To have a fast visual effect ;) try the following:

After that, experiment for your own.
### Some more details

Fractional trees have the general form
where those
a_{1},...,a_{m};b_{1},...,b_{n} are
again fractional trees. So they are built up recursively, starting
with 1. The order of the a_{i} and the order of the b_{i} does not matter. The text notation for the (above) fractional tree is
[a_{1},...,a_{m};b_{1},...,b_{n}],
which can be quite unmanageable for bigger expression. For
[;b_{1},...,b_{n}] we write short
[b_{1},...,b_{n}]. The natural numbers are short for
[...[1]...], for example 4 is short for [[[1]]]. To handle empty
sequences in the graphical representation we write according with the rules
The fraction intention is expressed by

You see that 1,o,~ form a group. By further playing

though be aware that parsing is not very robust. Note that expressions can become big very fast. And computations then rapidly increase in time, so the server will kill the process after some seconds and you retrieve an error page.

- Enter 4 in the 1st cell, enter ~5 in the 2nd cell, press then * (uncancelled multiplication) between them. A big expression on the right side (3rd cell) appears,
- now press % (cancel) on the 3rd cell
- and enjoy reducing the right side.

general equation | examples (the subscripts mean the assigned cell) |

xo(~x)=1 | enter [1,2,3]_{1}, ~[1,2,3]_{2}, press *, %_{3} |

(1+x)o(~x)=x+1 | enter [1,2,3,4]_{1}, press to2_{1}, ~_{1}, 1+_{1}, o |

~(xoy)=(~y)o(~x) | enter 4_{1}, ~5_{2}, press o, new cell, as3_{4},
swap, ~_{1}, ~_{2}, o, ~_{3}cell 3 and 4 should be equal, press x _{4} |

(xo(~y))o(yoz)=xoz | enter 4 * (~3)_{1}, press enter_{1}, %_{1}, enter 3 o (~5)_{2}, press enter_{2}, o, enter 4 o (~5)_{3}, press enter |

? | enter [1,2,3]_{1}, [2,3,4]_{2}, press ~_{2}, *, %_{3}enter [1,[1,1]] _{1}, [1,[1,1],[[1,1,1]];]_{2}, press *, %_{3}Caution: computation expensive! Could fail to finish,with error message. |

enter [1,2,[1,2]]_{1}, [1,2]_{2}, press ~_{2}, *, %_{3}enter [1,2,[1,2],[1,2,[1,2]]] _{1}, [1,2,[1,2]]_{2}, press ~_{2}, * , %_{3} (does not work with IE, expression too big)Notice that corresponds to the number n as defined in set theory, e.g. = {{},{{}},{{},{{}}}}=[1,2,[1,2]] enter 2o2o2o2 _{1}, 2o2o2_{2}, press ~_{2}, * , %_{3} |

Verify the following laws by some examples

1oa = ao1 = a

ao(~a) = (~a)oa = 1

ao(boc)=(aob)oc (associative)

a*(b*c)=(a*b)*c

a+(b+c)=b+(a+c) (left-commutative)

- Find out how ~ (inversion) works! (easy)
- Find out how the * (uncanceled multiplication) works!
- Find out how % (canceling) works! (terrible)

though be aware that parsing is not very robust. Note that expressions can become big very fast. And computations then rapidly increase in time, so the server will kill the process after some seconds and you retrieve an error page.

Fractional trees are a generalization of fractional numbers. If we call an algebraic structure (C,1,+,o,~) a *coppice* if (C,1,o,~) is a group and if (a+b)oc = (aoc) + (boc) then the fractional numbers are the free +-associative coppice and the fractional trees are the free +-left-commutative coppice (both generated by the empty set, because we have already the constant 1 in the signature).

For a theoretical foundation look at my phd Arborescent numbers and higher arithmetic operations.
To discuss this application feel free to post in the thread Fractional Tree Calculator of the Eretrandre mathematics forum.

Last modified 2008-02-14 17:22 GMT |
Comments and suggestions to bo198214 AT eretrandre DOT org |