Added Continued fractions (#6846)

* updating DIRECTORY.md

* added continued fractions

* updating DIRECTORY.md

* Update maths/continued_fraction.py

Co-authored-by: Caeden Perelli-Harris <caedenperelliharris@gmail.com>

* Update maths/continued_fraction.py

Co-authored-by: Caeden Perelli-Harris <caedenperelliharris@gmail.com>

---------

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
Co-authored-by: Caeden Perelli-Harris <caedenperelliharris@gmail.com>
Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com>

DIRECTORY.md

@@ -555,6 +555,7 @@
   * [Chudnovsky Algorithm](maths/chudnovsky_algorithm.py)
   * [Collatz Sequence](maths/collatz_sequence.py)
   * [Combinations](maths/combinations.py)
+  * [Continued Fraction](maths/continued_fraction.py)
   * [Decimal Isolate](maths/decimal_isolate.py)
   * [Decimal To Fraction](maths/decimal_to_fraction.py)
   * [Dodecahedron](maths/dodecahedron.py)

maths/continued_fraction.py

@@ -0,0 +1,51 @@
+"""
+Finding the continuous fraction for a rational number using python
+
+https://en.wikipedia.org/wiki/Continued_fraction
+"""
+
+
+from fractions import Fraction
+
+
+def continued_fraction(num: Fraction) -> list[int]:
+    """
+    :param num:
+    Fraction of the number whose continued fractions to be found.
+    Use Fraction(str(number)) for more accurate results due to
+    float inaccuracies.
+
+    :return:
+    The continued fraction of rational number.
+    It is the all commas in the (n + 1)-tuple notation.
+
+    >>> continued_fraction(Fraction(2))
+    [2]
+    >>> continued_fraction(Fraction("3.245"))
+    [3, 4, 12, 4]
+    >>> continued_fraction(Fraction("2.25"))
+    [2, 4]
+    >>> continued_fraction(1/Fraction("2.25"))
+    [0, 2, 4]
+    >>> continued_fraction(Fraction("415/93"))
+    [4, 2, 6, 7]
+    """
+    numerator, denominator = num.as_integer_ratio()
+    continued_fraction_list: list[int] = []
+    while True:
+        integer_part = int(numerator / denominator)
+        continued_fraction_list.append(integer_part)
+        numerator -= integer_part * denominator
+        if numerator == 0:
+            break
+        numerator, denominator = denominator, numerator
+
+    return continued_fraction_list
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+
+    print("Continued Fraction of 0.84375 is: ", continued_fraction(Fraction("0.84375")))