text:
A-level Mathematics/OCR/C1/Indices and Surds

The indices in this chapter heading refers to the power to which a number is raised. Thus $$x ^ 2$$ is a number with an index of 2. This is never normally said as people prefer the phrase "x to the power of 2". Likewise, surds is a heading that is not often used. Occasionally you wil be asked to give an answer in surd form; this implies that you should give the answer in terms of constants, trig functions and square roots instead of working out an imprecise decimal approximation.

Indices
As you should know from GCSE, $$x^3$$ is a shortening of $$x \times x \times x$$. In the same way, any number to the power of n is that number multiplied by itself n - 1 times. To describe more detail, in the expression $$x ^ 3$$, the $$x$$ is referred to as the base, and the 3 as the exponent.

By always reducing the expression back to its basic parts it is possible to come to conclusions about how to treat numbers raised to powers in algebra.

Multiplication
When you multiply indices you add the exponents together.

As you can see $$x^3 \times x^2 = x^{3+2} = x^5$$ .(law1)

Division
Division is, as expected, the opposite to multiplication. When you divide indices you subtract the exponents from each other.

$$ \frac{x^4}{x^2} = \frac{x \times x \times x \times x}{x \times x} = \frac{\not \! x \times \not \! x \times x\times x}{\not \! x \times \not \! x} = x \times x = x^2$$

Again this shows that $$\frac{x^4}{x^2} = x^{4-2} = x^2$$

Negative powers
Having done the above division, we begin to ask ourselves what happens when we have a fraction that has a higher exponent on the bottom than the top.

Using the trusted method of showing each x separately we obtain:

$$\frac{x^2}{x^3} = \frac{\not \! x \times \not \! x}{\not \! x \times \not \! x \times x} = \frac{1}{x}$$

Using the just demonstrated method of subtracting the exponents:

$$\frac{x^2}{x^3} = x^{2-3} = x^{-1} $$

This implies that x to the power of a negative number is one divided by x to the power of that number. The specific case of $$\frac{1}{x} \; \left( x^{-1} \right) $$ is referred to as the reciprocal of x or more often as the inverse of x.

Base raised to two powers
When you have a base raised to two powers you multiply the powers.

$$\left(x^2 \right)^3 = \left(x \times x \right) \times \left(x \times x\right) \times \left(x \times x\right)=x \times x \times x \times x \times x \times x = x^6$$.

As you can see that $$\left(x^2\right)^3 = x^{2 \times 3} = x^6$$.

Multiple bases to the same power
When you have two bases to the same power, you can raise both bases to the same power and multiply them.

For example: $$\left(xy \right)^3 = xy \times xy \times xy = x \times y \times x \times y \times x \times y = x \times x \times x \times y \times y \times y$$ which is the same as $$x^3y^3\,$$. Here is an example with numbers: $$\left(2 \times 5\right)^2 = \left(10\right)^2 = 100 = 4 \times 25 = 2^2 \times 5^2$$. There is a similar situation with division: $$\left(\frac{x}{y}\right)^2 = \frac{x}{y} \times \frac{x}{y} = \frac{x \times x}{y \times y} = \frac{x^2}{y^2}$$

Fractional powers
What if the power isn't even an integer? Suppose you wanted to find $$x^\frac{1}{2}$$, you could say that $${x^\frac{1}{2}} \times {x^\frac{1}{2}} = x^1$$ (by law 3, addition of powers) which means that $$x^\frac{1}{2}$$ must be $$\pm \sqrt x$$. However it is customary to only use the positive root and so $$x^\frac{1}{2}$$ is defined as $$\sqrt x$$. You can use a similar argument for other such fractions, for example $$\left(x^\frac{1}{3}\right)^3=x$$ so $$x^\frac{1}{3} = \sqrt[3]{x}$$. In cases when the numerator is not 1 we need to use other laws of indicies to prove the square definition, for example $$x^\frac{2}{3} = \left(x^2 \right)^\frac{1}{3}$$ (using law 3), and $$\left(x^2\right)^\frac{1}{3} = \sqrt[3]{x^2}$$ (using the definition above). It's useful to remember the general rule that $$x^\frac{a}{b} = \sqrt[b]{x^a}$$.

The 0 and 1 Power
You may well have realised that $${x^1} = x\,$$. You can prove this by $$\frac{x^n}{x^{n-1}}$$ which is clearly $$x$$ which is $$x^{n - \left (n - 1 \right )} = x^1 = x$$ by law 2. Also with $$x^0\,$$ we can prove that it is equivalent to 1, $$\frac{x^n}{x^{n}} = 1$$, but by law 2 it is also equivalent to $$x^{n-n} = x^0$$

The Laws of Indices
The rules that have been suggested above are known as the laws of indices and can be written as:

1. $$x^ax^b = x^{a+b}\,$$

2. $$\frac{x^a}{x^b} = x^{a-b}$$

3. $$x^{-n}=\frac{1}{x^n}$$

4. $$\left(x^a\right)^b = x^{ab}$$

5. $$\left(xy \right)^n = x^n y^n$$

6. $$\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$$

7. $$x^\frac{a}{b} = \sqrt[b]{x^a}$$

8. $$x^0 = 1\,$$

9. $$x^1 = x\,$$

10. $$ (x^m)^\frac{1}{m}= x\,$$

Surds
In mathematics, a Surd is an expression containing a root term, with an irrational value, that can not be expressed exactly – for example, $$\sqrt{3} = 1.732050808...$$.

Sometimes it is useful to work in surds, rather than using an approximate decimal value. Surds can be manipulated just like algebraic expressions and sometimes it may be possible to eliminate the surd (called rationalising the expression), which may have not been possible if you tried to work with the approximate value. When asked to give the exact value, approximate decimal answers will not do and you will have to manipulate surds in order to give a final answer in simplified surd form.

Simplification of surds
Because surds can be manipulated like algebraic expressions, you can easily multiply out the terms and add the like terms. However, there are also a few rules that will be useful when simplifying surds.

Basic rule of surds
Because $$\sqrt {x} \times \sqrt {x} = x$$, it is useful to know that it can be rearranged to give $$\sqrt {x} = \frac {x} {\sqrt {x}}$$ and $$\frac {1} {\sqrt {x}} = \frac {\sqrt {x}} {x}$$.

Surds as indices
Because $$\sqrt[n]{x} = x^\frac{1}{n}$$ the laws of indices also apply to any n-th root. The most frequently used instances of this are laws 5 and 6 with surds:

 * $$\left(xy \right)^n = x^n y^n$$ becomes $$\sqrt{xy} = \sqrt{x} \times \sqrt{y}$$
 * $$\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$$ becomes $$\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}}$$

The first of these points is often used to simplify a surd, for example $$\sqrt{200}=\sqrt{100 \times 2}=\sqrt{100} \times \sqrt{2}=10\sqrt{2}$$. In an exam, you will be expected to write all surds with the smallest possible number inside the surd (i.e. the number inside the root shouldn't have any square factors).

Note: This rule only applies to non-negative numbers, attempts to involve negative numbers might give absurd results; $$ -1 = \sqrt{-1}\cdot \sqrt{-1} = \sqrt{(-1)(-1)} =\sqrt{1} = 1 $$

Rationalising the denominator
Another technique to simplify expressions involving surds is to rationalise the denominator. This means removing a surd from a fraction. In the case of a fraction such as $$\frac{5}{\sqrt{3}}$$, both numerator and denominator can be multiplied by $$\sqrt{3}$$ to give $$\frac{5\sqrt{3}}{3}$$.

If the fraction is of the form$$\frac{a}{b+\sqrt{c}}$$ the strategy used in the previous paragraph will work if it is slightly modified. This time you should multiply the numerator and denominator by $${b-\sqrt{c}}$$. If you are familiar with the standard difference of two squares expansion you should already know what happens next:

$$\frac{a}{b+\sqrt{c}} \times \frac{b-\sqrt{c}}{b-\sqrt{c}} = \frac{ab-a\sqrt{c}}{b^2 + b\sqrt{c} - b\sqrt{c} - \sqrt{c}^2} = \frac{ab - a\sqrt{c}}{b^2-c} $$.

As you can see the denominator now does not contain any surds. For example: $$\frac{2}{\sqrt{3} - 1} \times \frac{\sqrt{3}+1}{\sqrt{3}+1} = \frac{2\left(\sqrt{3}+ 1\right)}{3-1} = \sqrt{3}+1 $$

Splitting up roots
A common mistake is to split $$\sqrt{x + y}$$ into $$\sqrt{x} + \sqrt{y}$$ or $$\left(x+y\right)^2$$ into $$x^2 + y^2$$, usually whilst moving it to the other side of the equals. Trying a few examples will quickly convince you that this is not possible:  $$\sqrt{64}\ \ne \sqrt{32} + \sqrt{32}$$ $$\sqrt{25}\ \ne \sqrt{12} + \sqrt{13}$$  And so on

What is the value of $$0^0$$?
The short answer is that for this course you don't need to know, and you can safely skip this section. If you're still interested then read on:

The question arises because $$x^0=1$$ for any $$x$$ and yet you would expect that $$0^y=0$$ for any $$y$$ as $$0 \times 0 \times 0 \dots = 0$$. It turns out using a value of 1 is quite useful (perhaps even necessary) in various parts of algebra, whereas making it zero doesn't help at all. Almost all mathematicians would therefore either say that $$0^0$$ is 1 or that it is undefined (that is, it can't be given a value). A more technical discussion can be found at http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0to0/


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Talk:How to Write a Program

21st December 2004 (Winter/Summer Solstice)
 * Three jewels:
 * Sequence
 * Selection
 * Repetition

Test cases
I have had a list of common test cases transwikied from Wikipedia because I thought it could be put to good use in this or a similar book. The list can be found at Transwiki:Common Test Cases. It needs a little cleanup, of course, but is in pretty good shape. Hopefully, someone for familiar with this book than me can put it to good use. Thanks --ThaddeusB (talk) 00:33, 6 June 2009 (UTC)

The Style Leaves a Lot to Be Desired - what should I do about it?
Hi all!

I learned about this book, from a link someone placed on the my Optimizing Code for Speed wikibook. Now I agree that reading a good introduction on how to write a program should come before you start optimizing a working code senselessly. (I should note that I haven't encouraged such unwise behaviour in my article, just possibly not stressed it enough.)

In any case, I feel the style of this wikibook and its tone leaves a lot to be desired. It's very unofficial, quite condescending not NPOVish (= Neutral Point Of View) enough (I admit that wikibooks need not be as NPOVish as the wikipedia and may be somewhat biased, but it's still an important partial factor). As a result requiring it as a pre-requisite to mine may leave a bad impression.

But naturally I should not complain if I don't volunteer to fix it myself, and I do. However, I'm not sure my modifications will be accepted by the originators of this wikibook, and so kindly ask for their permission, to fix the style and correct the manuscript. From my wiki-ing experience, I know that sometimes documents that I've started were taken to completely different directions that I've neither intended nor found desirable, which I didn't appreciate, due to a certain feeling of propriety. (Which I hope was not territoriality.). So I'm asking here whether it would be OK.

Shlomif (talk) 11:32, 10 November 2009 (UTC)

I find even the premise of the book a little skewed. According to the book title this is a book about learning how to program, but the contents assume familiarity with languages, so the topic is not how to use the languages to program, but about how to program in an industrial environment where you have Requirements Papers and such like already generated for you.

If we are going to discuss production programming, then shouldn't we cover more than just the basics such as refactoring, and perhaps cover disciplines like Top-Down, Extreme Programming, Agile Programming, etc. techniques? Should we accept the premise that language is not important to the book, or should we discuss the relative usefulness of a number of programming languages in order to give the programmer a background in modern languages other than the ones they learned at school, or should we discuss the use of Structured English, and Flow Charts as tools to organize the program. Perhaps a section on UML might be in order? How about a chapter on profiling and how to use it to determine where in the program you want to optimize?

Should we limit ourselves to just programming techniques or should we include Patterns, and links to the rich information sources that have been developed in relation to that discipline? I think that books that have to be used as pre-requisites for other books, should be held to a higher standard than this book seems to be.--Graeme E. Smith (talk) 15:09, 14 November 2009 (UTC)

 * Dear Shlomif,
 * I totally agree with practically everything you have said.
 * In particular, a good introduction on how to write a program should come before learning a bunch of optimization tricks.
 * I also agree that this particular wikibook is not yet a particularly good introduction.
 * Is there some other wikibook that would make a better introduction?

 * We have many Wikibooks about specific programming languages -- Python, Lua Functional Programming, etc.
 * I've been assuming, based on the title, that this wikibook "How to Write a Program" is intended to discuss the sorts of things that are important for every programmer to know, but are not specific to any one programming language -- and so get very little notice in those books about some particular programming language. Things like version control, automated testing, colored syntax highlighting, pair programming, handling bug reports, estimating time to fix a bug, estimating time to add a new feature, how to divvy up parts of a program among a bunch of programmers, etc.
 * Some teachers try to teach these things before getting to the language-specific stuff, but it seems that these things make more sense to students after they've learned to write a few small programs in some specific language.
 * shouldn't we cover more than just the basics such as refactoring, and perhaps cover disciplines like Top-Down, Extreme Programming, Agile Programming, etc. techniques? Yes, that is a great idea.
 * Should we ... discuss the relative usefulness of a number of programming languages...? While such a comparison would be really useful, I think this book should stick to non-language specific stuff -- perhaps the Computer Programming book or Programming Languages book is a better place for those comparisons.
 * How about a chapter on profiling Also an extremely useful technique, but -- perhaps the Optimizing Code for Speed book is a better place for that chapter.
 * I want everyone to feel free to make any and every kind of improvement to this book.
 * As long as you Wiki: AlmostNeverDeleteHumor.
 * --DavidCary (talk) 03:30, 20 November 2009 (UTC)

 * Well, I've started the rework of this book when I've ran into a little Browser-related Yak-shaving and now I'm back. However, in the meantime, I've concluded something else. I had written an essay titled "What Makes Software High-Quality?" (written in DocBook/XML and under CC-by - possibly wikibooking material) which contains four sections: an Introduction, a list of parameters of software quality, a list of ways to achieve software quality, and finally an evaluation of the quality of various Freecell solvers (the latter due to a quote that prompted the discussion in the first place in the introduction and my role as the primary author of a Freecell solver).
 * Now, while the section of the parameters of software quality was supposed to be as comprehensive as possible (and I would appreciate anything you believe is an omission), the other main section about ways of quality did not aim to be comprehensive, but rather give a taste of what people find useful. That's because it's
 * A huge subject. How to write quality software has been the subject of many books, and countless of articles, papers, essays, blog posts, mailing list posts and conversations.
 * A controversial subject. People don't seem to agree on what exactly is the right way to do things.
 * It doesn't appear to be absolutely necessary for one's success (or alternatively to ensure one's success). In the Joel Test, Joel Spolsky says that «Of course, these are not the only factors that determine success or failure: in particular, if you have a great software team working on a product that nobody wants, well, people aren't going to want it. And it's possible to imagine a team of "gunslingers" that doesn't do any of this stuff that still manages to produce incredible software that changes the world.». The plentyoffish.com creator taught himself .NET programming and became a millionaire a year later simply because he had the idea to create a completely-ad-supported web-site, and was the first to market it, but its code probably had left a lot to be desired. And on the other hand, thousands of well-managed projects, products, companies and teams fail each year, and many great web-sites (like my own Perl-Begin appear to be) still destined for obscurity.
 * Back to the topic, if we are going to write a "How to Write a Program" or "How to write Quality Software" or whatever you want to call it wikibook, then we'll have a huge and controversial task ahead of us. Do we cover what we think are the right measures for achieving quality? Do we cover all possible ways to achieve quality and discuss the various opinions about them? In which context do we discuss this: open-source software, commercial software, in-house software, embedded software, games; GUI software, command-line software, libraries and APIs, web-based software, terminal software, client/server; by freelancers, by businesses, out-sourcing, off-shore outsourcing, in-house development, for fun, for profit, etc. etc?
 * It's not an easy question, and I'm not sure we can start tackling it starting from "How to Write a Program" excuse for a wikibook. There are other much more commendable efforts in regards to FOSS development:
 * ESR's the "Cathedral and the Bazaar" and some other writings by him. (Open Publication License)
 * Karl Fogel's Producing Open Source Software - a comprehensive and good book about the topic (CC-by).
 * And naturally, there are many more books and resources about that for non-FOSS development. We should also ask ourselves if we want to add another resource. I think it would be more worthwhile to tackle particular aspects of achieving software quality at a time: speed and efficiency, modular code, readable code, skimmable code, automated tests, tact and social engineering, publicity, picking up a good name, writing comments (or not - some people think code should be self-documenting) bug tracking/triaging/fixing, and other aspects.
 * I think that unless we're planning on tackling that huge problem by writing a definitive resource, then I can safely remove the recommendation to read this book from Optimizing Code for Speed. I guess I can add some text to the introduction explaining the context and risk of optimisation and pointing to relevant resources about measuring and achieving software quality.
 * Would it be agreeable? Shlomif (talk) 23:22, 25 February 2010 (UTC)

 * possibly wikibooking material
 * Thank you for writing that essay.
 * if we are going to write a ... wikibook, then we'll have a huge and controversial task ahead of us.
 * Yes. This is also true for many (most? all?) wikibooks. Have you seen the C++ Programming controversy? :-(
 * In which context do we discuss this: open-source software, commercial software, in-house software, embedded software, games; GUI software, command-line software, libraries and APIs, web-based software, terminal software, client/server; by freelancers, by businesses, out-sourcing, off-shore outsourcing, in-house development, for fun, for profit, etc. etc?
 * Contexts where "Use version control, automated testing, and colored syntax highlighting" is good advice. That's all of them, right?
 * I think there's room for a small book that discusses these and similar widely-applicable programming tips, that we suggest people read just before the "optimization" book.
 * I don't think we should make a huge, definitive book on software quality a prerequisite to the "optimization" book.
 * There are other much more commendable efforts
 * I don't understand. I agree that the current version of wikibook, like the current version of every other wikibook, is inferior to some other work. But I hope that future versions rapidly improve and surpass other static works.
 * I'm not sure we can start tackling it starting from "How to Write a Program" excuse for a wikibook.
 * If you are suggesting starting from scratch, then I suspect you haven't actually read the links you mentioned.
 * One of them says "it's almost always easier to start from a good partial solution than from nothing at all." -- Eric Steven Raymond
 * --DavidCary (talk) 03:06, 18 March 2010 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:How to Write a Program


text:
User:Jyotibasu

Hi, I am Jyoti Basu, a software professional from India. I have been fascinated by this wonderful effort of a free encyclopedia and one which follows evolutionary methods for it's growth. Let me offer my 2 cents worth to this.

I'll keep adding to this later.

title: User:Jyotibasu


text:
Movie Making Manual/Keeping the crew happy

A few tips to keep the crew happy:

Food
Never underestimate the importance of good food. If your crew is working for free then you MUST feed them well and at well defined times. If you're going to overrun then stop and ASK your crew if they mind working for another hour before lunch. Always try to provide hot food. Sandwiches are your last resort.

Good food needn't cost a lot. See if you can get your Mum involved or your best friend's mate who's an out-of-work chef. Remember also that some of your cast and crew (especially the clapper-loader) may have to work though their food breaks and so will need easy-to-eat food. Oh, and don't forget to check on allergies.

Check here for advice on good food which is affordable and works on set - Ultra Low Budget production Recipes.

Also remember to have a snack table (called craft services table in the USA). This may seem insignificant, however, film sets bring together very different personalities and egos, both technical and artistic, in very intense situations over a short period of time. Tempers will flare. They always do. Humans when upset or dealing with any extreme emotion in all cultures like to do one thing: put something in their mouth. A drink, a smoke, a chocolate, a thumb... whatever. The craft services table provides this option. Coffee, tea, water, donuts, pastries, water, chocolates, sweets, juice, fruit and water. Don't underestimate it. Try to put the emphasis on healthy food like fruit rather than chocolates and crisps - junk food may give a quick 'high' but wont sustain that high so your crew will become irritable as the day progresses.

Another option (if you're lucky) is to get a deal with a local restaurant or pizza place, perhaps exchanging discounts on food for product placement in your film or advertising in the credits, brochures or your film's website. Local pizza places are a good first stop. This can get costly if done too often. Some consider this option selling out, which is fair, however, it's not a bad way to get some grub for the crew.

No matter what your position on set try to offer others a tea or coffee whenever you can. As a runner you will quickly gain the enthusiasm of others by offering tea and coffee. As a more senior crew member you will show that you aren't a snobby nosed, status-obsessed fool in it only for the power, but a good natured human who likes to get along with people and do the best job possible.

But - be warned - never eat in front of a hungry camera crew. Why? The camera crew don't have the option of quickly picking up a sandwich because their hands must be kept very clean whilst on set and handling camera kit. Once, on my first ADing job, I was casually nibbling on a sandwich when the DoP gave me a loud and public telling off. You have been warned!

Hours
Don't overwork your crew. Your job as a filmmaker is to get the very best out of your cast and crew - as a producer or director you actually do *very little* hands-on filmmaking. So the best thing you can do is create a productive, exciting and comfortable environment for your cast and crew. The one thing that will destroy the mood on set (and destroy your film) is exhaustion. If your schedule demands 20 hour days then cut the script or increase the number of shooting days.

It is important to remember that the hours worked also include the 'wrapping out' time: not all crew just walk away when 'wrap' is called, spark (electricians), camera crew, grips, makeup, wardrobe and others still have a lot of work to pack up from the day and clean and prepare for the next day. So an estimated wrap time of say, 22.00 hours really means an end of shooting at 21.00 to allow for the wrapping-out. Expecting a 12 hour on-camera day to then be followed by an hour's worth of putting away the gear is a surefire way to spark a mutiny!

Communication
Give the cast and crew as much information as possible, especially when things are going wrong. If you can then print and distribute the schedule to EVERY crew member at the start of each day, if not far earlier. If you're going run over schedule then first explain the situation to your crew, then ask very nicely if they mind working longer and then tell your crew the new estimate wrap time.

Drugs
You can't stop everyone from taking loads of drugs but do your best to dissuade your cast and crew.

Drugs and alcohol on set will void most insurance policies. You must get your crew to never ever take any kind of extreme mind altering substances while anywhere near set. I would advise firing anyone that does not heed this warning. This has nothing to do with moral or social issues it has to do with running a business. Until the insurance contracts change no drugs or alcohol on set.

Drinking
This is a dodgy one. Going out for drinks every night of the shoot is a good way to get to know everyone. But don't over do it. What you do as a director or producer is up to you. You'll certainly increase your chances of getting on well with your cast and crew if you go out drinking with them. But, then again, you've probably got to spend every evening preparing the next day's shot list.

Dealing with abusive characters

 * http://www.cinematography.net/abuse.htm

/Production /Crew


text:
A-level Mathematics/OCR/C1

This is the Core Mathematics 1 module of the A-level Mathematics Book. This module assumes that you are familiar with some of the concepts of GCSE Mathematics. You are not permitted to utilize a calculator in the Core Mathematics 1 examination. Please post all questions to A-level Mathematics/Discussion.

Contents

 * 1) /Indices and Surds/  Problems  Solutions
 * 2) /Equations/  Problems  Solutions
 * 3) /Polynomials/
 * 4) /Error bounds and Inequalities/
 * 5) /Coordinate Geometry and Graphs/  Problems
 * 6) /Differentiation/
 * 7) /Appendix A: Formulae/

title: A-level Mathematics/OCR/C1


text:
User:Lordscissorhand~enwikibooks

insert user page here.

title: User:Lordscissorhand~enwikibooks


text: User:Lucidish~enwikibooks

title: User:Lucidish~enwikibooks


text:
User talk:Lucidish~enwikibooks

Hello, It is nice to see someone new taking interest in the School of Philosophy. Please feel free to contribute anywhere you like. There is a lot of work ahead. (Alex beta 16:11, 21 Dec 2004 (UTC))

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Lucidish. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Lucidish~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:23, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 05:12, 19 April 2015 (UTC)

title: User talk:Lucidish~enwikibooks


text:
Movie Making Manual/Linux in film production

=What is Linux?= Linux is a piece of software known as a 'kernel.' The name has also been inaccurately applied to an entire operating system known as GNU. GNU/Linux is the preferred method of referring to the operating system. GNU/Linux is a computer operating system which can run on many different types of computer hardware. GNU/Linux is the most famous example of free software. It is developed and maintained by thousands of people around the world. Most GNU/Linux distributions are gratis. Please take a look at the Wikipedia entry for more info about GNU/Linux.

Advantages

 * Almost always free (or very inexpensive)
 * Open Source (most distributions, but not all)
 * Bugs are usually fixed very quickly
 * Stable
 * Fast
 * You can modify it to your heart's content, learn from it and even redistribute changes.
 * There is a lot of support available on-line through Wikis and Forums
 * Developers are usually very open to suggestions from users. Much more so than proprietary, closed-source, corporate software developers.  If you've got money to spend, then the developers will probably gladly design custom modules for you.
 * Many software options for similar tasks.
 * Many variations on the kernel for various uses depending on hardware, resources, and what you want to do on it.

Disadvantages

 * Documentation is often a little sketchy depending on the distribution
 * The Macintosh is the Industry standard for most aspects of film production
 * Is again its own operating system and takes time to get use to (depending on your own speed of picking things up)

=But I thought GNU/Linux was bad for content creation?= It used to be. But it's grown up. There are already lots of great tools for content creation on GNU/Linux (and, yes, lots of them are free).

=Why use GNU/Linux for film production?=
 * Lots of film companies (especially 3D and effects houses) already use GNU/Linux. So what?  Well, lots of these guys contribute code to the open-source world.  Free of charge.  So you and I can use code which has been developed at ILM.  Examples:
 * Cinepaint, forked from the GNU Image Manipulation Program (GIMP), is partially developed by Rhythm & Hues, ILM, and Sony Pictures. It's mainly used for cleanup, and dust and scratch removal.
 * OpenEXR is developed by ILM.
 * It's very stable.

=Who already uses GNU/Linux?=

Filmmakers

 * BUF
 * C.O.R.E.
 * Digital Domain (Titanic)
 * Dreamworks
 * Industrial Light and Magic
 * Intelligent Creatures
 * La ménagerie (in french)
 * Moving Picture Company, The (Harry Potter, Troy, Kingdom of Heaven, Charlie and the Chocolate Factory)
 * Pixar (Finding Nemo, Monsters Inc, The Incredibles, Toy Story)
 * Rhythm and Hues (Incredible Hulk, Wolfman, Alvin and Chipmunks, Cats & Dogs, Grinch, Sixth Day, Stuart Little, Planet of the Apes)
 * Rising Sun Pictures (Last Samurai, Harry Potter 4, Superman Returns, Charlotte's Web)
 * Sony Pictures Imageworks (Stuart Little II)
 * Warner Bros. Motion Picture Imaging (a 4k digital intermediate facility)
 * WetaDigital (Avatar, The Lord of the Rings, "I, Robot")

Others

 * Most world famous Computer Science Universities (MIT, Caltech, Paderborn)
 * The Government of the People's Republic of China
 * Google
 * NASA
 * CELCO
 * The NSA
 * Large part of French government, public service and administration.
 * The German government installed Suse on 11,000 police computers in Feb 2004
 * The German Railway! - 55,000 computers
 * The city of Vienna, capital of Austria uses a distribution of its own (Wienux) in administration.
 * Cuba 1,500 computers
 * South African public service and administration
 * Some Spanish provinces
 * The United States Navy
 * Venezuela government (60%)
 * Some schools
 * More than 90% (91.8% in December 2010) of the world's 500 most powerful supercomputers are GNU/Linux systems. Of these 500, the top-fastest are GNU/Linux systems.Top500.org.

=Film production tools available on GNU/Linux=

 * There is a huge list of media-related Linux programs here.

 * ... and a good guide to video editing on Linux here on Linux.com

 * Most Windows programs can be made to run under Linux using Wine (which is free, and costs no money).

Everything which isn't marked as "proprietary" is gratis!

Scriptwriting

 * KIT Scenarist Desktop versions are Free, licensed under GPL. Mobile versions are paid.
 * celtx (This has become a very robust tool over time with storyboarding, scheduling and web service capabilities having been added)
 * Trelby a simple, fast and elegantly laid out to make screenwriting simple and infinitely configurable.
 * OpenOffice with the Script template.
 * OpenOffice Script template by Carlos
 * LibreOffice which will replace OpenOffice.
 * Lyx with the Hollywood template (already included)

Budgeting

 * KBudget
 * Grisbi
 * QHacc
 * Quasar - free for one user
 * InfoFlex - proprietary
 * Money Dance - very low cost proprietary
 * GnuCash
 * LedgerSMB
 * SQL-Ledger - Free
 * Open Office Calc - a little like Microsoft Excel
 * Open Office Base - a database package a little like Microsoft Access
 * PC programmes running under Wine

Review of Linux Budgeting programs on Linux Planet.

Schedule / Project's Manager

 * Planner (MrProject) Free - Manage your project with great graphics
 * phprojekt Open source project manager
 * SunBird Virtual Calendar with lots of options

The program celtx listed in the "scriptwriting" section has added very nice scheduling capabilities, allowing you to tie in events with items in your scripts.

Storyboarding

 * Storyboard Lite (proprietary)
 * TVP Animation Pro (proprietary)
 * dastoryboard Open source.

 * Windows programs running under Wine

Besides dastoryboard, there are no free specialized storyboarding programs on Linux but see the "Still image Creation" section below.

A useful technique could be to photograph and outline your actors with a vector graphics program, then reposition them behind photographs of your locations. Check out the vector graphics section.

The program celtx, listed in the "scriptwriting" section, has storyboarding capabilities. You would still need to create the images as explained in the previous paragraph but they can be entered into a template within the celtx application. Notes can be added to the images as well.

The 2D animation software Tupi also has made steps to implement storyboard facilities.

Editing, compositing, grading and digital intermediate

 * Blender 3D (free, GPL) While it is known mostly as a 3D modeling/animation package, recent upgrades have added high-end compositing and editing features as well.
 * Cinelerra (free, GPL)
 * FilmLight (a DI tool, proprietary)
 * Gimp GAP (Gimp animation package), CVS version.
 * IFX Piranha HD (proprietary)
 * Jahshaka (free) combines editor, compositor, animator, effects, colour control and text creation.
 * kdenlive (free, GPL) -- Wikibook
 * Lightworks (currently closed-source) Professional target; currently a binary beta version for Linux is available; the source code will be released in future
 * LiVES (free, GPL) an advanced video editor which also provides VJ features
 * Lumiera (free)
 * Nuke (proprietary)
 * Pitivi (free, GPL)
 * Shake (proprietary)
 * Shotcut (free, GPL)
 * Various proprietary packages by Autodesk (formerly known as Discreet).

File conversion tools

 * Avidemux2 A clone of virtualdub
 * FFmpeg a complete solution to record, convert and stream audio and video, including frame to movie and movie to frame.
 * GStreamer Open source multimedia framework
 * Handbrake Convert DVDs to MPEG4/AVC
 * MPlayer/MEncoder MEncoder features a wide variety of audio/video filters and support (en)coding with most audio and video codecs
 * Transcode Transcode, another solution to convert and filter audio/video
 * WinFF is a GUI to the command line video converter FFmpeg

Audio editing

 * Ardour (free, GPL), Protools clone, uses JACK, video-sync via xjadeo.
 * Bitwig (proprietary)
 * Audacity (free, GPL), not very advanced, but relatively stable, lot of effects via LADSPA.
 * GNUsound (free, GPL), multitrack sound editor for gnome (1 & 2), support LADSPA for sound effects, and JACK for realtime processing.
 * Rezound (free, GPL) PowerFull audio editor
 * Tracktion (proprietary), uses JACK, version 4 and 5 are provided free of cost.
 * Wired (free, GPL)

Bitmap

 * Adobe PhotoShop CS2 will run under Wine(proprietary)
 * GIMP
 * ../IDruna Photopaint HDR/ (proprietary)
 * ../IFX Amazon paint/ (proprietary)
 * Krita
 * Pixel image editor (proprietary, but cheap)
 * TVP Animation (proprietary)

Vector

 * Cenon (free, vhf Public License)
 * gestalter (free, GPL) dead project
 * inkscape (free, GPL)
 * karbon (free, GPL)
 * skencil (free, GPL)
 * sodipodi (free, GPL) dead project
 * xara xtreme (free, GPL)

Free

 * Aqsis (rendering backend) (free, GPL)
 * Art Of Illusion (Java modeling and rendering) (free, GPL)
 * Blender (free, GPL) --Wikibook
 * Pixie (rendering backend) (free, GPL)
 * Wings 3D (free, GPL) --Wikibook
 * Yafray (rendering backend) (free, GPL)
 * SunFlow (open source Java rendering system for photo-realistic image synthesis) ( free, MIT License)
 * LuxRender (open-source, free software rendering system for physically correct, unbiased image synthesis) (free, GPLv3)
 * Kerkytea (Freeware)

Proprietary

 * 3Delight (rendering backend) (proprietary but first license is gratis)
 * AC3D (modelling software, proprietary)
 * AIR (rendering backend) (proprietary)
 * Gelato - nVidia (rendering backend) (proprietary)
 * Houdini (proprietary)
 * Jig (commercial)
 * Massive (proprietary)
 * Maya (commercial)
 * PhotoRealistic RenderMan (rendering backend) (proprietary)
 * Realsoft3D (proprietary)
 * Softimage (proprietary)

stop motion

 * Boats Animator (free, GPL)
 * Linux Stopmotion (free, GPL)
 * qStopMotion, a fork of Linux Stopmotion (free, GPL)
 * myrtille (free, GPL) no movement since 2009
 * stopmojo (Java, free, GPL)
 * eXposer (BeOS / Zeta, proprietary)

Free

 * Chalanam (free, GPL)
 * F4L (flash for linux) (free, GPL)
 * Gimp Animation Package (gimp-gap) (free, GPL)
 * Ktoon (free, GPL)
 * Tupi an active fork of KToon (free, GPL)
 * Synfig (free, GPL)
 * mvComicsMaker (free, GPL) dead project since 2001.
 * Pencil (free, GPL) hasn't changed for a couple of years.
 * Processing, procedural animation (free, GPL) and the JS version ProcessingJS
 * RaphaelJS (js, free, MIT License)
 * jQuery SVG (js, free, double license GPL + MIT)
 * uira (free, LGPL) development stopped in 2006, and officially postponed by author on January 05, 2008.
 * MyPaint-0.8 with animation patch (information for installing and getting it working) (free, GPL).
 * FlipBookCreator (free, GPLv3) more for animating flipbooks, still in very early stages.
 * OpenToonz (free, new BSD) based on the proprietary software "Toonz", which was developed by Digital Video S.p.A. in Italy and customized by Studio Ghibli.

Proprietary

 * Moho (proprietary) (discontinued: is now called Anime Studio)
 * Opus (proprietary)
 * Plastic Animation Paper (proprietary)
 * Toonz (proprietary)
 * TVP Animation Pro (proprietary)

LipSync

 * jlipsync (free, GPL)
 * papagayo (free, GPL)
 * yolo (free,GPL) and Compatible whit Moho and Magpie Timesheet

VJ

 * LiVES (free, GPL) an advanced video editor which also offers VJ functionality
 * FreeJ (free, GPL)
 * Gephex (free, GPL)
 * veejay (free, GPL) a visual instrument and realtime video sampler.

VCD & SVCD Authoring

 * GNU vcd imager (free, GPL) VCD (VideoCD) BIN/CUE CD image creation and ripping tool
 * qVCDgear (free, GPL) QT frontend to VCDgear
 * tovid (free, GPL) (S)VCD and DVD authoring utilities
 * VCDautomenu (free, GPL) VCD menu generation tool
 * VCDgear converts from and to (S)VCD and mpeg, and does mpeg corrections.

DVD Authoring
some tovid menu selections
 * dvdauthor (Free, GPL) (command line)
 * 'Q' DVD-Author More then just a dvdauthor frontend. Easy to use and intuitive. Key technologies: Templates, Animated Menus, MovieButtons, MovieBackgrounds, Transcode, Multi format DVDs, Slideshow creation, Subtitles.
 * dvdstyler Cross platform dvdauthor frontend (Windows and Linux)
 * ManDVD (Free, GPL)
 * DVDLab (using WINE) (proprietary)
 * KDE DVDAuthor Wizard wizard to create DVD from MPEG-2 files
 * kmediafactory template based dvd authoring tool
 * Linux Video Maker graphics interface (GUI) for transcode, mplex, DVDAuthor etc ...
 * polidori Development stalled in 2004. frontend to dvdauthor
 * tovid (Free, GPL) (S)VCD and DVD authoring utilities
 * varsha free, java.
 * videotrans a set of scripts that allow its user to reformat existing movies into the VOB format that is used on DVDs

DCP Authoring

 * DCP-o-matic (Free, GPL)

Streaming video

 * Fenice (free, GPL) Open Media Streaming Server formats: mpeg1,4 & ogg/theora for video
 * FFmpeg (free, GPL), ffmpeg read, convert and stream videos in various formats.
 * Flexcast (free, GPL) Stream most of video and audio format.
 * Flumotion (free, GPL), A streaming media server written in python.
 * GStreamer (free, GPL) streaming and convertion.
 * Icecast (free, GPL), stream video (ogg/theora) and audio in various formats.
 * jroar (free, [L]GPL) A java stream cache.
 * Open Media Streaming Project (free, GPL), A project about a whole streaming plateform.
 * Palantir (free, GPL) for streaming with low-end computer
 * VLC (free, GPL) from Videolan Project, to stream and broadcast any source on a Local Area Network.
 * VobStream (free, GPL) a DVD network streaming application

p2p video streaming for low bandwidth

 * freecast (free, GPL), p2p java streaming Ogg Theora video stream provided by Flumotion.
 * Peercast (free, GPL) Support for MP3, OGG Theora and Vorbis, WMA, WMV and NSV

Subtitles

 * Gaupol (GPL) is an editor for text-based subtitle files. It supports multiple subtitle file formats and provides means of creating subtitles, editing texts and timing subtitles to match video.
 * Gnome Subtitles is a subtitle editor for the GNOME Desktop. It allows subtitle editing, subtitle translation, video previewing and timings synchronization.
 * Ksubtile (GPL), an editor for the KDE environment to edit, make and save subtitles in the SRT subtitle format.
 * Orange Slice manipulates SSA, ASS, SUB, SRT subtitles format (abandoned project since 2005-01-14)
 * Sabbu, subtitle timing software that can handle ASS and SSA subtitles format.
 * Subtitle Editor (GPL3), can create new subtitles and transform, edit, correct, refine existing subtitles with formats such as ASS, SSA, SubRip, MicroDVD, MPL2, MPsub, SubViewer2.0.
 * Subtitlefix. . It corrects subtitles (*..srt, *..sub) which are not synchronized with the film.
 * USF. . The Universal Subtitle Format.

Web Authoring

 * bluefish
 * NVU
 * Screem

Audio/Video Container/Codec

 * Dirac Dirac is a video codec.
 * FLAC Free Lossless audio codec.
 * Matroska The extensible open standard Audio/Video container.
 * NUT A free container for the masses.
 * Ogg Vorbis Ogg Vorbis is an audio compression format.
 * Snow Snow is an experimental lossy video codec.
 * Theora Theora is an open video codec.
 * Wavpack WavPack is a completely open audio compression format providing hybrid compression mode.
 * XviD XviD is an ISO MPEG-4 compliant video codec.

Utilities

 * framecalc is an online timecode calculator; it calculates frames out of timecode, timecode out of frames or can calculate the difference between an IN and an OUT point. All this with a chosen frame rate.

=Specialized and "ready to use" distributions=

 * ArtistX (formerly MediainLinux) (Free) GNU/Linux LiveCD with additional multimedia packages.
 * AV Linux is a Debian GNU/Linux-based operating system wrapped with applications such as Cinelerra, Kdenlive, Ardour, GIMP, Inkscape, LibreOffice Suite among others.
 * Caster. . Caster is a LiveCD whose purpose is for streaming.
 * Dynebolic (Free, GPL) most of Linux free products for video, dj, animation, on a LiveCD.
 * Elive GNU/Linux LiveCD and distribution, includes Cinelerra CVS by default and more applications on their repositories.
 * Garbure. . (Free, GPL) Several specialized distributions on LiveCD, video, animation, audiovisual performances...
 * Ubuntustudio (Free) A multimedia creation flavor of Ubuntu.

=Personal opinions=

Wish List
I love Linux. Don't get me wrong. I think the whole project is perfect and I use Linux all the time. I use Linux for everything. Everything, including video editing.

My wish list is:

 * Flash MX 2004/2005 on Linux (for DVD Menus)
 * Adobe port Premiere over to Linux / Avid port DV Xpress to Linux
 * There are many very promising Open Source video editing packages that one day may become mature enough to compete with Premiere, but I'm porting this one right now with Xen.

Once one of these wishes becomes true then Linux will rock as an NLE platform.

GNU/Linux Multi-media forums
I'm somewhat disappointed that many of the forums on GNU/Linux multi-media are very quiet. The best forums that I know of are: Does anyone know of any other good multi-media forums for linux?
 * Doom9 Linux specific multimedia forum on a mostly windows site
 * Video Help.com Linux specific multimedia forum on a mostly windows site
 * Gentoo-Multimedia forum Hosted on a gentoo-specific forum, but has a lot of generic GNU/Linux multimedia information (and relatively very active)
 * MultimediaWiki All about Multimedia

=Getting help=

Software is often complex and has a steep learning curve. For ways of getting help online, take a look at the online resources page for links to forums and tutorials online. Also take a look at the training category index for more info on training.

/Tools and equipment /Post-production

Faire son film-Linux


text:
Abstract Algebra/Ideals

Motivation
In ../Rings/ we saw that the set of even integers $$2 \mathbb{Z} \subset \mathbb{Z}$$ was a subring of the integers.

We can also see very easily that the integers $$ \mathbb{Z} \subset \mathbb{Q} $$ are a subring of the rational numbers under the usual operations of addition and multiplication.

The even integers, when taken as a subring of the integers have a property that the integers when taken as a subring of the rationals do not. The even integers taken as a subring of the rationals also lack this property.

The property is that the even integers, taken as a subring of the integers, absorb multiplication. Let's call the even integers $$I = 2 \mathbb{Z} $$ for ease of notation.

Consider the following: For all $$i \in I$$, we can see by the definition of $$I$$ that $$i = 2k$$ for some $$k \in \mathbb{Z}$$.

For all $$a \in \mathbb{Z}$$ see that $$ai = a(2k) = 2ak \in I$$.

In Math, regardless of which even integer is chosen, multiplying it by any integer will give us a different even integer.

Definition of an Ideal
Definition: Given a ring $$R$$, a subset $$I \subseteq R$$ is said to be a left ideal of $$R$$ if it absorbs multiplication from the left; that is, if $$\forall i \in I, \forall r \in R, ri \in I$$.

Definition: Given a ring $$R$$, a subset $$I \subseteq R$$ is said to be a right ideal of $$R$$ if it absorbs multiplication from the right; that is, if $$\forall i \in I, \forall r \in R, ir \in I$$.

Definition: We define an ideal $$I$$ to be something that is both a left ideal and a right ideal. We also require that $$(I, +)$$ is a subgroup of $$(R, +)$$.

We write $$I \triangleleft R$$ as shorthand for this.

To verify that a subset of a ring is an ideal, it is only necessary to check that it is closed under subtraction and that it absorbs multiplication; this is because of the subgroup criterion from Abstract Algebra/Group Theory/Subgroup.

Definition: An ideal $$I \triangleleft R$$ is proper if $$I \subsetneq R$$.

Definition: An ideal $$I \triangleleft R$$ is trivial if $$I=\{0\}$$.

Lemma: An ideal $$I$$ is proper if and only if $$1 \notin I$$.

Proof: If $$1 \in I$$ then $$r = r1 \in I$$ so $$I = R$$.

The converse is obvious.

Theorem: In a division ring, the only proper ideal is trivial.

Proof: Suppose we have an ideal in a division with a nonzero element a. Take any element b in our division ring. Then a−1b is in the division ring as well, and aa−1b = b is in the ideal. Therefore, it is not a proper ideal.

Definition: Let S be a nonempty subset of a ring R. Then the ideal generated S is defined to be the smallest ideal in R containing S, which would be the intersection of all such ideals. We can characterize this ideal by the collection of all finite sums

$$\left \{ \sum_{i=1}^n r_{i1}s_ir_{i2}|r_{i1},r_{i2} \in R, s_i \in S \right \} $$

And one can easily verify that this is an ideal, and that all ideals containing S must contain this ideal. If it is commutative, then one can simply characterize it as

$$\left \{ \sum_{i=1}^n r_i1s_i|r_i \in R, s_i \in S \right \}$$

The ideal generated by a single element a is called a principal ideal. If the ring is commutative, it consists of all elements of the ring of the form ra where r is any element in the ring.

Example: Let $$R = \mathbb{Z}$$ be the ring of integers. The principal ideal $$(n)$$ is the subset of $$\mathbb{Z}$$ consisting of positive and negative multiples of $$n$$. For example $$(2)$$ is the subset of even integers. Then one can view the factor ring $$\mathbb{Z} / (n)$$ simply as the set $$\{0, 1, \ldots, n - 1 \}$$ under addition and multiplication modulo $$n$$.

Operations on Ideals
Given a collection of ideals we can generate other ideals. For instance it is easy to check that the intersection of any family of ideals is again an ideal. We write this simply as $$\bigcap_{j \in J} I_j$$.

Given any set $$S \subset R$$ we can construct the smallest ideal of $$R$$ containing $$S$$ which we denote by $$\langle S \rangle$$. It is determined by $$\langle S \rangle = \bigcap_{S \subset I \triangleleft R} I$$, though often we can be more explicit than this.

If $$I_{j \in J}$$ is a collection of ideals we can determine the sum, written $$\sum_{j \in J} I_j$$, as the smallest ideal containing all the ideals $$I_j$$. One can check explicitly that its elements are finite sums of the form $$\sum_{j \in J} x_j$$.

Finally if $$I,J$$ are two ideals in $$R$$ one can determine the ideal-theoretic product as the smallest ideal containing the set-theoretic product $$\{ i j \mid i \in I, j \in J \}$$. Note that the ideal-theoretic product is in general strictly larger than the set-theoretic product, and that it simply consists of finite sums of the form $$\sum i_r j_r$$ where $$i_r \in I \; j_r \in J $$

Example: Let $$(m)$$ and $$(n)$$ the principal ideals in $$\mathbb{Z}$$ just given. Then one can check explicitly that $$(m) \cap (n) = (r)$$, where r is the lcm of m and n. Moreover $$ (m)(n) = (mn)$$, and $$(m) + (n) = (s)$$ where s is the hcf of m and n. Observe that $$(m)(n) = (m) \cap (n)$$ if and only if s = mn if and only if m and n are co-prime if and only if $$(m) + (n) = (1)$$.

Homomorphisms and Ideals
Rings, like groups, have factor objects that are kernels of homomorphisms. Let $$f:R\to S$$ be a ring homomorphism. Let us determine the structure of the kernel of f which is defined to be all elements which map to the identity.

If a and b are in the kernel of f, i.e. $$f(a)=f(b)=0$$, and r is any element of R, then
 * $$f(a-b)=f(a)-f(b)=0$$,
 * $$f(ar)=f(a)f(r)=0$$,
 * $$f(ra)=f(r)f(a)=0$$.

Therefore $$\ker(f)$$ is an ideal of R.

Also note that the homomorphism will be a monomorphism i. e. it is injective or one-to-one when the kernel consists only of the identity element.

We also have the following

Theorem: If the only proper ideal of R is the trivial ideal {0}, then if f is a homomorphism from R to S, and it does not map all elements of R to the identity in S, then it is injective.

Proof: The kernel of the homomorphism must be an ideal, and since the only ideals are R and the trivial ideal, one of these two must be the kernel. However, since not all elements of R map to the identity of S, R is not the kernel, so the trivial ideal must be.

Since this condition is satisfied for all division rings, it is true for all division rings.

The construction of factor rings in the next section will prove that there exists a homomorphism with I as its kernel for any ideal I.

Factor Rings
Definition: Given a ring $$R$$ and an ideal $$I \subseteq R$$, the ring of cosets of $$R$$, r+I where r is within R, is written $$R/I$$, where each coset is defined to be the set {r+i|i is an element of I}, and by Lagrange's theorem, it partitions R. This set of cosets, called the factor ring (or quotient ring) of $$R$$ modulo $$I$$ is a ring with addition defined the same way as one would define it for groups (since the ring is a group under addition), and with multiplication defined as follows:
 * $$(a + I)(b + I) = ab + I$$.

To show that this is independent of the choice of a and b (or, the operations are well-defined), suppose that a' and b' are elements of the same respective coset. Then a'=a+j and b'=b+k for some element j,k within I. Then a'b'=ab+ak+jb+jk and since ak, jb, and jk are elements of I, a'b' and ab must belong to the same coset, so ab+I=a'b'+I. Obviously the cosets form a group under addition because of what was proved earlier about factor groups, and furthermore the factor ring forms an abelian group under addition because the ring forms an abelian group under addition. Since the product rs+I is analogous to the multiplication in the ring, it obviously has all the properties of a ring. Furthermore, if the ring is commutative, then the factor ring will also be commutative.

Observe that there is a canonical ring homomorphism $$\pi : R \rightarrow R / I$$ determined by $$\pi(r) = r+I$$, called the projection map. We record some properties of this homomorphism in the next section of the isomorphism theorems.

Ring Isomorphism Theorems
We have already proved the isomorphism theorems for groups. Now we can use analogous arguments to prove the isomorphism theorems for rings, substitution the notion of "normal subgroups" with ideals.

Factor Theorem
Let I be an ideal of a ring R, Let $$\pi(x)=x+I$$ be the usual homomorphism from R to R/I. Now let f be a homomorphism from R to S. Observe that if $$\tilde{\phi} : R/I \rightarrow S$$ is a ring homomorphism, then the composition $$\phi = \tilde{\phi} \circ \pi : R \rightarrow R/I \rightarrow S$$ is a ring homomorphism such that $$I \subset \ker{\phi}$$ (because $$x \in I \Rightarrow \phi(x) = \tilde{\phi} \circ \pi(x) = \tilde{\phi}(0_{R/I}) = 0_S$$). This characterizes all such morphisms in the following sense

Factor Theorem: Let $$\phi : R \rightarrow S$$ be a ring homomorphism such that $$I \subset \ker{\phi}$$. Then there is a unique homomorphism $$\tilde{\phi} : R /I \rightarrow S$$ such that $$\phi = \tilde{\phi} \circ \pi$$. Furthermore, $$\tilde{\phi}$$ is an epimorphism if and only if $$\phi$$ is an epimorphism, $$\tilde{\phi}$$ is a monomorphism if and only if its kernel is I.

Proof We prove it the same way we did for groups. Define $$\tilde{\phi}(x+I)$$ to be $$\phi(x)$$. To see that this is well-defined, let a+I=b+I, and so that a-b is an element of I, so that $$\phi(a-b)=0_S$$ so that $$\phi(a)=\phi(b)$$. Now $$\phi$$ is a homomorphism, implying that $$\tilde{\phi}$$ is also. The proofs of the additional statements can be carried over from the proofs of the additional statements of the factor theorem from groups.

First Isomorphism Theorem
Let R be a ring, and let f be a homomorphism from R to S with kernel K. Then the image of f is isomorphic to R/K.

Proof

Using the factor theorem, we can find a homomorphism from R/K to S, and since the kernel is the same as the ideal used in forming the quotient group, and since the f is an epimorphism over its image, this homomorphism is an isomorphism.

Second Isomorphism Theorem
Let R be a ring, let I be an ideal, and let S be a subring.
 * S+I, the set of all s+i with s within S and i within I, is a subring of R.
 * 1) I is an ideal of S+I.
 * 2) The intersection of S and I is an ideal of S.
 * 3) (S+I)/I is isomorphic to $$S/(S\cap I)$$.

Proof
 * 1) It can be verified that it contains 1, and is closed under multiplication.
 * 2) Of course, since I is an ideal of R, then it must be an ideal under any subring.
 * 3) From a similar argument for groups, it can only contain elements of I, but restricted to S, so it must be an ideal of S.
 * 4) Let $$f: R \rightarrow R/I$$ be a function restricted to the domain S, and define $$f(x)=x+I$$. It is obvious that its kernel is $$S\cap I$$ and that its image is (S+I)/I.

Third Isomorphism Theorem
Let I be an ideal of a ring R, and let J be an ideal of the same ring R that contains I. J/I is an ideal of R/I, and R/J is isomorphic to (R/I)/(J/I).

Proof Define the function $$f: R/I \rightarrow R/J$$ to be $$f(a+I)=a+J$$ which is well-defined because since I is an ideal that is within J. This is also obviously a homomorphism. Its kernel is all elements that map onto J, and is thus all a+I such that a is within J, or J/I. Moreover, its image is R/J, and so we can use the first isomorphism theorem to prove the result.

Correspondence Theorem
Let I be an ideal of a ring R. Define the function $$\pi$$ to map the set of rings and ideals containing I to the set of rings and ideals of R/I, where $$\pi (X)$$ = the set of all cosets x+I where x is an element of X. This function is one-to-one, and the image of rings or ideals containing I are rings or ideals within R/I.

Proof Define the function f from rings or ideals containing I to the rings or ideals of R/I, by f(A)=A/I. We have already proved the correspondence for addition because rings form an abelian group under addition. Thus, we need only to check for multiplication. Suppose S is a subring of R containing I. S/I is obviously closed under addition and subtraction. For multiplication, suppose that x and y are elements of S. Then (x+I)(y+I)=xy+I which is also an element of S/I, proving that it is closed under multiplication. The identity 1 is within S, and we have it that 1+I is also within S/I. Thus, S/I is a subring of R/I. Now suppose that S/I is a subring of R/I. Then it is also obvious that S is closed under addition and subtraction and multiplication, proving that S is a subring of R. Now suppose that J is an ideal of R containing I. Then by the third isomorphism theorem, J/I is an ideal of R/I. Now suppose that J/I is an ideal of R/I. Let r be any element of R, and let j be any element of J. Then since J/I is an ideal of R/I, (r+I)(j+I)=rj+I must be an element of J/I. This indicates that rj must be an element of J, proving that J is an ideal of R.

Definitions
Definition: Two elements $$a, b$$ are said to be congruent in an ideal $$I$$ if and only if they belong to the same coset in R/I. This is true when a-b is within I. Write $$a = b \bmod{I}$$ to mean that $$a$$ is congruent to $$b$$ modulo $$I$$.

Lemma: Given an ideal $$I \subseteq R$$, a subset of a ring $$R$$, the congruence class $$[r]$$ modulo $$I$$ of an element $$r \in R$$ is $$[0]$$ if and only if $$r \in I$$. To see this, simply note that $$a = b \bmod{I}$$ means $$a-b\in I$$; plugging in gives $$r-0\in I$$.

Definition: Two natural numbers are relatively prime when ax+by=1 for integers x and y. We do the same for rings - two ideals I and J are relatively prime when ai+bj=1 for ring elements a, b, and for an element i within I and an element j within J. In other words, two ideals are relatively prime if their sum contains the identity element i. e. if I+J is the whole ring R.

We will now prove the

Chinese Remainder Theorem
Let R be a ring, and let $$I_n$$ be n pairwise (i. e. when considering any two pairs) relatively prime ideals.
 * 1) Let a be a number from 1 to n. There exists an element r within R that is within all ideals $$I_n$$ such that $$n \neq a$$, and such that $$r = 1 \bmod{I_a}$$
 * 2) Let $$r_1, r_2, ..., r_n$$ be elements of R. Then there exists an element r within R such that $$r = r_i \bmod{I_i}$$ for all i=1,2,3,...,n.
 * 3) Let I be the intersection of the ideals. Another element of R, s satisfies $$s = r_i \bmod{I_i}$$ for all i=1,2,3,...,n if and only if $$r=s \bmod{I}$$.
 * 4) R/I is isomorphic to the product ring $$\Pi_{i=1}^n R/I_i$$

Proof
 * 1) Since $$I_1$$ and $$I_i$$ (i>1) are relatively prime, there will exist an elements $$b_i\in I_1$$ and $$c_i\in I_i$$ (i>1) such that $$b_i+c_i=1$$. This implies that $$\Pi_{i=2}^n (b_i+c_i)=1$$. Now we expand this product on the left side. All terms of the product other than $$c_2c_3c_4...c_n$$ belong to $$I_1$$ while $$c_2c_3c_4...c_n$$ itself belongs the set S of all finite sums of products $$x_2x_3x_4...x_n$$ with $$x_i\in I_i$$. Thus, it can be written in the form b+a=1, where b is an element of $$I_1$$, and where a is an element of S. Then $$a\equiv 1 \pmod{I_1}$$ and $$a\equiv 0 \pmod{I_i}$$ for i>1.

Prime and Maximal Ideals
There are two important classes of ideals in a ring - Prime and Maximal.

Definition: An ideal $$I \triangleleft R$$ is prime if it satisfies:

for any ideals A and B such that AB is a subset of I, implies A is in I or B is in I.

Definition: An ideal $$I \triangleleft R$$ is maximal if it is proper (i.e. $$I \subsetneq R$$ and it satisfies:

$$I \subset J \triangleleft R \implies I = J \text{ or } J = R$$

That is, there are no proper ideals between $$I$$ and $$R$$.

The following Lemma is important for many results, and it makes essential use of Zorn's Lemma (or equivalently the Axiom of Choice)

Lemma: Every non-invertible element of a ring is contained in some maximal ideal

Proof: Suppose $$x$$ is the non-invertible element. Then the first observation is that $$(x)$$ is a proper ideal, for if $$(x) = R$$, then in particular $$1 \in (x)$$ so $$1 = ax$$ contradiction the assumption. Let $$\mathcal{S}$$ be the set of proper ideals in $$R$$ containing $$x$$ ordered by inclusion. The first observation implies that $$\mathcal{S}$$ is non-empty, so to apply Zorn's Lemma we need only show that every increasing set of ideals contains an upper-bound. Suppose $$\{ I_j \}_j$$ is such an increasing set, then the least upper bound is $$\sum_{j \in j} I_j$$ as this is the smallest ideal containing each ideal. If one checks that the union $$\cup_{j \in J} I_j$$ is an ideal, then this must be $$\sum_{j \in J} I_j$$. To show it's proper, we need only show $$1 \notin \cup_{j \in J} I_j$$ for all $$j$$. But this follows precisely because each $$I_j$$ is proper.

Therefore by Zorn's Lemma there is a maximal element $$J$$ of $$\mathcal{S}$$. It is clearly maximal for if $$J'$$ were any ideal satisfying $$J \subset J^\prime \subsetneq R$$ then $$J^\prime$$ would be an element of $$\mathcal{S}$$, and by maximality of $$J$$ we would have $$J^\prime \subset J$$ whence $$J = J^\prime$$.

Properties of rings may be naturally restated in terms of the ideal structure. For instance

Proposition: A commutative ring $$R$$ is an Integral Domain if and only if $$(0)$$ is a prime ideal.

Proof: This follows simply because $$x = 0 \iff x \in (0)$$.

This explains why an Integral Domain is also referred to as a Prime Ring. Similarly, we may give a necessary and sufficient condition for a ring to be a field :

Proposition: A commutative ring $$R$$ is a Field if and only if $$(0)$$ is a maximal ideal (that is there are no proper ideals)

Proof: We only need to show that every element $$0 \neq x \in R$$ is invertible. Suppose not then by Lemma ... $$x$$ is contained in some (proper) maximal ideal, a contradiction.

Corollary: An ideal $$I \triangleleft R$$ is maximal if and only if $$R / I$$ is a field.

Proof: By the previous Proposition we know $$R / I$$ is a field if and only if its only proper ideal is $$(0)$$. By the correspondence theorem (...) this happens if and only if there are no proper ideals containing $$I$$.

Corollary: The kernel of a homomorphism f from R to S is a maximal ideal when S is a field. The proof of this follows from the first isomorphism theorem because S is isomorphic to R/ker f.

It's also clear that

Lemma: An ideal $$I \triangleleft R$$ is prime if and only if $$R / I$$ is an integral domain.

Proof: Write $$\bar{x}$$ for the element of $$R / I$$ corresponding to the equivalence class $$[x]$$. Clearly every element of $$R / I$$ can be written in this form.

$$\Rightarrow)$$ $$\bar{xy} = 0 \iff xy \in I \iff x \in I \text{ or } y \in I \iff \bar{x} = 0 \text{ or } \bar{y} = 0$$ where the second equivalence follows directly because $$I$$ is prime.

$$\Leftarrow)$$ This follows in exactly the same way.

Corollary: The kernel of a homomorphism f from R to S is a prime ideal when S is an integral domain. The proof of this follows from the first isomorphism theorem because S is isomorphic to R/ker f.

Lemma: A maximal ideal is also prime.

Proof: Suppose $$I \triangleleft R$$ is a maximal ideal, and $$xy \in I$$. Suppose further that $$x \notin I$$. Then the ideal $$I + (x)$$ is an ideal containing $$I$$ and $$x$$, so is strictly larger than $$I$$. By maximality $$I + (x) = R \ni 1$$. So $$1 = i + rx \implies y = iy + rxy \in I$$.

Alternatively, we can use the above two results, and the fact that all fields are integral domains to prove this.

Glossary
Please see the extensive Glossary of ring theory.

title: Abstract Algebra/Ideals


text:
Movie Making Manual/Mac in film production

Writing

 * Movie Magic Screenwriter
 * Final Draft
 * Celtx (free, Open Source)
 * Montage
 * Scrivener (free for 30 days, Universal) [includes Screenplay Template on disk image]

Budgeting

 * EP Budgeting
 * Gorilla
 * Movie Magic Budgetting

Scheduling

 * EP Scheduling
 * Gorilla
 * Company Move (Excellent piece of software!)
 * Movie Magic Scheduling

Storyboarding

 * StoryBoard Artist
 * Film Forge 3D
 * SketchBook Pro
 * Celtx (free)

Editing

 * Final Cut Pro
 * Avid
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File conversion tools

 * ffmpegX

Compositing

 * Shake
 * Motion
 * After Effects

Animation

 * Lightworks
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 * Maya
 * Blender 3D (free, Open Source)
 * Houdini

DVD Authoring

 * iDVD
 * DVD Studio Pro

Web Authoring

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 * Panic Coda

Audio Editing

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Getting help
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