Mathematics
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A note in abacus
There are several types of abacuses (or abaci) used throughout history and across different cultures. The most popular ones are from China, but there are other abacus that were developed independently. This article gives glimpse of some of the popular abacus. Types of abacus 1. Chinese Abacus (Suanpan) The Suanpan typically has 2 beads on…
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Derivation of Euler’s formula for imaginary numbers
The Euler formula, one of the most remarkable equations in mathematics, elegantly connects complex analysis, trigonometry, and exponential functions. It is named after the Swiss mathematician Leonhard Euler, who first introduced it in the 18th century. The formula is expressed as: $$e^{ix} = \cos{x} + i\sin{x}$$ Where: e is the base of the natural logarithm,…
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Derivation for sum of angles
The summation formulas for sine, cosine, and tangent are the key identities that govern trigonometric relationships. These formulas help us analyse combinations of angles in various scenarios, from simple geometric calculations to complex oscillatory systems. Let’s revise these fundamental concepts. Derivation Cos(A+B)=OP/OS=(OQ-PQ)/OS =OQ/OS-PQ/OS =OQ/OR*OR/OS-PQ/SR*SR/OS =CosA*CosB-SinA*SinB …………(since PQ=TR) Sin(A+B)=SP/OS=(ST+TP)/OS =ST/OS+TP/OS =ST/SR*SR/OS+TP/OR*OR/OS =CosA*SinB +SinA*CosB ………..(since TP=RQ) Tan(A+B)=Sin(A+B)/Cos(A+B)…
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The unification of calculus
The rivalry between Isaac Newton and Gottfried Wilhelm Leibniz over the invention of calculus is one of the most famous intellectual feuds in scientific history. Though both men made pioneering contributions to the field, their personal clash and the priority dispute that followed have become nearly as significant as their actual mathematical achievements. This article…
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Nowak’s Punishment Study
Martin A. Nowak's mathematical models challenge conventional theories in evolution, cancer, and HIV, now focusing on life’s origin, showing selection can exist before replication.
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Multiple linear regression analysis with constraints
Multiple linear regression is a statistical technique used to model the relationship between a dependent variable and two or more independent variables. It extends simple linear regression by allowing multiple factors to influence the outcome, making it useful for predicting or understanding the effect of several variables simultaneously. The model takes the form \[ y…
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Monte Carlo simulation to find value of PI
Inception While reading “Fooled by Randomness” by Nassim Nicholas Taleb, I came across a chapter discussing Monte Carlo theory. The author claimed that when bullets are fired inside a square and a circle is inscribed within it, the ratio of bullets inside the circle to the total number of bullets approximates a ratio involving π.…
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Derivation of combined angle- Version 2
In this post, I will show how to calculate the combined 3d angle. Such situation arises while designing a rigid 3d joints such as a pipeline. The coordinate-based approach can be employed to determine the actual deflection angle between two lines in 3D space. This method not only calculates the deflection angle, but also accurately…
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Derivation of 3D combined angle
Derivation of combined angle may become useful when you want to know the magnitude and direction of force due to bend in both XY and XZ plane. For eg in design of bend in pipeline.