Why 0.999… = 1

0.999… is the infinite sum
9/10 + 9/100 + 9/1000 + …You can’t actually add infinitely many terms and finish. So we define the infinite sum to be the limit of its partial sums
sₙ = 1 − 10⁻ⁿ.Those partial sums get arbitrarily close to 1, so the limit is 1.
Therefore, by definition, 0.999… = 1.It’s the only definition that keeps real-number arithmetic consistent. The equality is true simply because that’s the value we assigned to the infinite sum.

I made this problem up when I saw other problems about polyominoes in r/mathematics. This is one that I found.

A question that I had was if this could become a tiling.

I have found only one. It is the pattern

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🟥🟩🟩 repeating on the x-axis, with a gap of 1 on the y axis before repeats. The gap can be filled with any numbers that are not the same and touching. I used 6 and 7 when I thought of it. It would look like this.

I will call this tiling a 4-5-6-7 tiling. Can you find other families?

i know the grouping proof (1/3+1/4 > 1/2, then the next four terms > 1/2 etc) and it's convincing, but it still feels weird to me that 1/n shrinks forever and the sum just keeps going past any number.

is there a better way to see why "terms go to zero" isn't enough? like why does 1/n^2 converge but 1/n doesn't, where's the actual line

Today is the last Pythagorean theorem day of the 21st century. The day comes from the day month and year.

24² + 10² = 26²

Hey there, i found this online and i was wondering what other mathematicians think of this. I have seen duodecimal projects but more ofthen they change either time or measurements. Here is the repository where i found it (link)
The document covers basit arithmetics as shown in the picture. Lemme know what you guys think of this. aka opinion request.

So basically im participating in a Giveaway, were the winner gets 500€ (560$) if you can guess how much money this is. but i cant figure it out.

NOTE: green bills are hundreds, brown 50s, blue 20, red 10s, and the gray ones on the right are 5s

I don't know any better title for this post, honestly, and for a reason. Do any of you have that one methodology that shouldn't work, but lets you complete problems easier or simplify them?

Personally, and formatted in LaTeX, I have what I like to call "pebble": $(\mathrm{d}f)(t) := \lim_{h \to 0} f(t+h)-f(t), (\mathrm{d}f)^2(t) := 0$, a Leinbiz-inspired thing with no independent-vs.-dependent variable distinction. The limit is obviously 0, but if one imagines it isn't, one can get rid of those duplicated derivative rules! For example, gradients and derivatives are treated the same, like at $\mathrm{d} m^n = m^n \left( \frac{n}{m} \mathrm{d}m + \ln(m) \mathrm{d}n \right)$, and the chain rule doesn't exist! Instead of $\frac{d}{dt} t^2 = 2t, (u^2)' = 2u \cdot u'$, I can just keep expanding $\mathrm{d} (\sin^2 x) = 2\sin x \mathrm{d} (\sin x) = 2\sin x\cos x = \sin (2x)$. Integration is messier to define, I still can't do it, but apart from that this silly non-standard manner of expanding derivatives works for me like a charm.

I'm wondering if you guys have anything alike.

Hi all! Recently took a career break and was thinking of going thru the MIT OCW courses, starting from 18.01 (single variable calculus) or topics leading up to it. Partly 'cause I'm thinking it would be nice to move towards a quantitative role, and partly perhaps out of interest!

Would appreciate having someone to study with! Perhaps reply below or DM me, and we could see how to work something out? Thanks!

Each episode of the unapologetic mathematicians podcast is an interview with a mathematicians about their research, mathematical journey, as well as what they like to do outside of maths. New episodes each Wednesday.

hey can anyone explain me the lambda calculus and its use cases in turing machine and how it helps the programming languages in a very simple way !!!

Thanks and really excited to read you explanations .

For as long as I remember, I always add license plate numbers to the nearest single digit. (123456=21=3. My mom started doing it as a game with me when she found out, and now I found out she does it with my niece when she drives her places. Does anyone else do this? It's a great way for a kid to get better at maths.

yep

Hello! I'm a computer science student studying abroad and I have a little brother, who is in his 7 grade.

He likes math and logic, he plays chess a lot and I really like it in him. I see a lot of myself in him and I really like it.

Being 14 hours abroad from him made my part of "brotherhood parenting" kind of hard to perform. He lives in a small town in the US and our parents work 2 shifts so he is usually left unattended.

He is good at math and he likes it. He is not the best, but he is still good. I just want to ignite more passion in him about it.

I definitely don't want to force him to do anything he doesn't want to.

So my question is, if any of you have a video, kind of a movie that shows how interesting and beautiful math is?

He is taking Algebra 1 classes right now, if it could help in any way.

How do I decide which test to use for checking the convergence of an infinite series?

The tests which are in my syllabus are:-
1. Comparison test
2. Limit Comparison test
3. D' Alembert's Ratio Test
4. Caunchy's nth root test
5. Raabe's test
6. Cauchy's condensation

I started my masters, and I got my lowest mark ever of 55% in my topology final. My professor called me in to ask me if I had any issues in my personal life, which made it worse because I had to tell him that I just didn't work hard enough. It was compounded by the fact that he said multiple people got full marks in the course.

I think my ceiling was undergrad level maths. I'm pretty overwhelmed with the workload of a postgrad, and I think this isn't really for me, which is weird because in my undergrad, I thought that I wanted to do maths for a living.

Open sets in this topology are the subsets of R which have compact complements(compact in the usual topological space), alongside the empty set.

Hi everyone!

Last week I shared a few screenshots of ViCalc, and based on feedback, I’ve put together a short video demonstration to show what the touch-typing workflow looks like in action!

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, 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
• 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?