Physics & Chemistry

Madhava Series ✍️

Madhava of Sangamagrama was an Indian mathematician from Kerala around 1400. He made an important discovery: smooth, oscillating trigonometric functions like sine and cosine can be expressed as infinite sums of simple power terms. This allows you to calculate the exact value of any sine or cosine by adding enough terms from a straightforward arithmetic pattern. Each term includes a higher power of the angle divided by a rapidly growing factorial. The sine series uses odd powers of the angle with alternating plus and minus signs, while the cosine series uses even powers in the same alternating pattern. The factorials in the denominators, such as six, one hundred twenty, and five thousand forty, grow so quickly that each successive term contributes less. This means that just a few terms give very accurate results. This was groundbreaking because it eliminated the need for geometric construction or painstaking tables for computing trigonometric values. Instead, you could calculate them mechanically by simply multiplying and adding, which is how every calculator and computer works today.

The arctangent series has a similar alternating pattern but uses only odd numbers as denominators instead of factorials, causing it to converge more slowly. Its most notable result stems from a basic trigonometric fact: the arctangent of one equals forty-five degrees, or pi divided by four. Plugging one into the arctangent series simplifies everything down to the alternating reciprocals of odd numbers. This yields the surprising result that pi divided by four equals one minus one third plus one fifth minus one seventh, continuing indefinitely. This means pi the intriguing ratio of a circle's circumference to its diameter, which has digits that go on forever without a pattern is entirely defined by the simplest pattern involving odd numbers. The series converges very slowly in practice. Madhava recognized this and cleverly used other techniques to calculate pi to eleven decimal places, which was unprecedented for his time.

What is historically significant is that these findings were rediscovered in Europe by Gregory, Newton, and Leibniz about two centuries later. These results were taught for centuries as European discoveries. Historical evidence from Kerala manuscripts clearly shows that Madhava had them first. His work connects two seemingly unrelated areas: the geometry of circles and angles on one side, and pure arithmetic on the other. This reveals that they share the same mathematical language. The series also relate directly to Euler's formula mentioned earlier, as substituting imaginary numbers into the exponential series automatically produces the sine and cosine series. This shows that Madhava's insights hinted at deep mathematical unity that would not be fully recognized for hundreds of years

Displacement Current ✍️

It explains how changing electric fields can behave like real currents and allow electricity to “continue flowing” even where no physical charge is moving. Imagine a capacitor made of two metal plates separated by an insulating gap. When the capacitor is charging, electrons flow into one plate and out of the other, but nothing actually crosses the empty space between them.

At first glance, it seems like the current should stop at the gap. However, as the electric field between the plates rapidly builds up and changes, it creates an effect that acts just like a current in that region. This is called displacement current—not because charges are physically moving through the gap, but because the changing electric field produces the same magnetic effect as a real current.

This invisible “flow” ensures that magnetic fields remain continuous and consistent throughout the circuit, even across empty space. It allows Maxwell’s equations to stay balanced and makes it possible for electromagnetic waves, like light, to propagate through space without needing wires or particles.

Scientists use this idea to understand how electricity and magnetism are fundamentally linked, revealing that changing electric fields and magnetic fields are two parts of a single unified electromagnetic system.

Gödel’s Completeness Theorem ✍️

It explains how logical truths in a formal system are deeply connected to what can actually be proven using rules of deduction. Imagine a vast network of statements built from a fixed set of symbols and rules, like a game where you are only allowed to move according to strict logical steps. Some statements are true within the system’s meaning, while others can be derived step by step from axioms.

Gödel’s Completeness Theorem shows that, in first-order logic, these two ideas perfectly match: if something is true in every possible interpretation of the system, then there is always a finite sequence of logical steps that proves it. In other words, nothing “true” in the logical sense is left forever unreachable by reasoning alone.

You can think of it like a landscape of truth where every valid structure casts a shadow of provability. If a statement holds in all possible worlds allowed by the rules, then there must be a path of reasoning that leads to it inside the system. There are no hidden truths that escape formal proof in this setting.

This result gives logic a powerful sense of reliability: it ensures that the rules of deduction are strong enough to capture every universally valid statement. Mathematicians use it as a foundation for understanding the limits and strengths of formal reasoning systems.

On June 25, 1993, at the Isaac Newton Institute in Cambridge, Andrew Wiles delivered the third of three lectures and announced that he had proved Fermat’s Last Theorem (no positive integers a, b, c satisfy aⁿ + bⁿ = cⁿ for n > 2).

This 350-year-old problem, posed by Pierre de Fermat in 1637, had resisted all previous attempts. Wiles’ initial proof had a gap (discovered later by referee Nick Katz), which he and Richard Taylor fixed in 1994. The corrected proof was published in 1995 and is considered one of the greatest achievements in modern number theory.

DC Motor ✍️

It explains how electrical energy can be transformed into continuous mechanical motion by using the interaction between electricity and magnetism. Imagine placing a current-carrying wire inside a magnetic field. The magnetic field pushes on the wire, creating a force that makes it move. A DC motor uses this simple idea to produce rotation.

At its heart lies a coil of wire placed between the poles of a magnet. When direct current flows through the coil, each side experiences a force in opposite directions. These forces combine to create a turning effect, called torque, which starts rotating the coil.

But there is a challenge: after half a turn, the forces would reverse and stop the motion. To solve this, the motor uses a device called a commutator, which flips the direction of current every half-rotation. This keeps the torque acting in the same rotational direction, allowing the motor to spin continuously.

Scientists and engineers use DC motors in countless devices—from electric fans and toys to electric vehicles and robotics. By controlling the current, they can precisely adjust the speed and power, making the DC motor one of the most practical and widely used machines in modern technology.

The Quantum State of a Qubit ✍️

A classical computer bit is like a coin lying flat on a table, either heads or tails, zero or one, with no uncertainty. A qubit is more like a coin spinning in the air; it truly embodies both possibilities at once. This isn’t because we are unsure of which side it will land on, but because while it's spinning, it is neither one thing nor the other. When you measure it, it settles into a definite outcome like a spinning coin that eventually lands. Until that moment, it exists in a superposition of both zero and one, with specific proportions of each part of its physical state.

The most fascinating idea in this diagram is the Bloch sphere, a simple-looking ball that maps every possible state a single qubit can have. The north pole of the sphere represents the state of definitely zero, the south pole represents definitely one, and every other point on the surface represents a different quantum superposition, a different mix of the two. The red arrow pointing from the center to the surface shows the qubit's state, and everything about the qubit's physics is encoded in the direction that arrow points. How much the arrow tilts tells you the measurement probabilities: an arrow pointing straight up means you will definitely measure zero, straight down means definitely one, and pointing sideways along the equator means a perfect fifty-fifty chance. Importantly, the arrow always reaches the surface, never sitting inside the sphere. A qubit in a perfectly controlled quantum state always lives on the surface. If it drifts into the interior, it means the quantum information has been damaged by unwanted interactions with the environment, which is why building real quantum computers is so challenging.

The second piece of information in the qubit's state is the phase, which is the horizontal rotation angle of the arrow around the vertical axis of the sphere. This is where things get truly strange. Two qubits can sit at the same height on the sphere, giving them the same measurement probabilities, but they can face different directions horizontally. While this difference is completely invisible if you measure either qubit directly, it becomes crucial when quantum states combine and interfere. Just like waves on water, where crests meeting crests create bigger waves and crests meeting troughs cancel out, quantum states with different phases interfere when combined through quantum operations. A quantum algorithm is essentially a carefully arranged sequence of these interferences, designed so incorrect answers cancel out while correct answers reinforce certainty. Every quantum gate, every operation in a quantum computer, corresponds to a physical rotation of the Bloch sphere, spinning the arrow from one direction to another. Designing a quantum algorithm is like choreographing a precise sequence of rotations. The entire power of quantum computing comes down to manipulating these two angles: one controlling what you see when you measure and the other controlling how states interfere, for a single arrow pointing at the surface of an ordinary sphere.

De Moivre's Formula ✍️

Every complex number can be described in two equivalent ways: either by its real and imaginary components or by its distance from the origin and the angle it makes with the real axis. For complex numbers on the unit circle, which are at a distance of one from the origin, the angle alone defines them. De Moivre's Formula offers a clear insight into what happens when you raise one of these unit circle complex numbers to an integer power. The distance remains one, and the angle is multiplied by that power. If you raise a complex number at angle x to the power n, you end up with a complex number at angle nx. That sums up the formula in simple terms.

This makes sense geometrically because multiplying two complex numbers adds their angles. So, multiplying a unit circle number by itself once adds the angle to itself, resulting in 2x. Multiplying again gives 3x, and doing this n times gives nx. What would usually require n separate multiplications with a lot of algebra simplifies to one observation about angle multiplication.

The proof shown in the diagram is short and clear. It uses Euler's formula, which connects exponentials and trigonometry through the imaginary unit, to rewrite the complex number as e raised to an imaginary power. Then the basic rule of exponents applies: raising a power to another power multiplies the exponents. Finally, Euler's formula can be reversed to convert back to trigonometric form, with the result appearing immediately. Four steps, each fully explained, and the formula is proven. The benefits are immense. Computing a complex number raised to the hundredth power becomes a single step of angle multiplication instead of ninety-nine separate multiplications. All the trigonometric identities for multiple angles the formulas for cosine of three times an angle, four times an angle, and so on emerge automatically by expanding the formula algebraically. Finding roots of complex numbers works by dividing angles rather than multiplying them. De Moivre's Formula is ultimately a direct result of the same key insight Euler had: that rotation in the complex plane and exponential functions are essentially the same thing, making powers of complex numbers as simple as multiplying a single number.

Engineering Turbomachinery💧⚙️

One of the fundamental problems in turbomachinery is predicting when cavitation will occur.

In this problem, we determine the maximum flow rate that can be delivered before cavitation begins in a centrifugal pump system using Bernoulli’s equation, head losses, Reynolds number, and NPSH analysis.