Reals and Rationals

Mathematics - Teaching it, Learning it, Loving it.

[Reals and Rationals @realsrationals]

Pure mathematics is often considered an art, rather than science. Here, our own students use dance to express the essence of certain mathematical concepts.

https://t.co/vhvReE4Fuz

[Reals and Rationals @realsrationals]

The (almost) two-century-old prestigious Navier–Stokes mathematics problem has reportedly been solved using AI.

The Navier–Stokes equations were devised in the early 19th century (1820s–1840s) to describe the motion of fluids. These equations quintessentially demonstrate the power of mathematical language. Just a handful of equations (mathematical statements) model the intricacies of the physical world: atmospheric weather patterns, turbulence around airplanes, and the flow of blood through the human body, among other things.

As an applied mathematics problem, it asks whether a fluid modeled by the equations can start in a smooth (well-behaved) state, but develop infinite velocity in a finite time. Developing infinite velocity in a finite time is referred to as a blowup or singularity. Two mathematical possibilities exist: a proof that no smooth starting state for the fluid will ever cause a blowup, or a counterexample of a particular, albeit pathological, smooth starting state that causes a blowup.

In addition to its practical relevance, the problem has been of immense theoretical importance in mathematics. Over the centuries, it has driven advances in partial differential equations and other areas of mathematics.

The current news is that OpenAI claims to have found a counterexample: https://t.co/UfMOanLJzg

A blowup does not happen in practice, because a liquid does not spontaneously explode. So, what does discovering a counterexample mean? It implies that while the Navier–Stokes equations model most practical situations in fluid mechanics, they do not place enough constraints to disallow some unrealistic corner cases from creeping in.

If you would like to know more, there is a webinar hosted by Harvard University today (Friday, Sept 11th) at 2:30PM Eastern time. See here for details: https://t.co/poPJ0J8cDk

[Reals and Rationals @realsrationals]

Why learn math concepts when AI can solve the problems for you?

Humans work at the intersection of many disciplines (real-world, mathematical, human aspects, etc.). AI can solve a properly framed math problem, but humans are the ones translating an inter-disciplinary problem into an AI-solvable one.

And, to translate for AI, humans need a certain fidelity of language. Math concepts are the vocabulary of that language. For example, we cannot talk about Galois theory without understanding the concepts of groups, rings or factor rings.

On a related note, here is a nice talk by Prof. Ken Ono on a need for a sense of wonder, human aspects and critical thinking in K-12 mathematical learning....https://t.co/cVv6ITsu86

#AI #mathematics #learning

[Reals and Rationals @realsrationals]

Reals and Rationals wishes the math learning community a very Merry Christmas and a brilliant New Year! 🎄🪄

​Whether you’re solving complex equations or simply enjoying the geometry of a snowflake, we hope you take this time to recharge. ❄️

​Enjoy the cozy family vibes, a warm drink, and catching up on a good book. 📚☕

​Happy holidays and see you in the New Year! ⛷️

​#Mathematics #STEM #MathLearning #HappyHolidays #RealsAndRationals #Education

[Reals and Rationals @realsrationals]

Attempting a solution to a problem without the guidance of theory is like attempting to drive to a new place without a map or GPS.

Theory gives you a direction to make an optimal attempt. Also, just like a map tells you when a long scenic drive is about to hit a dead end, theory sets boundaries on what is actually achievable using a certain approach.

Leonardo Da Vinci drove this point home perfectly:

"Science is the captain, and practice the soldiers."

An excellent illustration of theory setting boundaries is the famous "Seven Bridges of Königsberg" problem.

The Dead End at Königsberg

The problem originated from a real physical situation in the city of Königsberg (Prussia), where landmasses were connected by seven bridges. The goal was simple: Walk across each bridge once and only once, returning to the start.

Walking attempts by people (the "Soldiers" practicing) yielded no solution.

Enter the Captain

The problem was put to rest by the mathematician Leonhard Euler. He argued that the actual geography (distances, shapes, scenic routes, etc.) didn't matter. Only the connections mattered.

Euler abstracted the city into points (land) and edges (bridges). By doing so, he proved mathematically that no solution exists.

The boundary that Euler's theory set was that of impossibility. No walk could ever be successful. The theory saved people from literally going around in circles.

Euler’s solution birthed Graph Theory and foreshadowed Topology.

Your Turn: What inspiring examples from mathematics come to your mind where theory saved mankind loads of labor?

#Mathematics #GraphTheory #Euler #DaVinci #Theory #Engineering

[Reals and Rationals @realsrationals]

Funny..

Coincidentally, we derived the area of a circle using integrals in one of our Calculus sections just this week..👇

[BURKOV @burkov]

A math professor noticed his kitchen sink at home was leaking.

He called a plumber.

The plumber came the next day, tightened a couple of nuts, and the sink worked perfectly again. The professor was delighted. But when, a minute later, the plumber handed him the bill, he was shocked.

“This is a third of my monthly salary!”
“Yeah, I get it…” said the plumber. “Why don’t you come work for our company as a plumber? You’ll make three times more than you do as a professor. Just remember: when you apply, say you only finished seventh grade. They don’t like hiring educated people.”

So the professor got a job as a plumber, and his life really did improve. All he had to do was tighten a nut here and there every so often, and his salary was much higher.

One day, the management of the plumbing company decided that every plumber had to attend evening classes to finish eighth grade. So our professor had to go too.

By chance, the very first class was math.

The evening school teacher, wanting to check what the students knew, asked for the formula for the area of a circle.

They called the professor up to the board, and he suddenly realized he’d forgotten it. He started frantically reasoning it out, covering the board with integrals, differentials, and all sorts of fancy formulas to re-derive the result. In the end, he got:
S = –π r²

He didn’t like the minus sign, so he started again.
Again he got a minus. No matter what he did, it kept coming out negative.

He cast a panicked look at the class, and all the plumbers were whispering:

“Swap the limits of integration!”

[Reals and Rationals @realsrationals]

Happy Halloween!

How do you make a classic pumpkin pie? With a classic Halloween joke..

What do you get if you divide the circumference of a jack-o-lantern by its diameter? A Pumpkin Pi (π)

[Reals and Rationals @realsrationals]

This week is the Nobel prize week, where the 2025 prizes will be announced.

The following is on a cute birthday tradition that mathematician and Nobel laureate Sir Roger Penrose has maintained ever since he turned 50. He is now 94.

[Oxford Mathematics @OxUniMaths]

Will you still need me, will you still feed me, when I'm 94?

Happy Birthday Roger.

[Reals and Rationals @realsrationals]

Today (3/14) is the International Day of Mathematics or Pi day. This is so because 3.14 is the familiar approximation of the math constant pi.

Here is a cute cartoon on pi:

[Reals and Rationals @realsrationals]

A nice article about the journey of the mathematical prodigy Ramanujan, and how his imaginative math ideas mysteriously appear in various branches of mathematics even today, about 100 years after he stated them.

https://t.co/5zjWoqOSpG

[Reals and Rationals @realsrationals]

This is an amazing piece of news. It is unconventional, but interesting to see that the pioneers from the fields of Computer Science and Mathematics were awarded Nobel prize in Physics.

[The Nobel Prize @NobelPrize]

BREAKING NEWS
The Royal Swedish Academy of Sciences has decided to award the 2024 #NobelPrize in Physics to John J. Hopfield and Geoffrey E. Hinton “for foundational discoveries and inventions that enable machine learning with artificial neural networks.”

[Reals and Rationals @realsrationals]

@divbyzero 😄 What was the "theorem" that you were after today?

[vanessa vakharia @TheMathGuru]

We don’t expect kids to WANT to learn how to play music when they’ve never heard an actual SONG before … so why do we expect them to WANT to learn math when they’ve never been exposed to ACTUAL MATHEMATICS?! Major mic drop moment for @mathyawp wow 🫳🎤

[Melanie Mitchell @MelMitchell1]

Stellar lineup of speakers, free to all on Zoom!

Organized by Stevan Harnad, U. Quebec

Large Language Models : Science and Stakes (June 3-14, 2024)

https://t.co/v5ZjzBBtRP

[Reals and Rationals @realsrationals]

@PoShenLoh @wtgowers This launch comes on the heels of Google DeepMind's recent announcement of AlphaGeometry, an AI system that was able to solve Math-Olympiad-level geometry problems. See: https://t.co/Ej5o6rV41U

[Reals and Rationals @realsrationals]

AI Math Olympiad prize (AIMO) launched to foster collaborative AI development for mathematical problem-solving and research...

https://t.co/F7ZUr7m1vR

Advisory committee has big names such as Terrence Tao, @PoShenLoh, @wtgowers.

[Reals and Rationals @realsrationals]

How do you make pumpkin pie?

You divide its circumference by its diameter.

[PUNS @ThePunnyWorld]

What do you get if you divide the circumference of a pumpkin by its diameter?

Pumpkin Pi.

[Reals and Rationals @realsrationals]

@Ananyo Frankly, for all the attention, the movie was disappointing. It was more political (covering obscure details of why Oppenheimer's security clearance was revoked) than biographical (covering the scientific journeys of him or his colleagues like Neumann, Feynman and Ulam).

[Reals and Rationals @realsrationals]

@ThebestFigen 112

Going in sequence after 5+7 = 12*5 = 60, we should have 6+8 = 14*6 = 84. Then, comes 7+9 = 16*7 = 112.

Of course, I am making the assumption that 7+9 = 9+7 (commutative property like with the usual addition).

[Reals and Rationals @realsrationals]

😀

[PUNS @ThePunnyWorld]

I never remember what people tell me at New Year's parties

It goes in one year and out the other.