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OpenCiv3: Open-source, cross-platform reimagining of Civilization III

https://openciv3.org/
568•klaussilveira•10h ago•160 comments

The Waymo World Model

https://waymo.com/blog/2026/02/the-waymo-world-model-a-new-frontier-for-autonomous-driving-simula...
885•xnx•16h ago•538 comments

How we made geo joins 400× faster with H3 indexes

https://floedb.ai/blog/how-we-made-geo-joins-400-faster-with-h3-indexes
89•matheusalmeida•1d ago•20 comments

What Is Ruliology?

https://writings.stephenwolfram.com/2026/01/what-is-ruliology/
16•helloplanets•4d ago•8 comments

Unseen Footage of Atari Battlezone Arcade Cabinet Production

https://arcadeblogger.com/2026/02/02/unseen-footage-of-atari-battlezone-cabinet-production/
16•videotopia•3d ago•0 comments

Show HN: Look Ma, No Linux: Shell, App Installer, Vi, Cc on ESP32-S3 / BreezyBox

https://github.com/valdanylchuk/breezydemo
195•isitcontent•10h ago•24 comments

Monty: A minimal, secure Python interpreter written in Rust for use by AI

https://github.com/pydantic/monty
197•dmpetrov•11h ago•88 comments

Show HN: I spent 4 years building a UI design tool with only the features I use

https://vecti.com
305•vecti•13h ago•136 comments

Microsoft open-sources LiteBox, a security-focused library OS

https://github.com/microsoft/litebox
352•aktau•17h ago•173 comments

Sheldon Brown's Bicycle Technical Info

https://www.sheldonbrown.com/
348•ostacke•16h ago•90 comments

Delimited Continuations vs. Lwt for Threads

https://mirageos.org/blog/delimcc-vs-lwt
20•romes•4d ago•2 comments

Hackers (1995) Animated Experience

https://hackers-1995.vercel.app/
450•todsacerdoti•18h ago•228 comments

Dark Alley Mathematics

https://blog.szczepan.org/blog/three-points/
78•quibono•4d ago•16 comments

PC Floppy Copy Protection: Vault Prolok

https://martypc.blogspot.com/2024/09/pc-floppy-copy-protection-vault-prolok.html
50•kmm•4d ago•3 comments

Show HN: If you lose your memory, how to regain access to your computer?

https://eljojo.github.io/rememory/
248•eljojo•13h ago•150 comments

An Update on Heroku

https://www.heroku.com/blog/an-update-on-heroku/
384•lstoll•17h ago•260 comments

Zlob.h 100% POSIX and glibc compatible globbing lib that is faste and better

https://github.com/dmtrKovalenko/zlob
11•neogoose•3h ago•6 comments

How to effectively write quality code with AI

https://heidenstedt.org/posts/2026/how-to-effectively-write-quality-code-with-ai/
228•i5heu•13h ago•173 comments

Show HN: R3forth, a ColorForth-inspired language with a tiny VM

https://github.com/phreda4/r3
66•phreda4•10h ago•11 comments

Why I Joined OpenAI

https://www.brendangregg.com/blog/2026-02-07/why-i-joined-openai.html
113•SerCe•6h ago•90 comments

I spent 5 years in DevOps – Solutions engineering gave me what I was missing

https://infisical.com/blog/devops-to-solutions-engineering
134•vmatsiiako•15h ago•59 comments

Introducing the Developer Knowledge API and MCP Server

https://developers.googleblog.com/introducing-the-developer-knowledge-api-and-mcp-server/
42•gfortaine•8h ago•12 comments

Female Asian Elephant Calf Born at the Smithsonian National Zoo

https://www.si.edu/newsdesk/releases/female-asian-elephant-calf-born-smithsonians-national-zoo-an...
23•gmays•5h ago•4 comments

Understanding Neural Network, Visually

https://visualrambling.space/neural-network/
263•surprisetalk•3d ago•35 comments

I now assume that all ads on Apple news are scams

https://kirkville.com/i-now-assume-that-all-ads-on-apple-news-are-scams/
1038•cdrnsf•20h ago•429 comments

Learning from context is harder than we thought

https://hy.tencent.com/research/100025?langVersion=en
165•limoce•3d ago•87 comments

FORTH? Really!?

https://rescrv.net/w/2026/02/06/associative
59•rescrv•18h ago•22 comments

Show HN: ARM64 Android Dev Kit

https://github.com/denuoweb/ARM64-ADK
14•denuoweb•1d ago•2 comments

Show HN: Smooth CLI – Token-efficient browser for AI agents

https://docs.smooth.sh/cli/overview
86•antves•1d ago•63 comments

Evaluating and mitigating the growing risk of LLM-discovered 0-days

https://red.anthropic.com/2026/zero-days/
47•lebovic•1d ago•14 comments
Open in hackernews

Product of Additive Inverses

https://susam.net/product-of-additive-inverses.html
17•blenderob•7mo ago

Comments

JadeNB•7mo ago
This is a formal justification, from the ring axioms, of the formula (−a)(−b) = ab. As the article mentions, this is often phrased as "the product of two negatives is positive," but, of course, the presence of a minus sign in front of a variable does not indicate a negative number (for example, if a = −3, then −a is positive); and the formula makes sense even in a ring with no notion of positive and negative numbers.
empath75•7mo ago
A simple example of how this is true _even if you don't have negative numbers_:

Let's use mod 5 arithmetic. You have 5 elements in the ring -- 0,1,2,3,4

The additive inverses are as follows:

  1 + 4 = 0
  2 + 3 = 0
Which is to say that 1 is the additive inverse of 4 and 2 is the additive inverse of 3, and vice versa. 0 is the identity, of course.

So what happens if you multiply 2 * -3 (2 times the additive inverse of 3).

The additive inverse of 3 is just 2, so the answer is 2 * -3 = 2 * 2 = 4.

The other way to calculate it is to find the additive inverse of the product:

2 * -3 = -(2 * 3) = -(1) which is the additive inverse of 1: 4 again.

CurtMonash•7mo ago
ab and (-a)(-b) can each be quickly proved to be the additive inverse of (-a)b. So they equal each other. No intermediate theorems are really needed.
susam•7mo ago
I am not sure how you can prove this more "quickly". Trying to do it any more quickly involves claiming some result (no matter how trivial) that is not directly present in the ring axioms. But the whole point of this post is to derive everything strictly from first principles, using nothing beyond the ring axioms themselves.

Here is your argument elaborated step by step.

STEP 1: First we want to show that ab is the additive inverse of (-a)b. This is Theorem 3 of the post.

STEP 2: Next we want to show that (-a)(-b) is the additive inverse of (-a)b. This follows similarly to the proof of Theorem 3: (-a)(-b) + (-a)(b) = (-a)(-b + b) = (-a)(0) and (-a)(0) = 0 by Theorem 2 of the post.

But nothing in the ring axioms directly says that the above results mean ab and (-a)(-b) must be equal. How do we know for sure that ab and (-a)(-b) are not two distinct additive inverses of (-a)b?

THEOREM 5: We now prove the uniqueness of additive inverse of an element from the ring axioms. Let b and c both be additive inverses of a. Therefore b = b + 0 = b + (a + c) = (b + a) + c = 0 + c = c.

Now from Steps 1 and 2, and Theorem 5, it follows that ab = (-a)(-b).

So what did we save in terms of intermediate theorems? Nothing! We no longer need Theorem 1 (inverse of inverse) of the post. But now we introduced Theorem 5 (uniqueness of additive inverse). We have exactly the same number of intermediate theorems with your approach.