Rosli Besar

[The Math Flow @TheMathFlow]

The geometry of eⁱᶿ = cos(θ) + isin(θ).

[Math Hub @mathhub_vn]

sin²x + cos²x =1

[ScieVision @scievision369]

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.

[SparkED @Afghan_Engineer]

Ohm's law as linear equation.

[Code Geek @codek_tv]

3D visualization of Euler’s formula

[MIT CSAIL @MIT_CSAIL]

A free MIT course about deep learning, 2026 edition: https://t.co/dEgRHps7Ro

Here, MIT CSAIL researcher Alexander Amini discusses the differences between supervised, unsupervised, & reinforcement learning approaches (Lecture 5).

[The Math Flow @TheMathFlow]

Imagine five of the most fundamental, unrelated concepts in mathematics.
Then, Leonhard Euler came along and brought them together in a single, flawless sentence.
It’s called Euler’s Identity, and it’s frequently voted the most beautiful equation in human history.

[The Math Flow @TheMathFlow]

The Anatomy of the Taylor Series Expansion of a Function around a point.

[Code Geek @codek_tv]

Fourier Series

[MIT CSAIL @MIT_CSAIL]

9 distance measures in data science w/algorithms (v/@MaartenGr).

[Rohit @ai_rohitt]

INSTEAD OF WATCHING AN HOUR OF NETFLIX TONIGHT.

This 60-minute Cambridge lecture by Demis Hassabis will teach you more about the future of AI than most people will learn in the next 5 years.

Bookmark it and give it an hour, no matter what.

[Math Hub @mathhub_vn]

Differentiation vs Integration

[ScieVision @scievision369]

Taylor Series ✍️

The Taylor Series is a way to approximate any smooth function using a polynomial, which is a simple equation made of powers of x. It works by taking a function f(x) and expanding it around a specific point 'a' as an infinite sum: f(x) = f(a) + f'(a)(x-a) + [f''(a)/2!](x-a)² + [f'''(a)/3!](x-a)³ + ... Here, f(a) is the starting value, f'(a) is the slope, f''(a) is the curvature, and so on. Each new term adds more detail to better match the original curve near point 'a'. The more terms you add, the more accurate the approximation becomes near 'a'. This useful concept is used in calculators, physics, and engineering to easily compute complex functions like sin(x), e^x, or cos(x). When the center point a = 0, it is called the Maclaurin Series.

[Cliff Pickover @pickover]

Mathematics, computer programming, engineering, motion, creativity.

This is a linkage-mechanism for converting Binary Numbers to Decimal Numbers.

Created by 上木 敬士郎/Keishiro Ueki, @KeishiroUeki, Used with permission.

[Sulekha Tripathi @sulekhat95]

A man spends 50 years teaching at MIT.

He knows his time is running out.

So he records one last lecture — everything he knows, distilled into a single hour.

He died 5 months later.

This is that lecture.

The most important hour you'll watch this week. 👇
Bookmark it for later

[Massimo @Rainmaker1973]

Fast Fourier Analysis in action.

Any complex waveform, sound, or shape can be perfectly reconstructed as the sum of simple rotating circles (epicycles).

[Code Geek @codek_tv]

3D visualization of Euler’s formula: e^{jθ} = cos(θ) + j sin(θ).

[divyansh tiwari @DivyanshT91162]

A man spends 50 years teaching at MIT.

He knows his time is running out.

So he records one last lecture — everything he knows, distilled into a single hour.

He died 5 months later.

This is that lecture.

The most important hour you'll watch this week. 👇Bookmark it for later

[AilaunchX @Ai_Tech_tool]

INSTEAD OF WATCHING NETFLIX TONIGHT.

Spend 1 hour with this.

Claude AI FULL COURSE that teaches you how to BUILD and AUTOMATE anything.

The people who watch this tonight will wake up tomorrow with a new skill.

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