Bridging the Past and Present: Implementing Ancient Indian Mathematical Techniques Using Python

INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
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ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Special Issue | Volume XIV, Issue XIII, October 2025

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Bridging the Past and Present: Implementing Ancient Indian
Mathematical Techniques Using Python

Pearly P Kartha*, Nikumbha Neha R

Department of Mathematics, Dr. D. Y. Patil Arts, Commerce and Science College, Pimpri, Pune, Maharashtra, India

DOI: https://doi.org/10.51583/IJLTEMAS.2025.1413SP035

Received: 26 June 2025; Accepted: 30 June 2025; Published: 25 October 2025

Abstract: Ancient Indian Mathematics has made significant contributions to arithmetic, trigonometry and algebra, many of which
continue to influence modern computational methods. Techniques such as Bhaskara I’s sine approximation and Vedic multiplication
were designed for rapid mental calculations and have inspired the development of various modern algorithms. This paper explores
the implementation and computational performance of two ancient Indian mathematical techniques—Vedic Multiplication and
Bhaskara I’s Sine Approximation using Python. Their efficiencies are evaluated against modern numerical libraries like NumPy in
terms of execution time, computational complexity and accuracy. The findings show that some ancient techniques are highly
efficient for specific tasks even today. This study connects traditional mathematical knowledge with modern computational
methods, emphasizing the lasting impact of Indian mathematical innovations.

Keywords: Bhaskara I’s Sine Approximation, Vedic Multiplication, Mean Absolute Error, Mean Squared Error, Computational
Efficiency, Absolute Error Analysis, Algorithm Development.

I. Introduction

Mathematics has always been a fundamental part of human development influencing everything from engineering to computer
science. Indian mathematicians have significantly contributed in early numerical techniques by developing methods that simplified
calculations and improved efficiency. The Vedic mathematical system, based on ancient Sanskrit texts, introduced quick mental
calculation tricks that made complex arithmetic much easier. Bhaskara I, in the 7th century, came up with a smart way to
approximate sine values, which was an important step in trigonometry. While these ancient methods contributed significantly in
their time, today’s numerical computing depends on modern algorithms and programming tools like Python, particularly with
libraries such as NumPy.

Even though these modern methods are optimized for computational efficiency, yet it remains unclear whether ancient Indian
techniques can still offer practical advantages in certain computational scenarios. With this study, we aim to bridge the gap by
implementing and analysing the performance of two ancient mathematical techniques: Vedic Multiplication and Bhaskara I’s Sine
Approximation using Python.

The primary objectives of this study are:

1. To implement these ancient techniques in Python.

2. To evaluate their execution time and computational complexity using NumPy.

3. To examine their applications in modern computing methods

This study explores whether these ancient techniques still hold computational advantages for specific tasks through a comparative
performance analysis. By connecting historical mathematics with modern computing, it highlights the enduring relevance of ancient
Indian mathematical knowledge.

II. Literature Review

Ancient Indian mathematics has had a lasting influence on modern computation, with many techniques being useful even today.
Historians like Bose [1] and Datta & Singh [2] have explored the evolution of Indian mathematical concepts, from basic numeration
systems to advanced algebraic techniques. Their work highlights how Indian mathematicians were way ahead of their time in
developing problem-solving methods which were efficient.

Vedic Mathematics and Fast Calculations

Vedic mathematics is well known for its sixteen sutras, which simplify calculations significantly. Williams [3] discusses how these
techniques make mental arithmetic calculations faster and more intuitive. Some researchers, like Plofker [8], have looked into the
historical context of these methods and their influence on early algorithmic thinking. In recent years, comparisons have been made
between these ancient techniques and modern numerical libraries like NumPy, with some studies suggesting that Vedic
multiplication, in particular, still holds up well for certain tasks [4].

Bhaskara I’s Sine Approximation

Bhaskara I is famous for his sine approximation formula, which is surprisingly accurate considering the tools that was available at

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that time. His contributions to trigonometry and astronomy have been discussed in depth by various scholars [8], [10]. More
recently, researchers have tested Bhaskara’s method using Python’s SciPy library and compared its efficiency to modern numerical
methods [6], [14]. Gupta [11] has examined the accuracy of ancient interpolation techniques and how they relate to modern
computational approaches.

Comparing Ancient Techniques with Modern Libraries

With the rise of high-performance computing, researchers have been able to evaluate traditional mathematical techniques against
modern computational tools. Libraries like NumPy and SciPy have been widely studied for their efficiency in handling large-scale
numerical computations [4], [6]. Van der Walt et al. [13] and Virtanen et al. [14] provide insights into how these libraries optimize
calculations. Stillwell [5] and Knuth [15] have also explored the mathematical foundations of modern algorithms, which is useful
when assessing ancient methods.

Overall, this review highlights how ancient Indian mathematical techniques continue to be relevant in today’s computational world.
By testing these methods with Python-based tools, we can better understand their strengths and limitations in modern applications.

III. Methodology

Vedic Multiplication Technique

Vedic multiplication is an ancient Indian technique designed for quick mental math calculations. It works by breaking large numbers
into smaller parts, calculating partial results, and then combining them to get the final answer. This method is particularly useful
for certain arithmetic operations and has even influenced some modern computational approaches.

Implementation in Python:

To evaluate the computational efficiency of Vedic multiplication, we implemented the method using Python and compared its
performance against NumPy’s built-in multiplication function.

Steps of Implementation:

1. Breaking Down the Digits: We split the given numbers into smaller parts using the divmod() function.

2. Performing Partial Multiplications: We then calculate the intermediate products.

3. Combining the Results: We then reconstruct the final result using place value adjustments.

Python Code Implementation:

The following figure presents the Python implementation of the Vedic multiplication technique.

Fig 3.1: Python Implementation of Vedic Multiplication

Bhaskara I’s Sine Approximation

Bhaskara I proposed an approximation formula for computing sine values, which provided a simplified yet effective way to estimate
sine values for small angles. His formula is given as:

��(��) =
4��(180−��)

(40500−��(180−��))
; Where θ is in degrees

Implementation in Python:

To analyze the accuracy of Bhaskara I’s sine approximation, we implemented his formula in Python and compared it with NumPy’s
“sin ()” function.

Steps of Implementation:

1. Using Bhaskara I’s Sine formula:

 Since modern computation requires handling negative angles, so we extend Bhaskara I’s formula using the symmetry

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property:

sin(−��) = −sin(��)

 This is achieved using np.abs() to apply the formula to absolute values and np.vectorize() for efficient computation over
arrays.

 We avoid division by zero using a small offset (epsilon).

2. Computing Values:

 We apply the formula to a range of angles, including negative values for a more detailed comparison.

3. Comparing with modern computation:

 We evaluate how well Bhaskara I’s method approximates NumPy’s sine function.

Python Code Implementation:

The following python function implements Bhaskara I’s sine approximation, extending it for negative angles and comparing it with
NumPy’s sine function.

Fig 3.2 (a) Python Implementation of Bhaskara I’s Sine Approximation

In the below python code 3.2 (b), we have taken positive angles (test_angles = [0, 30, 45, 60, 90, 120, 150, 180]) for computation
and computed the sine values at these angles using both Bhaskara I’s approximation and NumPy sin() function.

We then calculated the absolute error and checked the maximum and minimum error points

Fig 3.2 (b) Absolute Error Analysis

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IV. Result and Analysis

Vedic Multiplication:

The ‘timeit’ module in python was used for time measurements for different input sizes. The efficiency of Vedic Multiplication and
NumPy’s Multiplication was assessed by finding the execution time of both these methods.

Execution Time Comparison

Vedic multiplication was tested over 10,000 iterations, and its execution time was compared with NumPy’s built-in multiplication.
Refer fig 4.1 (a). The results provide insights into which method is faster for different input sizes. The time taken for Vedic
multiplication was about 0.0046275000 sec whereas NumPy took 0.01362930 sec. That is, Vedic Multiplication was faster than
NumPy.

Fig 4.1 (a): Execution Time Comparison

Accuracy & Performance Trends

Here in fig 4.1 (b) we have calculated the execution time for “vedic multiplication” and “NumPy multiplication” and plotted a bar
plot 4.1 (d) of comparison. For better readability we created a Dataframe using pandas. Refer fig 4.1 (c).

Fig 4.1 (b): Performance Evaluation as input size changes

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A comparative analysis of execution time is visualized using bar plots.

Here in below fig 4.1 (f) we have plotted a bar plot of the time taken by “vedic multiplication” and “NumPy multiplication” for a
bulk computation of 10000

Fig 4.1 (f): Bar Plot of Execution time of Bulk Computations

Bulk Computation (10,000 random pairs) shows that Vedic Multiplication is not optimized for large-scale operations.

NumPy's vectorized implementation achieves significantly faster execution times, confirming its superiority for large datasets. In
contrast, Vedic Multiplication is well-suited for mental calculations but lacks scalability.

The table 4.1 compares execution times of Vedic multiplication vs. NumPy multiplication for different input sizes.

Table 4.1: Execution Time Comparison of Vedic vs. NumPy Multiplication

Input Size(digits) Vedic Time (s) NumPy Time (s)
2 0.00002 0.00001
4 0.00004 0.00002
6 0.00009 0.00004
10 0.00022 0.00010

The results indicate that Vedic multiplication is efficient for small numbers but becomes slower as input size increases. NumPy's
vectorized operations outperform Vedic multiplication for larger inputs due to optimized low-level computations.

Bhaskar I’s Sine Approximation

Graphical Comparison with NumPy

Here in below fig 4.2 (a) we have implemented the code for visualization and in fig 4.2 (b) we have plotted the graph for the
approximations using both Bhaskara I’s and NumPy’s sine functions.

Fig 4.2 (a): Visualization of sine values

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Fig 4.2 (b): Graphical Comparison with NumPy

 The blue dashed line represents Bhaskara I’s formula, while the red solid line represents the actual sine values.

 As we can see Bhaskara I’s formula follows NumPy’s sine curve with a lot of precision.

 The approximation remains highly accurate, even in the extended range [−180°, 180°]. Thus, the symmetry is correctly
reflected, proving the validity of extending Bhaskara I’s formula to negative angles.

Numerical Error Analysis

Fig 4.2 (c): Computation of Min/ Max Errors

From the above fig 4.2(c) we see that the maximum error of 0.001224 occurs at 45 degrees. In below fig 4.2 (d) we have plotted
the maximum and minimum absolute errors.

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Fig 4.2 (d): Bar Plot of Absolute Errors

In below fig 4.2 (e) we have computed the mean absolute error 0.392785 and the mean squared error 0.6019371268

Fig 4.2 (e): Mean Absolute Error and Mean Squared Error

 Mean Absolute Error: Calculates the average absolute deviation between Bhaskara I’s method and NumPy.

 Mean Squared Error: Gives higher weight to larger errors, highlighting deviations at higher angles.

 The low values of 0.392785 and 0.6019371268 confirm that Bhaskara I’s approximation is fairly accurate.

The table 4.2 compares Bhaskara I’s sine approximation with NumPy’s sine values and computes absolute error.

Table 4.2: Absolute Error in Bhaskara I’s Sine Approximation

Angles (°) Bhaskara I’s Approx NumPy sin () Absolute Error

0 0.000000 0.000000 0.000000

30 0.500000 0.500000 0.000000

45 0.705882 0.707107 0.001224

60 0.864865 0.866025 0.001161

90 1.000000 1.000000 0.000000

120 0.864865 0.866025 0.001161

150 0.500000 0.500000 0.000000

180 0.000000 0.000000 0.000000

From our computations, we observe that Bhaskara I’s sine approximation closely follows the actual sine values computed using

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NumPy, with the absolute error remaining below 0.0013 across all test angles. The maximum error occurs at 45° (0.001224), while
the error is negligible at 0°, 30°, 90°, 150°, and 180°. Notably, the error is symmetric, with identical values of 0.001161 for 60° and
120°. These results confirm that Bhaskara I’s approximation is highly accurate within this range, offering a computationally efficient
alternative. However, the slight deviation at mid-range angles, particularly around 45° and 60°, indicates some limitations even
though the margin remains minimal.

Comparative Study:

The two mathematical techniques: Vedic Multiplication and Bhaskara I’s Sine Approximation serve different mathematical
purposes but share a common theme: computational efficiency using minimal resources. This section evaluates their computational
accuracy, execution speed, and practical relevance in modern applications.

Comparative Table of Strengths & Weaknesses

Table 5.1: Strength, Weaknesses and Best Use Cases

Technique Strengths Weaknesses Best Use Case

Vedic Multiplication
Fast for small numbers,
easy for mental
calculations

Inefficient for bulk
computation

Mental math, low-power
devices

Bhaskara I’s Sine
Approximation

Simple formula
provides a close
estimate across 0
to 180 degrees

Small but noticeable
errors, especially near 45
and 60 degrees

Quick sine estimations
in low-power
environments

Each method has some strength that makes it suitable for different computational purposes. Vedic multiplication is beneficial in
quick mental arithmetic but becomes less efficient for larger numbers. Bhaskara I’s sine approximation offers a simple and effective
way to estimate sine values, though minor errors are present.

Computational Performance Comparison

Table 5.2: Computational Performance Comparison

Technique Speed Accuracy Scalability

Vedic
Multiplication

Fast for small
numbers, slow for
large numbers

100%
accurate

Poor for large-
scale operations

Bhaskara I’s Sine
Approximation

Extremely fast
High accuracy
for most
angles

Suitable for quick
trigonometric
estimations

Bhaskara I’s sine approximation offers a computationally efficient method for estimating sine values, with small errors observed
primarily around 45° and 60°. Vedic multiplication remains highly efficient for small calculations but lacks scalability for larger
operations.

Real-World Applications of Each Method

Table 5.3: Real-World Applications of Each Method

Technique Where It Can be Used Today?

Vedic Multiplication
Used in low-power embedded systems where rapid arithmetic is
required with minimal computation. Can also serve as a foundational
concept for computational cryptography in modular arithmetic.

Bhaskara I’s Sine Approximation Useful for simplified physics engines in real-time graphics.

While these methods are historically significant, they still hold relevance in specific fields. Vedic multiplication finds applications
in embedded systems where computational power is limited. Bhaskara I’s sine approximation provides quick sine approximation
and can be useful in educational tools.

Summary of Comparative Study

Each of these techniques represents a key step in the evolution of mathematical computation. While modern methods are more

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accurate and efficient, their core principles still play a crucial role in algorithm development and education. This study shows how
ancient mathematical knowledge continues to inspire modern computing.

V. Discussion:

This study shows that even though ancient Indian mathematical techniques are computationally efficient in certain cases, they have
limitations when compared with modern numerical methods which are way more advanced and useful in today’s time. By
implementing and analyzing these two techniques, we have understood their efficiency, constraints, and practical applications.
Vedic multiplication is ideal for rapid calculations but is less efficient in large-scale numerical tasks whereas NumPy’s optimized
C-based implementation provides superior performance for larger datasets and bulk operations. Bhaskara I’s sine approximation is
highly effective for angle with very little error. While these techniques were revolutionary in ancient time but modern numerical
tools like NumPy offer way better scalability and accuracy. However, the principles behind these ancient methods continue to
influence modern algorithm design and educational frameworks.

Computational Complexity of Implemented Techniques

Table 6.1: Computational Complexities of Ancient and Modern Techniques

Technique Time Complexity Reasoning

Vedic Multiplication

O(n2)
Follows manual digit-by-digit multiplication,
making it slower for larger numbers.

NumPy Multiplication O(n)
Uses optimized low-level operations, performing
multiplication efficiently for large datasets.

Bhaskara I’s Sine
Approximation

O(1)
Uses a direct formula, making it computationally
efficient.

NumPy Sine Function
O(1) for scalar,

O(n) for arrays

Uses Taylor series approximations but is optimized
for batch computations

Real-World Relevance

Table 5.3 outlines the core applications of these techniques, but beyond their traditional use, there are additional computational
benefits in modern computing. Vedic multiplication can serve as a fundamental concept in low-power arithmetic circuits. Bhaskara
I’s sine approximation offers a computationally efficient way to estimate sine values with high accuracy, making it useful for real-
time physics engines and low-power computing environments.

While they have limitations compared to modern numerical tools, their conceptual simplicity provides valuable insights into
efficient computation continuing to inspire efficient mathematical algorithms in the digital age thus influencing fields such as
cryptography, algorithm design, and computational learning.

VI. Conclusion:

This research bridges the gap between ancient Indian mathematics and modern computational techniques. We have explored two
historical methods: Vedic Multiplication and Bhaskara I’s Sine Approximation and analyzed their computational performance.
While modern numerical libraries offer superior speed and accuracy, these ancient techniques remain valuable in educational
contexts and algorithm development.

Key Takeaways:

 Ancient Indian mathematical techniques remain computationally relevant for specific use cases.

 Vedic Multiplication is ideal for small-scale arithmetic calculations but lacks efficiency for bulk computations.

 Bhaskara I’s Sine Approximation works well for angles but accumulates errors at certain angles though it is very small
error.

 Modern numerical tools enhance efficiency but derive inspiration from these traditional methods.

By revisiting and reinterpreting ancient mathematical wisdom, we gain valuable insights that help drive the evolution of
computational mathematics."

This study paves the way to continue the research on combining ancient techniques with modern computational methodologies.

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