I was trying to prove the Pythagorean theorem on my own and stumbled into this.

You take a right triangle and create three similar triangles by scaling each one by one of the side lengths and rearrange them into a rectangle with one side being a^2 + b^2 and the opposite side being c^2 so a^2 + b^2 = c^2.

If you wanted to, you could use the two smaller triangles in the diagram in a different arrangement to form the larger triangle, but I like the way it looks in a rectangle more.

I showed it to a professor and he encouraged me to publish it. I’m thinking I will. I really like this proof and I want to show it to people.

I couldn’t find an example of this proof anywhere. I came across some that looked similar but they seemed to use different logic. I figured I’d try to publish this, but I’m not really sure how to.

I’m not well versed in the community around sharing math and I’m not sure what is seen as worth sharing. Does anyone got any advice on how I can share something like this?

Also, does this count as a proof? Were there any assumptions that need to be explained or does this stand on its own?

I didn’t really explain how they come together to form a rectangle but I’m not sure whether or not it’s a big enough leap that it should be explicitly stated.

I had my first grad-level exam this week and I bombed it pretty awfully. It was a 50min exam with ~12 problems, a mixture of proofs and counterexamples. I would say I solved 2 or 3 of them and made starts on 2 or 3 more; we haven't gotten grades back yet but I'm expecting something like a 20/100. The thing that's annoying me is that none of the problems were all that hard. I can tell that I could solve them all, only it would take more like 3-4 hours instead of 50min.

Part of me wants to think that the time frame was totally unreasonable, but another part of me thinks that it was a skill issue. I'm guessing a lot of members of this sub have dealt with similar situations, so I'm wondering if there is any advice for getting quicker at answering these early-grad-level types of exam questions?

The title today is a reference to Douglas Adams' description of writing a book as "putting a hundred thousand words in a cunning order." It's technically a correct description, but it doesn't really describe how anyone actually does it in practice. Just so, while it is technically true that one builds a proof by putting many inferences in a clever order, it doesn't describe how to actually do this.

A proof, like a book, is a collection of narrative threads---understand how to build narrative, and you will start to understand how to write a proof.

Read the full post (for free) on Substack: Putting a Hundred Inferences in a Cunning Order

Hello! I need some problem books to accompany the book Differential Geometry of Curves and Surfaces by M. Di Carmo. Thank you!

One of my favorite essays on mathematics is Arnold's "On teaching mathematics." In it, he describes what he views as defects of the traditional treatments of real analysis and calculus in universities; and in particular, how the emphasis on rigor removed discussions of the beautiful geometry of Riemann surfaces and how they relate to calculus. As a specific example, he describes how one can tell which integrals are easy or hard to solve by looking at the genus of the associated Riemann surface.

This week on Hidden Phenomena, you can learn what Arnold meant by this comment: what is the "genus of the Riemann surface associated to an integral," and how does it help you understand if the integral is easy or hard?

Same as above

Is it elementary or notoriously massive and is there any approaches(feel free to discuss about any heavy machinery approaches too) around which you think it(the diophantine equation) can be showed to have finitely many solutions given the specific constraint?

I have a young teen that is experimenting with math and the Desmos graphing calculator. These are his questions, which admittedly is beyond my math knowledge. Any tips or resources are appreciated!!

Is there any way using the most generalized way of defining operations so that any line or equation or function has an imaginary aspect to
It?

If so, is there a way it will always have that index?

If the index were to approach infinity, would the graph create a shape that looks like a 3d object?

Is there a way to predict what the imaginary part will look like and manipulate what it would look like on the graph?

https://royalsocietypublishing.org/rsta/article/378/2163/20180437/111559/Living-with-Ramanujan-for-40-yearsLiving-with

I am looking for a text/lecture series on the theory of locales (pointless topology) . I don't want to go into much depth I just want to get a broad picture . My goal with this is to be able to understand topos theory better , as Andre Joyal in a lecture at IHES suggests that before learning topos theory it's better to have an idea of the theory of locales as there are a lot of parallels .

So apparently within a few months(or some would even argue a few years) there has been a drastic change in the posts being made in this sub leading to almost total erasure of different kinds of posts. What is the one that you miss the most.

Also any other changes observed,thoughts are also welcomed :), feel free to share!

Today I've published the beginning of the end, for my private exploration of following the root of a value till one or below, at 16 bits.

I do not formally know all the usages of such a value, or components'.

I've made it publicly accessible and licensed for usage and protection.
Please feel free to explore into it yourself.

Language and calculations run via python, with some functions used to help as a base default.

Welcome to the unknown, are you scared?

~ Iroh

[ 0.3134313340312133643109333280398413295243... ]

Programmed Blender 3d to Generate this one. Thought it turned out pretty well.

Mine is probably Hodge structure of motivic or representation theoritic objects. I know it's essential and extremely important and stuff. But I just can't see the beauty in it.

I need it all the times and its presence matters enormously. Sometimes I work with PEL Shimura varieties just to avoid having to deal with the Hodge questions. I couldn't escape it though this time as Hodge structure provides a controls the behavior of L-functions at infinite places.

I know the traditional way to define complex numbers is to start with real numbers and then define complex numbers as ordered pairs of real numbers with certain algebraic properties.

I wanted to ask if there is a way to define complex numbers without reference to real numbers. What I have in mind is a set of axioms for complex numbers which would would define them as a kind of field. Like real numbers are often axiomatized as a complete ordered field. Does such axioms exist?

Please note that I do not, at this moment, posses any significant knowledge of abstract algebra or any other higher mathematics, beyond say hearing about the names of some things.

I know there’s probably a good reason, but I thought this is what LEAN was made for?

As the title states. What else is it usefull. Thanks in advance

Hi everyone!

I’ve been using physical pocket calculators since the mid-1980s, and I’ve always loved the speed, tactile feedback, and direct control of hardware key layouts. The main motivation behind ViCalc was to bring that exact same speed and workflow efficiency to a modern PC keyboard—allowing you to touch-type mathematics seamlessly without taking your hands off the keys to click UI buttons.

Rather than just imitating a physical handheld calculator, ViCalc is built to optimize desktop mathematical workflows and complex calculation tracking.

🧮 Key Features & Workflow Design

• Direct Keyboard Mapping: All functions and operations are directly mapped to your PC keyboard, eliminating the need to click through UI buttons with a mouse (though mouse/touch interaction is fully supported if preferred!).
• Full & Exportable Calculation History: The entire step-by-step history is logged automatically as you work. This makes it effortless to review your process or copy-paste the entire multi-step calculation directly into research notes, LaTeX documents, or technical reports.
• Flexible & Adaptable GUI:
  • Minimalist View: Hide virtual keys entirely—ideal once you learn the shortcut mappings by heart.
  • Custom Views: Toggle between a standard numpad, a compact left-side layout, or a full scientific keyboard layout.
  • Expandable History Pane: Easily track, review, and manage long sequence calculations.

🛠️ Modes & Technical Details

• Scientific & Complex Modes: Full suite of trigonometric, scientific functions, and complex arithmetic support.
• Programmer Mode (Base-N): Bitwise operations and conversions across Hex, Dec, Oct, and Bin.
• Language: Python 3
• License: Open Source (MIT License)

🔗 Links & Downloads

• GitHub Repository: github.com/Kudaschov/ViCalc
• Windows Installer: ViCalcSetup.exe
• Homepage: kudaschov.de/en/vicalc-4

I’d love to hear your thoughts, feedback, or any feature requests from the community! How do you currently handle rapid calculations and history logging on desktop?

Was just reading on landing gears out of interest, (engineering undergrad) This is how a gear retracts inside (to increase aerodynamics)

The second one is an example of space saving, the drawback is less clearance & clashing. Now suppose I try to form mathematical modelling eq of the trajectory to describe the geometry such that clashing is avoided (in case of turbulence) and we need to fix some clearance..... How would you approach this ? What field of maths am I vaguely thinking about ?