sqrtfractions

Python module to handle linear combinations of square roots (field extension of rational numbers over integer square roots) exactly. A single class, SqrtFraction, is provided, to create object of the form

$$ \frac{\sum_in_i\sqrt{r_i}}{d} \qquad n_i\in\mathbb{Z}, \ r_i\in\mathbb{N}^+, \ d\in \mathbb{Z}\setminus{0} $$

where the values are represented by Pythons built-in arbitrary size integers, meaning there is no theoretical limit in magnitude nor precision. It's meant to be a next step to Pythons fraction/Fraction.

Usage

A SqrtFraction object can be initialized in two ways:

• with the constructor SqrtFraction(n={}, d=1). The numerator n can be given as an integer or as a dictionary of keys:values that correspond to the radicand:factor terms (SqrtFraction{2:3, 5:-7}) for $\frac{3\sqrt{2}-7\sqrt{5}}{1}$).
• by the random factory SqrtFraction.random(N=10, precision=20).

>>> from _sqrtfractions import SqrtFraction
>>> SqrtFraction(5)
SqrtFraction{(+5√1)/1}
>>> SqrtFraction(5, 2)
SqrtFraction{(+5√1)/2}
>>> SqrtFraction({2:3, 5:7}, 2)
SqrtFraction{(+3√2+7√5)/2}

The objects get simplified automaticly on creation, except reduce=False is passed.

>>> SqrtFraction({4:1, 5:0}, -2, reduce=False)
SqrtFraction{(+1√4+0√5)/-2}
>>> SqrtFraction({4:1, 5:0}, -2)
SqrtFraction{(-1√1)/1}

The objects can be printed

• in Unicode by __repr__
• in Latex by _repr_latex_.

The objects can be casted to

• floats
• ints

>>> float(SqrtFraction(5, 2))
2.5
>>> int(SqrtFraction(5))
5
>>> int(SqrtFraction(5, 2))
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "sqrtfractions.py", line 93, in __int__
    raise ValueError('doesn\'t represent an integer')
ValueError: doesn't represent an integer

Basic arithmetic operations are implemented:

• unary negation - and inversion (reciprocal value) ~
• addition + and subtraction -, multiplication * and division / with other SqrtFractions and with ints.

>>> s, t = SqrtFraction({2:3}, 4), SqrtFraction({5:6}, 7)
>>> s
SqrtFraction{(+3√2)/4}
>>> t
SqrtFraction{(+6√5)/7}
>>> -s
SqrtFraction{(+3√2)/-4}
>>> ~s
SqrtFraction{(-12√2)/-18}
>>> s+t
SqrtFraction{(+21√2+24√5)/28}
>>> s+5
SqrtFraction{(+3√2+20√1)/4}
>>> s-t
SqrtFraction{(-21√2+24√5)/-28}
>>> s-5
SqrtFraction{(+3√2-20√1)/4}
>>> s*t
SqrtFraction{(+18√10)/28}
>>> s*5
SqrtFraction{(+15√2)/4}
>>> s/t
SqrtFraction{(-126√10)/-720}
>>> s/5
SqrtFraction{(+3√2)/20}

For more precise descriptions of the methods please refer to the docstrings.

Design choices

• Reducing a new SqrtFraction directly after initialization: Initially SqrtFractions didn't simplify themselves. It had been thought that the intermediate simplification during multiple consecutive operations would hinder performance. Eager simplification actually improved speed of an application. Actual testing would have to be done.

TODO

• Arithmetic with floats (cast self to float and then use float arithmetic, like fractions/Fraction)
• Complexity analysis. Especially of reduce.
• abs & sign methods. Might be difficult: Square-root sum problem - Wikipedia Sum of radicals - Wikipedia

License (MIT)

Copyright (c) 2024 Sebastian Gössl

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.