arXiv subjects
Publications and source records attributed to Rajinder Kumar Jagpal.
In this work, we prove the irrationality of $π$ motivated by the nested radicals with roots of $2$ of kind $c_k = \sqrt{2 + c_{k - 1}}$ and $c_0 = 0$. Sample computations showing how the rational approximation tends to $π$ with increasing the integer $k$ are presented.
In this work, we obtain an iterative formula that can be used for computing digits of $π$ and nested radicals of kind $c_n/\sqrt{2 - c_{n - 1}}$, where $c_0 = 0$ and $c_n = \sqrt{2 + c_{n - 1}}$. We also show how with the help of this iterative formula, the two-term Machin-like formulas for $π$ can be generated and approximated. Some examples with Mathematica codes are presented.
In this work, we consider the properties of the two-term Machin-like formula and develop an algorithm for computing digits of $π$ by using its rational approximation. In this approximation, both terms are constructed by using a representation of $1/π$ in the binary form. This approach provides the squared convergence in computing digits of $π$ without any trigonometric functions and surd numbers. The Mathematica codes showing some examples are presented.
In this work, we develop a new iterative method for computing the digits of $π$ by argument reduction of the tangent function. This method combines a modified version of the iterative formula for $π$ with squared convergence that we proposed in a previous work and a leading arctangent term from the Machin-like formula. The computational test we performed shows that algorithmic implementation can provide more than $17$ digits of $π$ per increment. Mathematica codes, showing the convergence rate for computing the digits of $π$, are presented.