4503599627370496}}\simeq {\frac {6}{5}}$ but anyway...