Consistency by Ciro Santilli 40 Updated 2025-07-16
A set of axioms is consistent if they don't lead to any contradictions.
When a set of axioms is not consistent, false can be proven, and then everything is true, making the set of axioms useless.
Wikipedia by Ciro Santilli 40 Updated 2025-07-16
The most important page of Wikipedia is undoubtedly: en.wikipedia.org/wiki/Wikipedia:Reliable_sources/Perennial_sources which lists the accepted and non accepted sources. Basically, the decision of what is true in this world.
Wikipedia is incredibly picky about copyright. E.g.: en.wikipedia.org/wiki/Wikipedia:Deletion_of_all_fair_use_images_of_living_people because "such portrait could be created". Yes, with a time machine, no problem! This does more harm than good... excessive!
Citing in Wikipedia is painful. Partly because of they have a billion different templates that you have to navigate. They should really have a system where you can easily reuse existing sources across articles! Section "How to use a single source multiple times in a Wikipedia article?"
Video 1.
What Happened To Wikipedia's Founders?
Source.
Video 2.
Inside the Wikimedia Foundation offices by Wikimedia Foundation (2008)
Source.
dumps.wikimedia.org/enwiki/latest/enwiki-latest-category.sql.gz contains a list of categories. It only contains the categories and some counts, but it doesn't contain the subcategories and pages under each category, so it is a bit pointless.
The SQL first defines the table:
CREATE TABLE `category` (
  `cat_id` int(10) unsigned NOT NULL AUTO_INCREMENT,
  `cat_title` varbinary(255) NOT NULL DEFAULT '',
  `cat_pages` int(11) NOT NULL DEFAULT 0,
  `cat_subcats` int(11) NOT NULL DEFAULT 0,
  `cat_files` int(11) NOT NULL DEFAULT 0,
  PRIMARY KEY (`cat_id`),
  UNIQUE KEY `cat_title` (`cat_title`),
  KEY `cat_pages` (`cat_pages`)
) ENGINE=InnoDB AUTO_INCREMENT=249228235 DEFAULT CHARSET=binary ROW_FORMAT=COMPRESSED;
followed by a few humongous inserts:
INSERT INTO `category` VALUES (2,'Unprintworthy_redirects',1597224,20,0),(3,'Computer_storage_devices',88,11,0)
which we can see at: en.wikipedia.org/wiki/Category:Computer_storage_devices
Se see that en.wikipedia.org/wiki/Category:Computer_storage_devices_by_company
so it contains only categories.
We can check this with:
sed -s 's/),/\n/g' enwiki-latest-category.sql | grep Computer_storage_devices
and it shows:
(3,'Computer_storage_devices',88,11,0
(521773,'Computer_storage_devices_by_company',6,6,0
There doesn't seem to be any interlink between the categories, only page and subcategory counts therefore.
The basic intuition for this is to start from the origin and make small changes to the function based on its known derivative at the origin.
More precisely, we know that for any base b, exponentiation satisfies:
  • .
  • .
And we also know that for in particular that we satisfy the exponential function differential equation and so:
One interesting fact is that the only thing we use from the exponential function differential equation is the value around , which is quite little information! This idea is basically what is behind the importance of the ralationship between Lie group-Lie algebra correspondence via the exponential map. In the more general settings of groups and manifolds, restricting ourselves to be near the origin is a huge advantage.
Now suppose that we want to calculate . The idea is to start from and then then to use the first order of the Taylor series to extend the known value of to .
E.g., if we split into 2 parts, we know that:
or in three parts:
so we can just use arbitrarily many parts that are arbitrarily close to :
and more generally for any we have:
Let's see what happens with the Taylor series. We have near in little-o notation:
Therefore, for , which is near for any fixed :
and therefore:
which is basically the formula tha we wanted. We just have to convince ourselves that at , the disappears, i.e.:
To do that, let's multiply by itself once:
and multiplying a third time:
TODO conclude.
Algebraic number field by Ciro Santilli 40 Updated 2025-07-16
The set of all algebraic numbers forms a field.
This field contains all of the rational numbers, but it is a quadratically closed field.
Like the rationals, this field also has the same cardinality as the natural numbers, because we can specify and enumerate each of its members by a fixed number of integers from the polynomial equation that defines them. So it is a bit like the rationals, but we use potentially arbitrary numbers of integers to specify each number (polynomial coefficients + index of which root we are talking about) instead of just always two as for the rationals.
Each algebraic number also has a degree associated to it, i.e. the degree of the polynomial used to define it.
E notation by Ciro Santilli 40 Updated 2025-07-16
What do you prefer, 1 \times 10^{10} or 1E10.
Agilent Technologies by Ciro Santilli 40 Updated 2025-07-16
In a way, Agilent represents the most grassroots electronics parts of HP from before they became overly invested in laptops and fell.
They spun out the electronics part as Keysight in 2014, becoming life science only.

Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
We have two killer features:
  1. topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
    Articles of different users are sorted by upvote within each article page. This feature is a bit like:
    • a Wikipedia where each user can have their own version of each article
    • a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
    This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
    Figure 1.
    Screenshot of the "Derivative" topic page
    . View it live at: ourbigbook.com/go/topic/derivative
  2. local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
    This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
    Figure 2.
    You can publish local OurBigBook lightweight markup files to either https://OurBigBook.com or as a static website
    .
    Figure 3.
    Visual Studio Code extension installation
    .
    Figure 4.
    Visual Studio Code extension tree navigation
    .
    Figure 5.
    Web editor
    . You can also edit articles on the Web editor without installing anything locally.
    Video 3.
    Edit locally and publish demo
    . Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
    Video 4.
    OurBigBook Visual Studio Code extension editing and navigation demo
    . Source.
  3. https://raw.githubusercontent.com/ourbigbook/ourbigbook-media/master/feature/x/hilbert-space-arrow.png
  4. Infinitely deep tables of contents:
    Figure 6.
    Dynamic article tree with infinitely deep table of contents
    .
    Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact