  Long before we learn to construct the field of rational numbers (out of the
ring of integers) at university, we learn how to calculate with fractions at
school. When it comes to "numbers", we are used to a commutative
multiplication, for example 2*3=6=3*2. On the other hand --even before we can
write-- we learn to talk (in a language) using words, consisting of purely
non-commuting "letters" (or symbols), for example "xy" is not equal to "yx"
(with the concatenation as multiplication). Now, if we combine numbers (from a
field) with words (from the free monoid of an alphabet) we get non-commutative
polynomials which form a ring (with "natural" addition and multiplication),
namely the free associative algebra. Adding or multiplying polynomials is easy,
for example (2/3*xy+z)+1/3*xy=xy+z or 2*x(yx+3*z)=2*xyx+6*xz. Although the
integers and the non-commutative polynomials look rather different, they share
many properties, for example the unique number of irreducible factors:
x(1-yx)=x-xyx=(1-xy)x. However, the construction of the universal field of
fractions (aka "free field") of the free associative algebra is highly
non-trivial (but really beautiful). Therefore we provide techniques (building
on the work of Cohn and Reutenauer) to calculate with free fractions
(representing elements in the free field or "skew field of non-commutative
rational functions") to be able to explore a fascinating non-commutative world.
