text:
Nanotechnology/About

Vision
We hope to use the Wikibooks format to make an Open Source Handbook on Nanoscience and Nanotechnology, freely accessible for everyone, that can be updated continuously.

Wikipedia is growing fast and one of the most visited websites on the net – a valuable resource of information we all use.

In science and technology we often need more detailed information than what can be presented in a brief encyclopedic article – and here wikibooks.org, a sister project to Wikipedia, can help us with this newly started handbook.

Though the book is still in its infancy, it has been elected book of the month December 2006, and we hope this will provide PR and more people contributing to the project!

The plan to create the book:

1: First to create smaller articles to ‘cover’ the entire area of nanotechnology and achieve a well defined structure the book (some parts could be revised thoroughly in this process,for instance the materials chapter).

2: Once the structure is reasonably well defined, to begin refining the articles with in-depth material so we reach lecture-note level material.

3: Since everybody can contribute, a continuous contribution of material is expected and a backing group of editors is needed to maintain a trustworthy level of information.

An voluntary editorial board is being put together to oversee the book, support, contribute and follow its development.

Discussion about the content of the book can be found on the main talk page talk:Nanotechnology

As with Wikipedia, we hope to see a solid information resource continuously updated with open source material available for everyone!

References in Wikibooks
Add references whenever possible, with reference lists at the end of each page. Please try to make links to the articles with the DOI (digital object identifier) because that gives a uniform and structured access for everyone to the papers.

All papers get a DOI - a unique number like a bar code in a supermarket. All DOIs are registered by www.doi.org and in the reference list you can add links like https://doi.org/10.1039/b504435a so people will be able to find it no matter how the homepage of the journal or their own library changes.

The References section has an example reference.

Add links to the Wikipedia whenever possible - and for the beginning I will rely extensively on Wikipedia's pages on the subjects, simply referring to these. This textbook could be simply a gathering of Wikipedia pages, but an encyclopedia entry is brief, and for a handbook it is preferable to have more in-depth material with examples and the necessary formulas. So, some information in this textbook will be very much like the Wikipedia entries and we might not need to write it in the book but can simply refer to Wikipedia, but the hope is that this will be more a text book as is the intention with Wikibooks.

Multiple references, see w:Help:Footnotes

Links
There's a shorthand way to make links to Wikipedia from Wikibooks: Wikipedia on Quantum Tunneling gives the link Wikipedia on Quantum Tunneling.

History
The book was started by Kristian Molhave(wiki user page) 13. Apr. 2006. Initially it was named Nanowiki, and later changed to Nanotechnology. Kristian is currently slowly uploading material to the book and looking for people who would like to contribute that can and substantial material to specific sections under the GNU license. I hope we can make an 'editorial panel' of people each keeping an eye on and updating specific sections.

The Summer 2008 Duke Talent Identification Program (TIP) eStudies Nanotechnology students will be adding to the content of this Wikibook. From June-Aug 2008 there will be content additions with references that will add to this great resource.

Authors and Editors

 * The Opensource Handbook on Nanoscience and Nanotechnology was started by Kristian Molhave(wiki user page)

Editors

 * An editorial board is currently being organized.

Support and Acknowledgments
Starting this book is supported by the Danish Agency for Science, Technology and Innovation through Kristian Mølhave's talent project ’NAMIC’ No. 26-04-0258.

How to Reference this Book
I am not currently sure how work on wikibooks or wikipedia can be referenced reliably in published literature.

Three suggestions:

1) Reference the references from the wikibook. Wikibooks are not intended to be the publication channel for new results, but should be based on published and accepted information with references and these references can be used. But this of course does not give credit to the book, so I recommend then adding an acknowledgement about the book to give it PR and credit.

2) Reference the book with a specific page and date - the previous versions of the pages are all available in the history pane and can easily be accessed by future users. You can also hit "permanent version" on the left side of the webpage (it is under "toolbox"). That sends you specifically to the selected version of the wikipage with a link to it that will never change.

3) Reference the PDF version and its version number. Once the book achieves a reasonable level, PDF versions will become available for download and they will have a unique version number and can be retrieved.

Other suggestions are most welcome!

Edwards, Steven A.,The Nanotech Pioneers Christiana, USA: Wiley-VCH 2006, pg 2

title: Nanotechnology/About


text:
DD-WRT

DD-WRT is an alternate firmware available for Wi-Fi routers. This software is usually not officially supported by the original manufacturer of the router and the use may void your warranty. DD-WRT is free, open source software licensed under the GNU General Public License version 2. The official website is http://www.dd-wrt.com/.

Table of Contents

 * Features
 * Supported devices
 * Flashing
 * User interface
 * References

Motivation
This book is intended to give a compact overview of the possibilities and usage of DD-WRT, especially for users who are not primarily concerned with the topic of routers. Therefore, it can not and shall not replace the wiki and the forum of the official website, which provide a high amount of detailed information.

In some fields requires router software specific background knowledge, so that it is not always possible to avoid using technical terms. If possible and requirded, these should be shortly explained here.

Anyone can contribute to this book.

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

title: DD-WRT


text:
Category:Book:Linear Algebra

he:קטגוריה:אלגברה לינארית pt:Categoria:Álgebra linear

title: Category:Book:Linear Algebra


text:
Nanotechnology/Overviews

=Internet Resources=

Handbooks and Encyclopedias
These are only accessible for subscribers (which is one reason this Wikibook on Nanotechnology was started):
 * Encyclopedia of Nanoscience and Nanotechnology, 10-Volume Set
 * Handbook of Theoretical and Computational Nanotechnology, 10-Volume Set
 * Springer Handbook of Nanotechnology

Websites and newsletters

 * International Council on Nanotechnology (ICON) a multi-stakeholder group dedicated to the safe, responsible and beneficial development of nanotechnology. ICON serves as an aggregator for news and research related to nanotechnology environment, health and safety.
 * ICON Virtual Journal of Nanotechnology Environment, Health and Safety compiles papers in peer-reviewed journals that address EHS issues in nanotechnology.
 * Virtual journal of nanotechnology compiles nano-related papers in peer reviewed journals that do not specialize solely in nanotechnology.
 * ACS Nanotation
 * Nanotechweb.org
 * Nanoforum
 * Nanowerk
 * nanoRISK
 * Physorg.com
 * Foresight Nanotech Institute
 * Center for Responsible Nanotechnology
 * A-Z of nanotechnology
 * Google Directory - Nanotechnology (sourced from the Open Directory Project)
 * Scientific American Nanotechnology Page
 * NanoEd Resource Portal managed by the National Center for Learning and Teaching in Nanoscale Science and Engineering (NCLT)
 * NanoHub.org
 * UnderstandingNano

Search engines
There are many ways to find information in scientific literature and some that even specialize in nanotechnology. Apart from the free search engines and useful tools such as Google scholar and Google Desktop, there are several more dedicated commercial services:
 * ICON Virtual Journal of Nanotechnology Environment, Health and Safety (VJ-NanoEHS) compiles papers in peer-reviewed journals that address EHS issues in nanotechnology.
 * ISI - Web of Science Database contains peer reviewed journals, their references and citations. It also has very useful tools such as 'find related papers' that searches for papers sharing the same references as the entry you're looking at. This is the database behind the compilation of the journal citation factors.
 * Knovel - Online Handbook Collection and Database is an extensive collection of handbooks and tables.
 * PROLA - the Physical Review Online Archive searches Physical Review journals.
 * Rubber Bible Online is a physical chemistry handbook which contains tables of physical and chemical data.
 * Spin AIP Scitation searches related journals.
 * Web of Science by ISI generates the impact factors (see journals below).
 * Virtual Journal of Nanotechnology collects nanotech related papers from non-nano specialized journals.
 * Derwent Patent database

=Peer reviewed Journals= Overview of the nanotechnology related journals and their impact factors (2007 values):

 * Impact factors are only guides to how much a papers is referenced in the years just after publication.
 * Please add comments about the journals and update impact factors!
 * The Physics and Astronomy Classification Scheme PACS2006 and Nanoscale Science and Technology - Collection of Applicable Terms from PACS 2006

=Conferences=

 * NSTI Nanotech 2008
 * TNT - Trends in Nanotechnology 2007;2006
 * MNE - Micro and Nanoengineering 2007;2008
 * Virtual Conference on Nanoscale Science and Technology
 * Foresight Unconference Vision Weekend NanoBioInfoCognoSocioPhysical technologies
 *
 * International Microprocess and Nanotechnology Conference, Japan
 * Nanosafe 2008: International Conference on Safe production and use of nanomaterials

=Nanotech Products=

Please add more products, comments and more info about the products if you have any!

See also the List of nanotechnology applications in wikipedia

Woodrow Wilsom Center for International Scholars is starting a Project on Emerging Nanotechnologies (website should be under construction at www.nanoproject.org) that among other things will try to map the available 'nano'products and work to ensure possible risks are minimized and benefits are realized.

Emerging products

 * 2008 MultiProbe’s AFM Nanoprober is now qualified for 32nm technology nodes.
 * Intel will make products with 45 nm linewidth transistors available from 2008
 * Batteries are increasingly incorporating nanostructures.
 * Flexible, cheaper, or more luminous Flat screen displays
 * Pressure-sensitive mobile devices

Available in 2006

 * Surface coatings: TCnano, Nanocover, Stay clean.

Available in 2005

 * Molybdenum disulfide catalytic nanoparticles in Brimm catalysts made by Haldor Topsøe
 * Forbes top ten nanoproducts in 2005
 * Apples IPod with sub 100nm elements in its memory chips
 * Choleterol reducing nanoencapsulated oil,Shemen Industries Canola Active.
 * Nanocrystals improve the consistency of chocolate
 * Zelen Fullerene C-60 Day Cream
 * Easton Stealth CNT baseball bat
 * Nanotex textiles once again
 * ArcticShield polyester socks from ARC Outdoors with 19nm silver particles that kill fungs to reduce odor.
 * NanoGuard developed by Behr Process for improved paint hardness.
 * Pilkingtons self-cleaning 'Activ Glass'.
 * NanoBreeze Air Purifier from NanoTwin Technologies, where the UV light from a fluorescent tube cleans the air by photochemical reactions in nanoparticles.

Available in 2004

 * Cold cathode carbon nanotube emitters for X-ray analysis by Oxford instruments[]
 * Forbes has an overview in 2004 of what they consider the top ten nanotech products:
 * Footwarmers with nanporous aerogel for 3-20 times lighter than comparable insulating materials used in shoes (produced by Aspen Aerogels).
 * Matress covers with nanotex fibres that can be washed (Simmonos bedding company).
 * Better golf drivers with carbon nanotube enforced metal composites (produced by Maruman & Co) and nanocomposite containing golf balls (produced by NanoDynamics)
 * The company 'Bionova' apparently adds some nanoproducts to their 'personalized product line'.
 * EnviroSystems make a nanoemulsive disinfectant cleaner, called EcoTru, that is EPA Tox category 4 registered (meaning very safe to use)
 * EnviroSystems also make a spray-on version of this product.
 * BASF makes a nanoparticle coating for building materials called Mincor, that reduces their wettabililty.
 * A nanostructured coating produced by Valley View, called Clarity Defender, improves visibility through windscreens in rain. Another company, Nano-Film, makes a similar coating on sunglasses.
 * Flex-Power makes a gel containing nanoscale liposomes for soothing aching muscles
 * 3M espe Dental adhesive with silica nanoparticle filler.

Available in 2003

 * NanoGuard Zink Oxide nanoparticles for sunscreens FDA approved
 * Forbes 2003 top ten nanoproduct includes:
 * High performance ski wax, Cerax Nanowax.
 * Nanotex textiles in ski jackets from Ziener
 * Nanotex textiles
 * Plenitude Revitalift antiwrinkle cream by L'Oréal contains nanocapsules with vitamin A
 * organic light-emitting diodes (OLEDs) in Sony camera flat screen display
 * Nanofilm coatings for ani-reflection and scratch resistance
 * Zink oxide nanoparticles in Sunscreen by BASF
 * carbon nanotube enforced tennis rackets and nanopolymer enforced tennis balls

Available in 2000
Nanotex makes textiles where the clothing fibres have been coating in nanoscale fibres to change the textile wettability. This makes the textile much more stain resistant.

Companies making nanotech research equipment

 * MultiProbe Manufacturer of a 1-to-6 head Atomic Force nanoprobing tool used in failure analysis, that combines multi-scan fault isolation imaging with nanoprobing electrical capabilities. For process technology node measurements of 32nm, 45nm, 65nm, 90nm or larger.
 * Veeco AFM and related equipment
 * Zyvex nanomanipulation equipment
 * Nanofactory in-situ TEM manipulation equipment
 * SmarAct nanomanipulators
 * Capres micro four point conductance measurement probes
 * ImageMetrology SPIP software for SPM analysis
 * QuantumWise software for simulating nanosystems
 * AFM and related equipment

Products that have been nanostructured for decades

 * Catalysts
 * Haldor Topsøe
 * Computer processesors are increasingly made of nanoscale systems
 * Intel

Non-nanotech products and a warning
Not everything that says nano is nano - and given the hype surrounding nanotechnology you will see an increasing number of 'nano' products that have nothing to do with it. It is worrying when sometimes problems arise with non-nano products and this adds to the 'scare' that is present in the public, fuelled by the newspapers where they are just waiting for a nice scandal... an example was the product Magic Nano from a German company that made a number of users sick when inhaling the aerosol cleaning product - which in the end turned out to have nothing 'nano' in it. There is good reason to be very alert to such issues. Not all countries have legislation in place to secure the consumers against the possible dangers present in nanoparticles and some products could end being marketed before having been tested well enough. Though this example turned out to be 'non-nano', we will probably meet new cases shortly that are truly 'nano'. On this background environmental and health aspects will be an important part of this book.

=Suppliers=

Nanomaterials
 * Sigma-Aldrich
 * Zyvex
 * overview of companies making nanoparticles and related equipment

Nanolithography
 * NIL Technology sells stamps for nanoimprint lithography (NIL) and provides imprint services.

Quantum Dots
 * Evident technologies

=A nano-timeline=

Overview of some important events in nanotechnology

See also History of Nanotechnology in Wikipedia

=A nano-scale overview=

Just to get a sense of proportion

 * Distances between objects can be measured with sub Å precision with STM, laser interferometry and its even done continuously in a standard airbag acceleration sensor chip that costs a few dollars and senses the vibrations of a micro-inertial mass element with femtometer precision (10^-15 m).

=Bibliography=
 * G. Ali Mansoori, Principles of Nanotechnology, Molecular-Based Study of Condensed Matter in Small Systems, (New Jersey: World Scientific, 2006).
 * Monthioux, Marc; Kuznetsov, Vladimir L. (2006). "Who should be given the credit for the discovery of carbon nanotubes?". Carbon 44. doi:10.1016/j.carbon.2006.03.019. Retrieved on 2007-07-26.

=References=

See also notes on editing this book Nanotechnology/About.

the nanotechnology pioneers by Steven A. Edwards Engines of Creation 2.0: The Coming Era of Nanotechnology by K. Eric Drexler

title: Nanotechnology/Overviews


text:
User:Sarephina

Hi. :o) http://med.cuny.edu xanga.com/sarephina

title: User:Sarephina


text:
User:Anastasios~enwikibooks

Hello there! My name is Anastasios. I'm a 24 year old university student living in Petroupolis, Athens, Greece. I study Electrical and Computer Engineering at the National Technical University of Athens (NTUA).

For more information, see my page in Wikipedia.

title: User:Anastasios~enwikibooks


text:
User talk:Bin Poopin

This user has placed a floating pornographic picture, Image:Strech.png, on Main Page, effectively. This is a serious hostile turn against Wikibooks and a permanent banishment would be the mildest action to recommend. Ramir 07:18, 25 September 2006 (UTC)

title: User talk:Bin Poopin


text: User talk:70.48.13.132

title: User talk:70.48.13.132


text:
User:HJHerbert

Q: How to delete a ogg-File?
Add in its description: Reason

As mentioned in Deletion_policy

Q: Which files do I want to delete from the englisch wikibooks
All ogg files, *after* I have them moved to the commons.wikimedia.org, because all that files will be used by translations of the english chinese lessions.

This applies to all ogg files out of Chinese/Lesson_1 Chinese/Lesson_2 Chinese/Lesson_3 Chinese/Lesson_4 Chinese/Lesson_5 Chinese/Lesson_6 Chinese/Lesson_7 Chinese/Lesson_8 Chinese/Lesson_9 ...

Who is HJHerbert?
He is Hjherbert at de-Wikibooks.

title: User:HJHerbert


text:
User:Alexsh

ja:User:Alexsh zh:User:Alexsh zh-min-nan:User:Alexsh

title: User:Alexsh


text:
User:Alexsh/link

 com | spec | meta | mw | incu | enwp | enwt | enwb | enwq | enwn | enws | Wikia

title: User:Alexsh/link


text:
User:Mirtone/myskin.css

body { background-color: white; }

{ margin-left: 2%; }
 * 1) bodyContent

{ margin-left: -2%; }
 * 1) bodyContent > h2

{ margin-left: -2%; }
 * 1) bodyContent > h3

{ font-weight: normal; font-size: 1.0em; margin-left: -2.0em; font-size: 0.9em; }
 * 1) siteSub

{ margin-top: 2.25em; }
 * 1) content

table { clear: both; }

h1 { font-size: 1.5em; }

h2 { font-size: 1.3em; }

h3, h4, h5, h6 { font-size: 1.1em; }

{ position: fixed; top: 0; left: 0; z-index: 4; width: 50%; padding: 0.2em; }
 * 1) p-search

 * 1) p-cactions h5, #p-search h5 { display: none; }

input[type="submit"] { border: 1px outset #369; background-color: #369; color: white; }

input[type="text"], textarea { border: 1px inset grey; }

textarea { padding-left: 0.1em; }

a:link, a:visited, a:hover { /* font-weight: bold;*/ text-decoration: underline; }

a:link { color: #369; }

a:visited { color: #636; }

a.new:link, a.new:visited { color: #739900; }

a:hover, a.new:hover { color: #CC0000; }

h1, h2, h3 { margin: 0; }

h1 { border-bottom: 1px solid #888888; }

h2 { border-bottom: 1px dotted #888888; }

{ display: none; }
 * 1) jump-to-nav

{ float: right; width: 30%; background-color: #eeeeee; padding: 0.5em; margin: 0.5em; }
 * toc

{ background-color: #eeeeee; padding: 0.1em 0.75em 0.1em 0.75em; margin: 1em 0 0.5em 0; }
 * 1) catlinks

.portlet { width: 32%; float: left; }

.portlet h5 { margin-top: 0.5em; margin-bottom: 0; border-bottom: 1px solid silver; }

{ clear: both; }
 * 1) footer

.printfooter { border-top: 1px solid black; padding-top: 1em; margin-top: 1em; }

{ position: fixed; top: 0; right: 0; width: 100% !important; background-color: ButtonFace; padding: 0.5em; padding-bottom: 3px; border-bottom: 1px solid ButtonShadow; display: block; z-index: 3; font-size: smaller; }
 * 1) p-cactions

{ display: block; margin: 0; padding: 0; text-align: right; }
 * 1) p-cactions ul

{ display: inline; }
 * 1) p-cactions li

{ display: inline; padding: 0.1em 0.3em 3px 0.3em; border: 1px outset #eeeeee; border-bottom: 1px solid ButtonShadow; text-decoration: none; color: MenuText; }
 * 1) p-cactions li a

{ background-color: white; border-bottom: 1px solid white; }
 * 1) p-cactions li.selected a

{ background-color: ButtonHighlight; }
 * 1) p-cactions li a:hover

img { border: none; }

div.thumb { border: 1px solid #dddddd; padding: 0.5em; background-color: white; }

.tright { margin: 0.25em 0 0.5em 0.5em; float: right; }

.tleft { margin: 0.25em 0.5em 0.5em 0; }

{ display: none; }
 * 1) f-poweredbyico,
 * 2) f-copyrightico

pre { background-color: #f7f7f7; padding: 0.2em; border: 1px solid #eeeeee; }

td.diff-otitle { background:#cccccc; } td.diff-ntitle { background:#cccccc; }

td.diff-addedline { background:#ccffcc; }

td.diff-deletedline { background:#ffffaa; }

td.diff-context { background:#eeeeee; font-size: 0.9em; }

span.diffchange { color: red; font-weight: bold; }

{ width: auto; clear: both; }
 * 1) p-lang

{ padding-left: 0; }
 * 1) p-lang ul

{ list-item-style: none; display: inline; }
 * 1) p-lang li

{ font-size: 0.9em; }
 * 1) footer

title: User:Mirtone/myskin.css


text:
Arithmetic

Welcome to the Arithmetic Handbook!
Arithmetic is a field of mathematics that deals with number manipulation. This book will teach you the ins and outs of arithmetic, including fractions, radicals, exponents, bases and more! Although it is recommended (and assumed, considering that you can read this) that you understand basic mathematics, you do not need to know any math to start reading this book! Just select the topmost page and start reading from there. There will be examples on the bottom of each page to help you understand the concepts further. Enjoy!

Prologue

 * /How This Textbook Is Organized/

Part I: Beginning Arithmetic

 * /Introduction to Arithmetic/
 * /Number Operations/
 * Addition
 * Subtraction
 * Multiplication
 * Division
 * /Types of Numbers/
 * /Introduction to Natural Numbers/
 * /Introduction to Operations/
 * /Zero and Numbers Greater Than Nine/
 * /Numbers Less Than Zero/
 * /The Number Line/
 * /Spelling Out numbers/
 * /Reading Decimal Numerals/
 * /Rounding/
 * /Truncating/
 * /Part I Review and Test/

Part II: Basic Arithmetic

 * /Working with Integers/
 * Addition and Subtraction
 * /Multiplication and Division/
 * /More About Multiplication/
 * /More About Multiplication/Fundamental Theorem of Arithmetic
 * /Properties of Operations/
 * /Order of Operations/
 * /Estimation/
 * /Putting It Together/
 * /Putting It Together/Equations/
 * /Putting It Together/Inequalities/
 * /Putting It Together/Aren't Inequalities Easy with Infinity?/
 * /Putting It Together/More Complicated Grouping/
 * /Putting It Together/Filling in the Blanks/
 * /Putting It Together/A Look Ahead at Algebra/
 * /Putting It Together/More Domains: A Look Ahead at Smaller Than One/
 * /Putting It Together/Review and Quiz/
 * /Part II Review and Test/

Part III: Intermediate Arithmetic

 * /A Look Ahead at the Decimal System/
 * /Introduction to Fractions/
 * /Introduction to Fractions/What Is a Fraction?/
 * /Introduction to Fractions/Fractions with Pies and Pizzas/
 * /Introduction to Fractions/Fractions with Numbers/
 * /Introduction to Fractions/Numerators and Denominators/
 * /Introduction to Fractions/The Least Common Denominator/
 * /Introduction to Fractions/Fractions on the Number Line/
 * /Introduction to Fractions/A Look Ahead: What Kinds of Numbers Are Fractions?/
 * /Introduction to Fractions/Review and Quiz/
 * /Working with Fractions/
 * /Working with Fractions/Reducing Fractions/
 * /Working with Fractions/Equality/
 * /Working with Fractions/Inequality/
 * /Working with Fractions/Improper Fractions/
 * /Working with Fractions/Mixed Fractions/
 * /Working with Fractions/Too Many Conversions!/
 * /Working with Fractions/Review and Quiz/
 * /Operations With Fractions/
 * /Operations With Fractions/Positive and Negative Fractions/
 * /Operations With Fractions/Absolute Value/
 * /Operations With Fractions/Addition/
 * /Operations With Fractions/Subtraction/
 * /Operations With Fractions/Multiplication/
 * /Operations With Fractions/Division/
 * /Operations With Fractions/Fractions as Division in Themselves/
 * /Operations With Fractions/Review and Quiz/
 * /Revisiting the Decimal System/
 * /Revisiting the Decimal System/Places as Multiplication by 10/
 * /Revisiting the Decimal System/Going Backwards: Places as Division by 10/
 * /Revisiting the Decimal System/The Decimal Point/
 * /Revisiting the Decimal System/What Does 0.1 Really Mean?/
 * /Revisiting the Decimal System/How Many Digits Long Can a Number Be?/
 * /Revisiting the Decimal System/Review and Quiz/
 * /Working with Decimals/
 * /Working with Decimals/Positive and Negative Decimals and Absolute Value/
 * /Working with Decimals/Rounding Decimals/
 * /Working with Decimals/Addition/
 * /Working with Decimals/Subtraction/
 * /Working with Decimals/Multiplication/
 * /Working with Decimals/Division/
 * /Working with Decimals/Review and Quiz/
 * /Introduction to Rational Numbers/
 * /Introduction to Rational Numbers/What Are Rational Numbers?/
 * /Introduction to Rational Numbers/A Look Ahead at Irrational Numbers/
 * /Introduction to Rational Numbers/Putting Rational Numbers in Context/
 * /Introduction to Rational Numbers/Review and Quiz/
 * /Part III Review and Test/
 * /Part III Review and Test/Review/
 * /Part III Review and Test/Test/

Part IV: Advanced Arithmetic

 * /Introduction to Exponents/
 * /Introduction to Exponents
 * /Introduction to Exponents
 * /Introduction to Exponents
 * /Introduction to Exponents
 * /Introduction to Exponents
 * /Introduction to Exponents
 * /Very Large and Very Small Numbers/
 * /Very Large and Very Small Numbers/What Is Scientific Notation?/
 * /Very Large and Very Small Numbers/Why Do We Use Scientific Notation?/
 * /Very Large and Very Small Numbers/Scientific Notation Terminology/
 * /Very Large and Very Small Numbers/Working with Scientific Notation/
 * /Very Large and Very Small Numbers/For Fun: What Are Some Really Large Numbers?/
 * /Very Large and Very Small Numbers/Review and Quiz/
 * /Introduction to Radicals/
 * /Introduction to Radicals/A Focus on Square Roots/
 * /Introduction to Radicals/What Are Radicals?/
 * /Introduction to Radicals/A Focus on Cube Roots/
 * /Introduction to Radicals/Working with Radicals/
 * /Introduction to Radicals/Domains: What Is the Square Root of 2?/
 * /Introduction to Radicals/Another Look Ahead at Irrational Numbers/
 * /Introduction to Radicals/Domains: What Is the Square Root of a Negative Number?/
 * /Introduction to Radicals/Review and Quiz/
 * /Exponents and Radicals in Depth/
 * /Exponents and Radicals in Depth/Radicals as Exponents/
 * /Exponents and Radicals in Depth/Negative Exponents/
 * /Exponents and Radicals in Depth/More Properties of Radicals and Exponents/
 * /Exponents and Radicals in Depth/Operations with Radicals and Exponents/
 * /Exponents and Radicals in Depth/Revisiting the Properties of Radicals and Exponents/
 * /Exponents and Radicals in Depth/Review and Quiz/
 * /Introduction to Real Numbers/
 * /Introduction to Real Numbers/Introduction to Irrational Numbers/
 * /Introduction to Real Numbers/Putting Rational and Irrational Numbers Together/
 * /Introduction to Real Numbers/What Are Real Numbers?/
 * /Introduction to Real Numbers/A Look Ahead: If There Are Real Numbers.../
 * /Introduction to Real Numbers/Working with Irrational Numbers/
 * /Introduction to Real Numbers/Real Numbers on the Number Line/
 * /Introduction to Real Numbers/A Look Ahead: Some Special Irrational Numbers/
 * /Introduction to Real Numbers/Review and Quiz/
 * /Putting it All Together/
 * /Putting it All Together/Types of Numbers/
 * /Putting it All Together/Equalities and Inequalities/
 * /Putting it All Together/Operations/
 * /Putting it All Together/Review and Quiz/
 * /Part IV Review and Test/
 * /Part IV Review and Test/Review/
 * /Part IV Review and Test/Test/

Part V: Beyond Arithmetic

 * /Bases/
 * /Bases/Another Look at Decimal (Base 10)/
 * /Bases/What Are Bases?/
 * /Bases/A New Base: Octal (Base 8)/
 * /Bases/Bases and Places: How They Work/
 * /Bases/A Special Base: Binary (Base 2)/
 * /Bases/Converting Between Bases/
 * /Bases/Operations in Other Bases/
 * /Bases/When You Need More Numbers: Hexadecimal (Base 16)/
 * /Bases/Operations in Hexadecimal/
 * /Arithmetic with Imaginary Numbers/
 * /Arithmetic with Imaginary Numbers/What Are Imaginary Numbers?/
 * /Arithmetic with Imaginary Numbers/Another Look Ahead at Algebra/
 * /Arithmetic with Imaginary Numbers/Working with Imaginary Numbers/
 * /Arithmetic with Imaginary Numbers/Introduction to Complex Numbers/
 * /Arithmetic with Imaginary Numbers/Working with Complex Numbers/
 * /Revisiting Types of Numbers/
 * /Revisiting Types of Numbers/Types of Numbers in Context/
 * /Revisiting Types of Numbers/Where Do Natural Numbers come From?/
 * /Revisiting Types of Numbers/Are There Other Types of Numbers?/
 * /Prime Numbers Revisited/
 * /Prime Numbers Revisited/What Are Prime Numbers?/
 * /Prime Numbers Revisited/Prime Numbers in Factoring/
 * /Prime Numbers Revisited/For Fun: Why Are Prime Numbers Important?/
 * /Arithmetic With Machines/
 * /Arithmetic With Machines/Calculators and Computers/
 * /Arithmetic With Machines/How Computers Do Arithmetic/
 * /Arithmetic With Machines/Significant Places: Precision in Arithmetic/
 * /Where to Go From Here?/
 * /Where to Go From Here?/Where Does Arithmetic Fit Into Mathematics?/
 * /Where to Go From Here?/Algebra, Geometry and Calculus/
 * /Where to Go From Here?/Number Theory and Logic/
 * /Part V Review and Test/
 * /Part V Review and Test/Review/
 * /Part V Review and Test/Test/

Appendices

 * /Comprehensive Exam/
 * /Answers to Quizzes, Tests and Exam/

Fractions

 * /Fractions/
 * /Adding Fractions/
 * /Multiplying Fractions/
 * /Percents Decimals and Fractions/
 * /Lowest Common Multiple/
 * /Highest Common Factor/

Advanced Arithmetic

 * /Absolute Values/
 * /Exponents/
 * /Radicals/
 * /Factoring/
 * /Scientific Notation/

Les quatre operacions bàsiques 算术

title: Arithmetic


text:
Arithmetic/Introduction

Arithmetic is the study of numerical quantities. At least a basic understanding of arithmetic is fundamental in the study of algebra and many other mathematical studies.

Basic Operations
There are four basic operations one can perform on different numeric quantities (as well as other mathematical objects which you will encounter later). These are:
 * addition
 * subtraction
 * multiplication
 * division

Equals
It is also important to understand that the equals sign (in this book, '$$=$$' ) means that the things on both sides are of equal value, and that $$=$$ is not an operator.

When you see an $$=$$ sign, say to yourself Same value, different appearance.

title: Arithmetic/Introduction


text:
Arithmetic/Types of Numbers

We can classify numbers into several different kinds.
 * 1) /Natural Number/
 * 2) /Integer Number/
 * 3) /Fraction/ and /Mixed Fraction/
 * 4) /Real Number/
 * 5) /Imaginary Number/
 * 6) /Complex Number/

title: Arithmetic/Types of Numbers


text:
Arithmetic/Multiplication and Division

Multiplication
Multiplication is denoted by an asterisk (*), $$\times$$, or $$\cdot$$ sign. However, the “ × ” sign is normally not used in algebra, and is instead limited to very basic elementary math, as it can easily be confused with an “x” variable. The generic multiplication operator will take any two numbers, called factors, as operands. The result is called the product of the two numbers. If the multiplicants are not both written as numbers, the multiplication sign can be left out. Thus, the following example expressions are equivalent:

$$3*\!(a*b) = 3 \times \!(a \times b) = 3\! \cdot \!(a \!\cdot \! b) = 3 \!\cdot \!(ab) = 3(a \!\cdot \! b) = 3(ab) = 3ab$$

Multiplication is a form of repeated addition. For example $$ 3 \times 5 $$ means

$$ 3 + 3 + 3 + 3 + 3 \quad \operatorname{or} \quad 5 + 5 + 5 $$

Multiplication is also commutative. This means that the multiplication of two numbers (factors) will give the same product regardless of the order in which the numbers are multiplied together. The following expressions are also equivalent:

$$3 \!\cdot \!(ab)= a \!\cdot \! (3b) = b \!\cdot \! (3a)$$

Numbers with exponents that are whole numbers larger than 1 indicate the number of factors to be multiplied, thus that number is multiplied by itself as many times as the exponent shows. Numbers with an exponent of 1 have only one factor, and therefore are equal to the number. Any number with an exponent of 0 has no factors at all, and the result is 1. Examples:

$$5^3 = 5 \times 5 \times 5 \qquad\qquad 5^1 = 5 \qquad\qquad 5^0 = 1$$

Long Multiplication
Long Multiplication is the multiplication of numbers larger than 12, but usually only the facts from 1 through 9 are used. Before you attempt long multiplication, please make sure you know the facts 1 through 9. The others are optional, but makes long division a bit easier. The steps for the vertical multiplication method are:

1. Write the numbers down.

52  19  -- 2. Multiply 9 times 2. If there is a tens place for the answer, regroup. Multiply 9 times 5, add the regroup, and write the numbers down. You should have:

1  52   19 --  468

3. Multiply 1 times 2, and 1 times 5, regrouping if needed, but this time shift the answer one space to the left. If you want to, you can put a zero under the 8 instead. You should now have:

52  19 --  468  520 4. Now add.

52 19 - 468 520 - 988

If you are multiplying decimals, then multiply without the decimal point. Count the number of decimal places in both numbers, and add the number of decimal places. In the answer, count that number of spaces to the left. Put the decimal point there.

5. In a summary:

Write the problem down, vertically. Multiply the last digit of the second number to the last digit of the first number. Regroup if there is a tens place in the answer. Multiply it by the second to last number, and ADD the regroup. Repeat the process for the second to last digit of the second number, but put a 0 at the end of the line under the number you got before, if a third line put 2 0s, and repeat until the problem is done.

Fast Multiplication
Fast Multiplication is the method in which you can reasonably multiply numbers greater than 10 and reasonably less than 1000 by simply multiplying by the method of "10's." This is done by recognizing how many digits there are in the numbers. Here are some steps which are useful for multiplying numbers really fast

1. See if you can recognize any zeros on the end and simply "add them" to your answer.

45,300 x 5 The easy way to do this is "taking away" the 2 zeros for now and reserving them for later. the number is now 453 x 5, which is much too mind boggling to do. Now here comes the interesting part of the method of "10's"

2. Break down the number into its "10's" parts What this means is basically breakup the number by its place value.

453 = 4 (hundreds place) + 5 (tens place) + 3 (ones place)

Knowing that, this become 400, 50 and 3.

3. Multiply and apply the "10's part" okay now simply multiply:

So, here is a step where we essentially take out the "0's" out for a bit and put it back in when were done.

so, its now 5 x 400. in order to make it easier, "take out" the zeroes for now and multiply 4 x 5 = 20. Now heres the magic. Since you magically took away the 2 zeroes, you will now suddenly make the 2 zeroes reappear! 20 + "00" = 2000! AMAZING! (the quotes means they're magic zeroes, and simply not the value   of zero!)

50 is done the same away. Take away the "0" and multiply 5 x 5 = 25. Now add it back, 25 + "0" = 250

Simply 3 x 5 = 15

4. Now abracadabra, Add them all together!

2000    250     15   -    2265!

5. Now take the two zeroes you reserved in the beginning (from the original 45,300), and tack them onto the end to get your answer: 226,500.

Step 2: Multiplying numbers that are not zero friendly

2102 x 52

Using the step from before recognize that:

2102 = 2 (thousand) + 1(hundred) + 0 (tens) + 2 (ones) 52 = 5 (tens) + 2 (ones)

Now, to make it easier on yourself, circle the number 2 of "52" and put it in your magic hat. (2)

Now the problem becomes 2102 x 50. Look familiar? First of all, take out the magic "0" and put it in the hat, too. Since we recognized that 50 is basically 5 with an added magical "0" to it, we now see the problem as

2102 x 5!

Now break down the bigger, uglier number and start multiplying: 2000 x 5 (take away the magic zeroes) = 2 x 5 = 10 + "000"(now put them back!) = 10,000 (notice it has 4 zeroes) 100 x 5 (take away the magic zeroes) = 1 x 5 = 5 + "00" (now put them back!) =   500 (2 zeroes) 2 x 5 (sadly, no magical zeroes)  = 2 x 5 = 10                             =    10 (1 zero)

Remember, after every step, be sure to put your friendly magical "0" back in: 10,000 + "0" =100,000      500 + "0" =  5,000      10 + "0" =    100    (notice how the number of zeroes on the left side equal the number of zeroes on the right    side)

Now add them all together:

100000   5000     100  -  105100....... That's not all yet folks! Do you remember the 2 in your magic hat? Lets get it to work:

2 x 2102 = 2000 x 2 = 2 x 2 = 4 = 4000 100 x 2 = 1 x 2 = 2 = 200 2 x 2            =    4 total: 4204

So the answer should be   105,100 4,204   ---    109,304! Wow!

Outside Resources
To practice this concept, I recommend Developmental Mathematics, volume 8, along with cards to memorize multiplication facts. This is my favorite for developing multiplication skills, but if you prefer something else, try Miquon Math. This is a complete program for grades 1-3 and is an excellent program for developing advanced math skills. Another option is Progress in Mathematics, a standards-based math program for Sadlier-Oxford. Of course, advance multiplication skills until the student/s is doing 5-digit multiplication with ease.

Division
Division uses the ÷ sign. It may also be signified by the slash(/), :, or the fraction bar. The generic division operator will take any two numbers as operands. The number before the ÷ sign is called the dividend and the number after the ÷ sign is called the divisor. The result is called the quotient of the two numbers.

Division is not a commutative operation. Switching the dividend and the divisor will likely give a different quotient (but sometimes not). The division with a divisor of 0 is not defined (to be more precise, it's indeterminate, which means that there is more than 1 answer, therefore we don't give it a value). There is no answer for it.

Example:
 * $$4 / 2 = 2$$ and $$2 / 4 = 0.5$$

Long Division
In arithmetic, long division is an algorithm for division of two real numbers. It requires only the means to write the numbers down, and is simple to perform even for large dividends because the algorithm separates a complex division problem into smaller problems. However, the procedure requires various numbers to be divided by the divisor: this is simple with single-digit divisors, but becomes harder with larger ones.

A more generalized version of this method is used for dividing polynomials (sometimes using a shorthand version called synthetic division).

In long division notation, 500 ÷ 4 = 125 is denoted as follows:

\begin{matrix} \quad 125\\ 4\overline{)500}\\ \end{matrix} $$

The method involves several steps:

1. Write the dividend and divisor in this form:

 * $$4\overline{)500}$$

In this example, 500 is the dividend and 4 is the divisor.

2. Consider the leftmost digit of the dividend (5). Find the largest multiple of the divisor that is less than the leftmost digit: in other words, mentally perform "5 divided by 4". If this digit is too small, consider the first two digits.

In this case, the largest multiple of 4 that is less than 5 is 4. Write this number under the leftmost digit of the dividend. Divide the multiple by the divisor (in this case, 4 divided by 4) and write the result (in this case, 1) above the line over the leftmost digit of the dividend.

\begin{matrix} 1\\ 4\overline{)500}\\ 4 \end{matrix} $$

3. Subtract the digit under the dividend from the digit used in the dividend (in this case, subtract 4 from 5). Write the result (remainder) (in this case, 1) underneath and in the same column, then drop the second digit of the dividend (in this case, the first zero) to the right of it. This gives you a new number to divide by the divisor.

\begin{matrix} 1\\ 4\overline{)500}\\ \underline{4}\\ \;\,10 \end{matrix} $$

4. Now apply the same steps 2 and 3 to this new number, and write the results in the corresponding columns (in this case, the unit column is aligned with the second digit of the original dividend): multiple and remainder underneath the new number, and the answer above the line.

\begin{matrix} \,\,12\\ 4\overline{)500}\\ \underline{4}\\ \;\,10\\ \quad\underline{8}\\ \quad\;\,20 \end{matrix} $$

5. Repeat step 4 until there are no digits remaining in the dividend. The number written above the bar is the quotient, and the last remainder calculated is the remainder for the entire problem.

\begin{matrix} \quad 125\\ 4\overline{)500}\\ \underline{4}\\ \;\,10\\ \quad\underline{8}\\ \quad\;\,20\\ \quad\;\,\underline{20}\\ \qquad0 \end{matrix} $$

title: Arithmetic/Multiplication and Division


text:
Arithmetic/Order of Operations

Order of operations
The order of operations is the order in which all algebraic expressions should be simplified. Oftentimes, the meaning of a complex expression changes depending upon the order in which it is calculated. The order of operations is:

 * Parentheses means brackets
 * Exponents (and Roots) means power
 * Multiplication & Division
 * Addition & Subtraction

EXAMPLE: 2 + 2 × 5 is equal to 12

This means that expressions within parentheses are evaluated first, then exponents (including roots, i.e. radicals), then multiplication and division (at the same level), and finally addition and subtraction (at the same level). If there are multiple operations at the same level on the order of operations, move from left to right.

There is a number of different abbreviations for memorizing the order. PEMDAS, BEDMAS and BODMAS (B is Brackets) are common. Another common way to remember the order is the mnemonic

 * "Please Excuse My Dear Aunt Sally,"

with the beginning letters standing for each operation. Whichever mnemonic you use, be aware that multiplication does not always come before division, and addition does not always come before subtraction. For example:

If you have an expression like

 * 3 × 3 - 5 + 2

you work like this: First notice that, there are no Parentheses or Exponents, so we move to Multiplication and Division. There's only the one multiplication, so we do that first and end up with 9 - 5 + 2. Now we move to Addition and Subtraction, working left to right. So firstly we do the subtraction to get 4 + 2, and finally the addition to give 6. If we had blindly done the addition first, we would have got the answer 2, which is wrong!

The rationale for the grouping (apart from parentheses, which are obviously first) is that multiplication is repeated addition and exponentiation is repeated multiplication. Also, division is multiplication by the reciprocal and subtraction is addition of the negative, so these operations are equivalent. In fact PEMA would be a better phrase ("Please Excuse My Aunt"), but in lower arithmetic courses MDAS is often taught without explaining reciprocals.

Parentheses are curved symbols, ( and ), that are put around part of an expression in order to convey that the expressions inside them should be evaluated first. Within a set of parentheses, the order of operations should be followed. Square brackets, [ and ], are sometimes used around parentheses to avoid confusion: [(3+5)×2]2 means the same as ((3+5)×2)2. The fraction bar and radical bar (often called a vinculum) groups expressions like parentheses.

For example, the expression 2×(6+7)-82 should be solved in the following order:

If the desired order for solving the expression were different (based on the initial problem), parentheses would be positioned differently, or even omitted.

The meaning of the fraction and radical bars must be deciphered carefully. The part of the expression directly below or above the bar is to be treated as parenthesised. (Care must be taken in writing expressions with a bar.)

The expression $$\sqrt{a + b} \times c$$ means c times the root of a + b, not the root of a + b × c or even the root of c times the sum a + b, since the bar is above the a and the b, but not the c.

The expression $$\frac{4 + 5}{1 + 2}$$ could be written in one line as (4+5)/(1+2) = 9/3 = 3, not as 4+5/1+2 = 4+5+2 = 11. As you see, the expression above the bar is evaluated, as is the expression below the bar. Finally we can divide.

Because of order of operations -22 = -(22) = -4, not (-2)2 = +4: the negative sign can be considered to have an implicit 0 in front, making the expression 0 - 22.

When it comes to distributing a power, use the raise a power to a power rule. Example: (xy^2)to the fourth power (^4) =(x)^4 (y^2)^4 =x^4×y^8 (originally it was y^6; this would only be true for y^2 * y^4, where you add the exponents)

The order of operations is very important, and you must remember the order when using simple calculators. Expressions such as 2+3 × 5 vary on the order used. Entering [2] [+] [3] [×] [5] on most calculators would result in adding three to two and then multiplying by five, resulting in 25; the proper evaluation sequence would be 2+(3×5), multiplying three and five and then adding that to two to get 17. Some scientific calculators and most graphing calculators use the proper order of operations, but four-function calculators typically use "left-to-right" evaluation, which can return incorrect results.

title: Arithmetic/Order of Operations


text:
Arithmetic/Fractions

Fractions
A fraction consists of one quantity divided by another quantity. The fraction "three divided by five" or "three over five" or "three fifths" can be written as: $$ \frac{3}{5} $$

or as:   3/5

In this section, we will use the notation 3/5. Note that some fractions have special names.

 * 1/2 - Instead of calling this a “twoth”, or a ”second”, it's called a half.
 * 3/4 - As well as three fourths, it can also be called three quarters.

Numerator and Denominator
The first quantity, the number on top of the fraction, is called the numerator. It tells you the number, or how many of something you've got. The other number, on the bottom, is called the denominator. The denominator tells you the denomination of the fraction, which is really just a fancy way of telling you the type of the fraction. In exactly the same way, we know a £5 note and a £10 pound note are different because they are different types.

So, look at a number like 3/5. The numerator is 3. So whatever we've got, we've got three of them. The denominator is 5, so we've got fifths, whatever they are. Put the two together, we've got three of those things called fifths. We've got three fifths, we've got 3/5. Same value, different appearance.

Several rules for calculation with fractions are useful:

Changing the type of a fraction
If both the numerator and the denominator of a fraction are multiplied or divided by the same number, then the fraction does not change its value.

This means we can make some fractions simpler, by making the numbers involved smaller. When the numbers are as small as possible, the fraction is said to be expressed in its lowest terms, or reduced. In a reduced fraction, the numerator and denominator are not divisible do not share any common divisors greater than one. Let's look at an example: 4/6 = 2/3.

Remember what that ' = ' sign means? It means that 4/6 and 2/3, although they look different, have the same value. Same value, different appearance.

This is because

 * $$ \frac{4}{6} = \frac{4/2}{6/2} = \frac{2}{3}$$.

Try it out yourself with a calculator or with long division. From here, there is nothing we can divide both numbers by, so the fraction is reduced, which is our goal. To find out what to divide them by, you need to find a common divisor, which is a number they are both divisible by.

Mixed fractions
Sometimes you may meet 'mixed fractions' like $$1\frac{3}{5}$$. This really means $$1+\frac{3}{5}$$, or in words one and three-fifths. Mixed fractions are hard to work with... how would you go about working out the following?

 * $$1\frac{3}{5} \times 2\frac{2}{7}$$

The answer is to convert the mixed fraction into a standard fraction of the form numerator/denominator. For example,

$$1\frac{3}{5}=1+\frac{3}{5}=\frac{5}{5}+\frac{3}{5}=\frac{8}{5}\,\,\,\,$$

and

$$2\frac{2}{7}=2+\frac{2}{7}=\frac{14}{7}+\frac{2}{7}=\frac{16}{7}$$

Now, it's simple to multiply the fractions together:

$$1\frac{3}{5} \times 2\frac{2}{7}=\frac{8}{5} \times \frac{16}{7}=\frac{128}{35}=\frac{105}{35}+\frac{23}{35}=3\frac{23}{35}$$

Keep in mind that with mixed fractions, the whole number represents the quotient whereas the fraction represents the remainder. If you were to divide 25 by 5, you would be left with only 5 (and no remainder). However, if you were to divide 27 by 5, you would be left with 5 2/5.

Exercises
 {Which of the following is the same as $$3\frac{3}{5}$$?} -5/18 -9/5 -18/3 +18/5 -11/5

title: Arithmetic/Fractions


text:
Arithmetic/Adding Fractions

Adding fractions
Before we go into fractions, let's have a think about what addition is.  {Answer these really simple questions; and don't forget units!}

{What is 1 bird + 5 birds? { 6 birds (i) }
 * type="{}"}

{What is 3 elephants + 9 elephants? { 12 elephants (i) }
 * type="{}"}

{What is 6 birds + 2 elephants? { 8 animals (i) }
 * type="{}"}
 * Hopefully you could answer the first two, but you might have had trouble with the third. Why?  Because we can't add up two different things!  However, before we add them up, we can change them into something else.  We can say that 6 birds are also 6 animals, and 2 elephants are also 2 animals, so now    6 birds + 2 elephants = 8 animals.

You probably did all this in your head without even thinking about it - so what has it all got to do with fractions?

To add or subtract two fractions, you first need to change the two fractions so that they have the same type. The simplest way to do this is to multiply the numerator and denominator of each fraction by the denominator of the other.

For instance,

$$\frac{2}{3} + \frac{1}{4} = \frac{2 \times 4}{3 \times 4} + \frac{1 \times 3}{4 \times 3} = \frac{8}{12} + \frac{3}{12} = \frac{(8+3)}{12} = \frac{11}{12}$$

A more advanced way is to use the LCM of the denominators, which will be explained later in this section. Then you can add or subtract the numerators and put the common denominator as the denominator of the solution.

Adding Mixed Numbers
There are two things you should know before you learn about how to add mixed numbers.

This is a mixed number, 5 3/4

We will be using 5 3/4 and 3 2/3 in our problem

Set = one mixed number

To add two mixed numbers, first you can turn the mixed number into an improper fraction. To do that you multiply the whole number by the denominator of your fraction. For this problem you would get 20 for the first set and 12 for the second set. Before you move on don’t forget to add the numerator to your other number such as 20+3 for the first problem. After that you change those numbers into fractions like this 23/4 and 12/3. Next you add them. You would get 35/9 after you add in this problem. Finally, you can turn that back into a mixed number by dividing the denominator by the numerator. After you divide you should get 3 8/9 as your mixed number. Also, no you cannot simplify. Now you know how to add mixed numbers!

title: Arithmetic/Adding Fractions


text:
Arithmetic/Multiplying Fractions

Multiplying fractions
To multiply two fractions:
 * multiply the numerators to get the new numerator, and
 * multiply the denominators to get the new denominator.

For instance,

 * $$\frac{2}{3} \times \frac{1}{4} = \frac{2 \times 1}{3 \times 4} = \frac{2}{12} = \frac{1}{6}$$.

Dividing fractions
To divide one fraction by another one, flip numerator and denominator of the second one, and then multiply the two fractions. The flipped-over fraction is called the multiplicative inverse or reciprocal.

For instance,

 * $$\left(\frac{2}{3}\right) / \left(\frac{4}{5}\right) = \frac{2}{3} \times \frac{5}{4} = \frac{2 \times 5}{3 \times 4} = \frac{10}{12} = \frac{5}{6}$$.

To simplify a compound fraction, like $$\frac{\left(\frac{3}{5}\right)}{\left(\frac{1}{4}\right)}$$, just remember that a fraction is the same as division, and divide (3/5) ÷ (1/4), which comes to 12/5.

title: Arithmetic/Multiplying Fractions


text:
Arithmetic/Absolute Values

Absolute value
The absolute value of a number is found by applying a simple rule: If you see a negative sign in front of the number, change it to a plus sign. If you see a plus sign, leave it alone. So, for example, the absolute value of -17 is +17. The absolute value of +36 is +36.

Another way to understand the absolute value of a number is to think about the number line:

The absolute value of a number is the distance from zero to that number on the number line.

The absolute value of x is usually written as |x|. On calculators and computers it is sometimes written as abs (x).

 {Questions: (note: if you don't see an answer box, do it on paper)}

{Calculate the absolute value of the following numbers:}

{
 * type="{}"}
 * -5|={ 5_7 }

{
 * type="{}"}
 * 9|={ 9_7 }

{
 * type="{}"}
 * -3.8|={ 3.8_7 }

{
 * type="{}"}
 * -139,462|={ 139,462|139462_7 }

{
 * type="{}"}
 * 5/8|={ 5/8|0.625_7 }

{What is the absolute value of 0?
 * type="{}"}
 * 0|={ 0_7 }

{Why?
 * type="{}" coef="2"}

{Calculate the following:}

{
 * type="{}"}
 * 27|={ 27_7 }

{
 * type="{}"}
 * -1.9|={ 1.9_7 }

{
 * type="{}"}
 * 3 - 7|={ 4_7 }

{
 * type="{}"}
 * 36| - |-11|={ 25_7 }

{
 * type="{}"}
 * 3 - 0.5|={ 2.5_7 }

{ abs (-6)={ 6_7 }
 * type="{}"}

{Draw a graph of abs(x) from -5 to +5.
 * type="{}" coef="2"}

{Can abs(x) ever be less than zero? -yes +no
 * type=""}

{How can you see that from your graph?
 * type="{}" coef="2"}
 * /Answers to "Why" and graphical questions/

title: Arithmetic/Absolute Values


text:
Arithmetic/Radicals

Radicals
A radical is a special kind of number that is the root of a polynomial equation. First, let us look at one specific kind of radical, the "square root". It is a special kind of number related to squaring.

When we have a number, say 2, what positive number will give us, when we square it, the number 2?

 * 1 × 1 = 1, so the number must be higher than 1.
 * 2 × 2 = 4, so the number must be lower than 2.
 * 1.5 × 1.5 = 2.25, so the number must be lower than 1.5.

We could continue like this forever, and with each step get closer and closer to the answers (this kind of process for refining the answer is called iteration). The number we are looking for is approximately 1.41421...

Obviously this is very difficult for us to work with, so we have a special notation. We write for a number a, $$\sqrt{a}$$ to represent the number when squared will give us a back.

Since this is the inverse operation from squaring, it can also be denoted as a1/2 and $$(a^{1/2})^2=a^1=a$$.

We can extend this idea to other kinds of radicals. $$\sqrt[3]{a}$$ indicates the number $$x$$ such that $$x^3=a$$. For example, $$\sqrt[3]{7}=1.91293...$$

Note that it is not possible to find any real number whose square would be negative. Multiplying with a negative number changes the sign of the number being multiplied and two negative signs thus eliminate each other. For example, -7 × -7 = 49 and also 7 × 7 = 49. Therefore the square root of a negative number is an undefined operation unless imaginary numbers are allowed as answers.

Simplifying Radicals
To simplify a radical, you look for the largest perfect square factor of the number under the radical.
 * $${\sqrt{162}}$$

Once you find a perfect square factor, you can express the number under the radical as the product of two factors.
 * $${\sqrt{81*2}}$$

Then you can take the square root of the perfect square factor out of the radical and place it outside.
 * $$9{\sqrt{2}}$$

Once you have factored out all perfect square factors, you have simplified the radical.

Rationalizing the Denominator
"Rationalizing the denominator" is simply taking the roots out of the denominator. This is needed because it is not proper to leave roots in the denominator. To rationalize the denominator you simply multiply the numerator and the denominator by the root that is in the denominator. For example:

$${4\over\sqrt{5}}={4\sqrt{5}\over(\sqrt{5})^2}={4\sqrt{5}\over5}$$

If you can simplify, do so.

 * $${10\over\sqrt{5}}={10\sqrt{5}\over(\sqrt{5})^2}={10\sqrt{5}\over5}={2\sqrt{5}}$$
 * $${4\over\sqrt{8}}={{4}\over{\sqrt{(4*2)}}}={{4}\over{2}{\sqrt{2}}}={{4{\sqrt{2}}}\over{2{{{\sqrt{2}}}^2}}}={{4{\sqrt{2}}}\over{4}}=\sqrt{2}$$

Addition and Subtraction
Adding and subtracting radicals can only be done when the numbers inside the radicals are the same. For instance, consider the following expression:
 * $${\sqrt{7}+\sqrt{28}}$$

You can not add those two terms as they are. However, the following equation simplifies the second term, and you get this:
 * $${\sqrt{7}+\sqrt{4*7}}={\sqrt{7}+2\sqrt{7}}$$

These terms can be added (or subtracted, for that matter). Also, coefficients of 1 are usually not written, so:
 * $${\sqrt{7}+2\sqrt{7}}={1\sqrt{7}+2\sqrt{7}}={3\sqrt{7}}$$

You can subtract radicals using the same procedure as adding, just instead of adding the coefficients, you subtract them.

The reason we can do this is the distributive property. Proving it is quite simple, and would look like this:
 * $${\sqrt{7}+2\sqrt{7}}$$

Remember that coefficients of 1 are not written, so this equation could also be written as:
 * $${1\sqrt{7}+2\sqrt{7}}$$

We then extract the term $${\sqrt{7}}$$ from the expression:
 * $${\sqrt{7}*(1+2)}$$

We all know that $${1+2}={3}$$, so:
 * $${\sqrt{7}*(3)}={3\sqrt{7}}$$

Multiplication and Division
Multiplying and dividing radicals is quite simple. Multiplication requires very little work. When you are given two radicals to multiply, all you do is multiply the numbers inside the radicals and put the product under a radical. Consider the following equation:
 * $${\sqrt{11}*\sqrt{5}}={\sqrt{55}}$$

Division is a bit different, however. One way to divide radicals that uses a concept mentioned earlier on this page is to set up the division problem as a fraction. The following equation illustrates this concept using constants (regular numbers):
 * $${{1}\div{2}}={{1}\over{2}}={0.5}$$

The same concept applies to the following equation:
 * $${{\sqrt{10}}\div{\sqrt{3}}}={{\sqrt{10}}\over{\sqrt{3}}}$$

You can then rationalize the expression $${{\sqrt{10}}\over{\sqrt{3}}}$$ using the technique found earlier on the page under Rationalizing the Denominator

Exercises
 {Simplify the Radicals}

{ $${\sqrt{50}}=$${ 5_5 }$$\surd$${ 2_5 }
 * type="{}"}

{ $${\sqrt{392}}=$${ 14_5 }$$\surd$${ 2_5 }
 * type="{}"}

{ $${\sqrt{290}}=$${ 1_5 }$$\surd$${ 290_5 }
 * type="{}"}

{Rationalize the Denominators}

{ $${2\over\sqrt{4}}=$${ 1_5 }$$\surd$${ 1_5 }
 * type="{}"}

{ $${3\over\sqrt{7}}=$${ 3/7_5 }$$\surd$${ 7_5 }
 * type="{}"}

{ $${11\over\sqrt{72}}=$${ 11/12_5 }$$\surd$${ 2_5 }
 * type="{}"}

{ $${396\over\sqrt{11}}=$${ 36_5 }$$\surd$${ 11_5 }
 * type="{}"}

{Calculate}

{ $${\sqrt{2}+\sqrt{8}}=$${ 3_5 }$$\surd$${ 2_5 }
 * type="{}"}

{ $${\sqrt{60}-\sqrt{15}}=$${ 1_5 }$$\surd$${ 15_5 }
 * type="{}"}

{ $${\sqrt{13}\times\sqrt{7}}=$${ 1_5 }$$\surd$${ 91_5 }
 * type="{}"}

{ $${{\sqrt{17}}\div{\sqrt{5}}}=$${ 1/5|0.2_5 }$$\surd$${ 85_5 }
 * type="{}"}

title: Arithmetic/Radicals


text:
Arithmetic/Estimation

Estimation
Estimation involves working out a rough answer to a calculation. The most common way to estimate a solution to a calculation is to round the numbers up or down to numbers which are easier to calculate with.

An example: Estimate the answer to 99 × 9. In this case we can easily see that 9 is almost 10, and multiplying by 10 is really easy, so we can replace our calculation with 99 × 10 which is far more easy to calculate. As we rounded a number up we know that our estimate is going to be larger than the real answer.

Why don't we just call this a guess? The difference is that a guess is just that, a completely wild guess. An estimate is based on some extra information. So whilst you might guess that 99 × 9 is something around 1000 by just pulling a number out of the air, we estimate that 99 × 9 is close to 99 × 10, then we work out 99 × 10 exactly, which gives 990 as an estimate of 99 × 9.

When we estimate, we want our estimate to be close enough to the actual value so as to be useful. (This also applies to approximation.) In our example above, the answer deviates from the actual value by about 10%, which might not be acceptable. Estimation often involves some guesswork, so we might not actually know how accurate our estimate is. (And yes, that figure 10% is itself an estimate.)

A better way to estimate 99 × 9 is to say that 99 is close to 100, then we work out 100 × 9 exactly, which gives 900 as an estimate of 99 × 9. This estimate is 1.0101...% off. The reason this is a better estimate is that 99 deviates from 100 much less than 9 deviates from 10 in percentage terms (although the absolute difference is one in each case).

title: Arithmetic/Estimation


text:
Arithmetic/Percents Decimals and Fractions

Conversions between percents, decimals and fractions
The perhaps easiest way to convert from:

Decimal to fractions is by simply moving the period over 2 and putting it over a 100

(ex) Express .50 as a fraction

.500  = 50.0 = 50/100 = 1/2    . >. (this means the moving of the dot)

usually the best way to set it all up is the following --                            | fraction | Decimal | percent | |   ?     |   ?     |   50%   |                               --            (move 2 to the right)    <--   x       -->(move dot over 2 to the right) (and put over 100)

When converting decimal to fraction                 When converting from decimal to percent always be sure it has 2 place to the                this always works right of the (.)                                    ? = 50.% if its not known, do the "reverse" (ex)  .5 = .50   = ? = 50                         .50 <--.             .-->   ---  ---                    100  100

When reading decimals, it is similar to verbalizing a fraction. For example, 0.2 is spelled out as "two-tenths", hence 2/10, which simplifies to 1/5. What if you had the real number 0.002? Same principle: "two-thousandths" implies 2/1000, which reduces to 1/500.

Previous: ../Grammar/ Next: ../Simple symbolic manipulation/ Return to: Algebra Outline


text:
Arithmetic/Exponents

Exponents
Exponents or 'powers' are a process of repeated multiplication, in much the same way that multiplication is a process of repeated addition.

Exponents are normally written in the form $$a^b$$, where $$a$$ is the base and $$b$$ is the exponent. In contexts where superscripts are not available, such as in many contexts in computers, $$a^b$$ is commonly written as "a^b" or less often as "a**b". If you're not familiar with algebra, you can just imagine the letters a and b as representing numbers. We pronounce $$a^b$$ as a to the power of b, a to the b or a exponent b.

Integer exponents
When the exponent is a positive integer, then it is just a simple case of multiplying the base by itself a certain number of times. For example,

$$3^4 = 3\times3\times3\times3 = 81\,$$

Here, 3 is the base, 4 is the exponent (written as a superscript), and 81 is 3 raised to the 4th power. Notice that the base 3 appears 4 times in the repeated multiplication, because the exponent is 4.

Some more examples:

$$ \begin{matrix} 12^2=12\times12=144\\ 2^8=2\times2\times2\times2\times2\times2\times2\times2=256\\ 1^5=1\times1\times1\times1\times1=1 \end{matrix} $$

Multiplying exponents
If you have two or more exponents with the same base, then multiplying them has the same effect as adding their exponents.

For instance $$(a^b)*(a^c)\,$$ is the same as $$a^{b+c}\,$$. For example,

$$(3^4)*(3^2)=(3*3*3*3)*(3*3)=(3*3*3*3*3*3)=(3^6)=(3^{4+2})\,$$

Dividing exponents
If you have two or more exponents with the same base, then dividing them has the same effect as subtracting their exponents.

For instance $$(a^b)/(a^c)\,$$ is the same as $$a^{b-c}\,$$. For example,

$$(3^6)/(3^2)=(3*3*3*3*3*3)/(3*3)=(3*3*3*3)=(3^4)=(3^{6-2})\,$$

Exercises
 {What is?}

{ $$4^3=$${ 64_3 }
 * type="{}"}

{ $$3^4=$${ 81_3 }
 * type="{}"}

{ $$1^{250}=$${ 1_3 }
 * type="{}"}

{ $${250}^1=$${ 250_3 }
 * type="{}"}

{Write these numbers as powers of 2}

{ $$128=2^\wedge$${ 7_2 }
 * type="{}"}

{ $$8=2^\wedge$${ 3_2 }
 * type="{}"}

{ $$1024=2^\wedge$${ 10_2 }
 * type="{}"}

{What is? $$(2^3)*(2^2)=$${ 32_2 }
 * type="{}"}

{What is? $$(2^6)/(2^2)=$${ 16_2 }
 * type="{}"}

{Harder: Why does $$3^0=1\,$$ (clue: think about $${3^2}/{3^2}\,$$, for example) (answer on paper)
 * type="{}" coef="2"}
 * /Exercise Answers/

Negative exponents
Negative exponents work slightly differently. Let's say you want to calculate $$3^{-2}$$. To do that, you take $$1/3^2$$ to get your answer. We do the exponent first, see Order of Operations

$$3^{-2}=\frac{1}{3^2}=\frac{1}{9}$$

The commutative property doesn't apply in exponents. See for yourself! Try to calculate 23, and then see if it is the same as 32 (answer here). The distributive and associative properties don't apply either.

Exponents do, however, have their own set of axioms that they consistently follow. Consistent with the preceding examples, one can state generally that:

(a^b)\times(a^c)=a^{b+c} \,\,\,\,$$ and

(a^b)/(a^c)=a^{b-c} \,\,\,\,$$

It's also easy to see that $$ \begin{matrix} (a^b)^c=a^{b\times c} \end{matrix} $$

Fractional exponents
So far, we have only seen exponents as whole numbers, but exponents can be fractional as well. With a fractional exponent, the numerator acts as a normal whole-number exponent, while the denominator acts as a root.

In general, $$a^{p/q}=\sqrt[q]{a^p}\,\,\,\,$$ for any real number $$q$$ ≠ 0.

Let's look at $$8^{2/3}$$ as an example. First, we raise 8 to the power of the numerator, 2. Then, since the denominator is 3, we take the third root of this number. The expression is read as the cubed root of eight squared, and written as:

$$8^{2/3}=\sqrt[3]{8^2}=\sqrt[3]{64}=4$$

It should then be evident that when the numerator of the fractional exponent is 1, the expression is a simple root. That is, $$1/2$$ is a square root, $$1/3$$ is a cubed root, $$1/4$$ is a fourth root, etc.

For example, $$9^{1/2}$$ would be read as the square root of nine, and written as:

$$9^{1/2}=\sqrt[2]{9^1}=\sqrt{9}=3$$

title: Arithmetic/Exponents


text:
Arithmetic/Lowest Common Multiple

To find the Lowest Common Multiple (LCM) of several numbers, we first express each number as a product of its prime factors.

For example, if we wish to find the LCM of 60, 12 and 102 we write

$$ \begin{matrix} 60=2^2 \cdot 3 \cdot 5 \\ 12=2^2 \cdot 3 \\ 102=2 \cdot 3 \cdot 17 \end{matrix} $$

The product of the highest power of each different factor appearing is the LCM.

For example in this case, $$2^2\cdot3\cdot5\cdot17=1020$$. You can see that 1020 is a multiple of 12, 60 and 102 ... the lowest common multiple of all three numbers.

Another example: What is the LCM of 36, 45, and 27?

Solution: Factorise each of the numbers

$$ \begin{matrix} 36=2^2\cdot3^2\\ 45=5\cdot3^2 \\ 27=3^3 \end{matrix} $$

The product of the highest power of each different factor appearing is the LCM, i.e.;

$$2^2\cdot5\cdot3^3=540$$

Properties of the LCM
If the LCM of the numbers is found and 1 is subtracted from the LCM then the remainder when divided by each of the numbers whose LCM is found would have a remainder that is 1 less than the divisor. For example if the LCM of 2 numbers 10 and 9 is 90. Then 90-1=89 and 89 divided by 10 leaves a remainder of 9 and the same number divided by 9 leaves a remainder of 8.

title: Arithmetic/Lowest Common Multiple


text:
Arithmetic/Highest Common Factor

The highest common factor (HCF) or greatest common divisor (GCD) can be found in a similar fashion. In this case, the product of the prime factors common to all the numbers gives the HCF.

For example, if we wish to find the HCF of 60, 12 and 102 we write

$$ \begin{matrix} 60=2^2 \cdot 3 \cdot 5 \\ 12=2^2 \cdot 3 \\ 102=2 \cdot 3 \cdot 17 \end{matrix} $$

Now the HCF is $$2 \cdot 3=6$$.

Two numbers which have an HCF of 1, for example 12 and 5, are said to be coprime - they have no factors in common (except 1).

title: Arithmetic/Highest Common Factor


text:
Biblical Studies/Christianity/Mormonism

Official church websites: Mormon.org and lds.org.

See also: ../The Church of Jesus Christ of Latter-day Saints/

title: Biblical Studies/Christianity/Mormonism


text:
Arithmetic/Properties of Operations

Why Do Operations Have Properties?
Before continuing to learn about the properties of operations you must understand two basic questions:

What is the point of operations having properties? → Without operations having properties, we would not know their usage. Understanding their usage helps lay the foundation for solving word problems later on.

Why do operations have properties? → Operations have properties which define their usage. The usage of operations is the very essence of them, without usage properties are useless (vice versa).

By understanding properties we are able to enter the realm of higher-level thinking.

This is because properties illustrate general cases which allow us to lead to more mathematical generalizations.

Definitions of Mathematical Properties:

Addition: Multiplication: In general, the inverse property of multiplication can be stated more rigorously as saying that two numbers are multiplicative inverses of each other, if and only if their product is 1.
 * 1) Commutative Property: $$a + b = b + a$$
 * 2) Associative Property: $$(a + b) + c = a + (b + c)$$
 * 3) Identity Property of zero: $$0 + a = a (= a + 0)$$
 * 4) Inverse Property: For every $$a$$, there exists a $$-a$$ such that $$a + (- a) = 0$$
 * 1) Commutative Property: $$ab = ba$$
 * 2) Associative: $$(ab)c = a(bc)$$
 * 3) Identity: $$a \cdot 1 = 1 \cdot a = a$$
 * 4) Inverse Property, for every $$a \ne 0$$, there exists a $$(\dfrac{1}{a})$$ (or $$a^{-1}$$), such that $${a}(\dfrac{1}{a}) = 1. $$

Remember that combining addition and multiplication yields the... → Distributive Rule: $$a(b + c) = ab + ac.$$

Most rules can be derived by paying attention to expressions and clever use of the properties.

Example:

Show that $$a \cdot 0 = 0$$.

Proof. $$a = a \cdot 1 = a \cdot (1 + 0) = a \cdot 1 + a \cdot 0.$$

is equivalent to:

$$a = a + a \cdot 0$$ subtract a from both sides, yielding

$$a \cdot 0 = 0$$.

QED

Properties of Addition and Subtraction
Sure, you've added 4 + 3 = 7, but have you tried 3 + 4? You get the same answer right? Yes, because of the commutative law of addition. What about 5 + 6 = 11? That's right, 6 + 5 = 11 as well.

There is another property of addition, the associative property. Here is an algebraic example:

For any numbers $$a, b, c$$: $$a + (b + c) = (a + b) + c$$

Can the same be said about subtraction? Well, let's try it... 7 - 5 = 2, does 5 - 7 = 2? Well... no. If you look at the number line, you will notice that when you do 5 - 7, you go below 0 into the realm of the negative numbers. Specifically, your solution is -2. Even though -2 is same in absolute value as 2, it isn't the same number. Therefore, subtraction is not commutative. Is it associative? No, because the associative law depends on the commutative law in order to work (since it really is just an extension of the commutative law.)

Exercises
 { 8 + 9 = { 17_2 }
 * type="{}"}

{ 50 + 30 = { 80_2 }
 * type="{}"}

{ 45 + 9 = { 54_2 }
 * type="{}"}

{ 36 + 11 = { 47_2 }
 * type="{}"}

{ 2 + 3 = { 5_2 }
 * type="{}"}

{ (5 + 7) + (4 + 3) + (8 + 2) + 5 = { 34_2 }
 * type="{}"}

{ 6 - 3 = { 3_2 }
 * type="{}"}

{ 77-11 = { 66_2 }
 * type="{}"}

{ 66 - 5 = { 61_2 }
 * type="{}"}

{ (80 - 13) - (36 - 5) - 11 + (36-11) = { 50_2 }
 * type="{}"}

Associative property
The associative property of real numbers is: $$(a+b)+c = a+(b+c) $$ for all real numbers $$a, b, c$$. This implies that the order in which addition is done does not matter.

Note however that this only applies to addition and multiplication. In fact, multiplication shares the same property. $$ (ab)c=a(bc)$$ for all real numbers $$a, b, c$$.

The position of the parenthesis does change but the order by which you operate does not.

title: Arithmetic/Properties of Operations


text:
Systems Of Logic/Mathematical Logic

Logic is the strict following of several simple rules. These rules have to do with persistence and relationships. In logic, a statement can only be true or false - there is nothing in between. Following this, a statement can only be formulated so that everyone can state whether it is true or false independently of your relationship with the statement. Lets illustrate this with some examples:
 * Statement: 1 + 1 = 2. Value: True.
 * Statement: 1 + 1 = 5. Value: False.
 * Statement: Jonny is a nice fellow. Value: ''There is no logical truth value as it depends on your relationship with Jonny. It is also not clear who Jonny is and what you mean by "nice" (friendly, honest, cooperative or a completely different meaning), so in logic you cannot give a truth value to this statement.

In logic, the truth value of a statement remains unchanged. If a statement is true, it remains true. The truth value of a statement can lead to the truth value of another statement, which in the same way remains unchanged.

Let's take a look at the relationship between two statements named p and q. The relationship is: If p is true then q is true too. (p = True => q = True) Can q be false? Not as long as p is true. What about when p is not true? (p = False) Since q is only dependent on p when p is true then q is unsolved as long as p is false.

Relationship 1:
 * p = True => q = True
 * p = False => q = unsolved

If we do not know the result of p but know the result of q, then can we tell if p is true or not? If p is true then q has to be true, but if p is not true then q is unsolved (can be true or not). Following this we can say that if q is not true then p can also not be true as a true p leads to a true q (p = True => q = True). If q is true (q = True) p can be true but it can also be not true as q is independent of p when p is False. We have now used relationship 1 to set up a new relationship.

Relationship 2:
 * q = False => p = False
 * q = True => p = unsolved

How about the persistence of p  or  q?

p is  p.  We know this much. And we know that q is q. Do we really? Well can't they change? We must define these statements as persistent. If they change, then they are defined as new statements.

Examples
What does this mean? Let's look at some real world statements. Here are two:

The sky is blue. The water is blue.

Now if the sky is blue, then is the water blue? Well in many natural areas this is the case, however in many natural areas, the water is muddy and brown or glacial and green and even these colors are a matter of perspective. Some might say that the water is clear and any color in the "water" comes from something that is not H2O at all. There are too many assumptions and conditions to state any useful logical relationship about these two statements except that they do not necessarily "follow" one another logically.

Here are two different statements:

The sky is blue. There are no clouds in the sky.

Does one of these statements follow from the other? I'll suggest that in order to be certain of something we must assume that we are in the day without special circumstances such as a dust storm, volcanic ash haze, or a solar eclipse. Then I can safely say that:

If there are no clouds in the sky then the sky is blue. If p  then  q.

What about the other way around?

If q  then  p. If the sky is blue, then there are no clouds in the sky.

I don't think that works. Have you ever seen a brilliant blue sky and one cloud in the sky at the same time? Sure! So p  doesn't follow  q.  But  q  does indeed follow  p  (under our special assumptions about volcanoes and such).

What about if q  is false? Is p?

If the sky is not blue, then there are clouds in the sky.

Yes! This seems to ring true in all cases. (Again under our special assumptions.) Logicians call this statement the contrapositive of the first statement.

Statement: If p  then  q.

Contrapositive: If not q  then not  p.

Statements
There are four types of statements in logic. A bar over a letter negates the statement. For example, "My name is Tristan" would become "My name is not Tristan". The four types of statements are the:

 * Statement: p implies q
 * $$p \to q$$

 * Contrapositive: not q implies not p
 * $$\bar{q} \to \bar{p}$$

 * Converse: q implies p
 * $$q \to p$$

 * Inverse: not p implies not q
 * $$\bar{p} \to \bar{q}$$

The statement and contrapositive are said to be "logically equivalent". The converse and inverse are also logically equivalent. What this means is that if the statement is true, the contrapositive is also true. The converse and inverse are not necessarily true, but can be. Here are some basic statements to illustrate this:

p = Today is New Year's Day. q = Today is a holiday.

The statement, which is $$p \to q$$, would read, "If Today is New Year's Day, then Today is a holiday".

The contrapositive, $$\bar{q} \to \bar{p}$$ is also true, "If Today is not a holiday, then Today is not New Year's Day".

In this case, the converse, $$q \to p$$, is not true. It would read, "If Today is a holiday, then Today is New Year's Day.

Since the converse and the inverse are logically equivolent, the inverse, $$\bar{p} \to \bar{q}$$ is not true, either. It reads, "If Today is not New Year's Day, then Today is not a holiday".

title: Systems Of Logic/Mathematical Logic


text:
Probability/The Counting Principle

The Counting Principle
Before we can delve into the properties of probability and odds, we need to understand the Counting Principle. We use the Counting Principle to determine how many different ways one can choose/do certain events. It is easier to define the Principle through examples:

Independent Events
Let's say that John is at a deli sub restaurant. There are 3 different breads, 3 different cheeses, 3 different condiments, and 3 different vegetables he can place on his sandwich, assuming that he can only place one from each category on to his sandwich. How many different ways can he set up his sandwich?

Since choosing a cheese doesn't affect the number of choices of vegetables, condiments, or bread, these events are called independent events. For this problem, we will multiply $$3\times3\times3\times3$$, so $$3^4$$ , which is 81. So there are 81 different possible combinations to form this sandwich.

Practice Problems
1) Thomas goes to a McDonalds' restaurant and chooses to create his own burger. He has 2 different kinds of cheese, 3 different breads, and 3 different sauces he can choose from, but he can only choose one of each category. How many different ways can he create this burger?

2) Diane is ordering pizza for her family. There are 4 different possible sizes of the pizza. Also, she has to choose one of 5 toppings to place on the pizza and one of 3 different types of cheese for the pizza. In addition, she must choose one of 3 different kinds of crust. How many different ways can she have her pizza?

3)
 * a) How many 3-digit numbers can be formed from the digits 2, 3, 4, 5, 7 and 9?

 * b) How many of these numbers are less than 400?

Dependent Events
Assume that John is now working at a library. He must put 5 books on a shelf in any order. How many different ways can he order the books on the shelf? Unlike the independent events, when John puts a book on the shelf, that eliminates one book from the remaining choices of books to put next on the shelf; thus these are referred to as dependent events. At first, he has 5 different choices, so our first number in the multiplication problem will be 5. Now that one is missing, the number is reduced to 4. Next, it's down to 3, and so on. So, the problem will be
 * $$(5)(4)(3)(2)(1)$$

However, there is a symbol for this very idea. A $$!$$ represents the term factorial. So, for example, $$3!=(3)(2)(1)$$. Factorials are very useful in statistics and probability.

Therefore, the problem can be rewritten as $$5!$$, which ends up being equal to 120.

However, not all dependent event problems are that simple. Let's say that there are 10 dogs at a dog competition. How many different ways can one select the Champion AND the Runner-Up? This problem could actually be considered simpler than the last one, but it doesn't involve a factorial. So, how many different ways can the judge determine the Champion? Of course, there are 10 different dogs, so 10 different ways. Next, how many dogs are left to select the Runner-Up from? Well, you removed one, so it's down to 9. Instead of placing a factorial, though, you will only multiply 10 and 9, resulting in 90.

Independent Or Dependent?
To help you differentiate between the two, we will do a few more examples, but we will have to decide if it is dependent or independent before solving the problem.

Let's say that you are creating a 5-digit garage door opener code (the digits would include 0-9). If there were absolutely no restrictions, would the events be independent of each other or dependent on each other? Of course, there are no restrictions, since you could have five 4's for the code, so to solve the problem, you multiply 10 by itself 5 times, resulting in 100000.

Alternatively, suppose that the first number of the code cannot be 0, and that there can be no repeated numbers whatsoever. Obviously these events are dependent on each other, because there cannot be any repetitions. Let's look at this problem one number at a time.

The first number can be all the numbers except 0, reducing the possible numbers to 9. The second number can be 0 this time, so the possible numbers returns to 10. However, it cannot be a repeat of the previous number, so there are 9 possible choices again. After that, the numbers will reduce by one each time, due to the fact that there cannot be repeats, so the problem would look like this
 * $$(9)(9)(8)(7)(6)=\frac{(9)(9!)}{(5!)}=27216$$

Now, just one more example. Let's say that you were choosing your schedule for school. There are 8 periods each day, and there are 7 classes to choose from. Nonetheless, you must have a lunch period during the 4th period. We can think of 4th period as non existent because it is in a constant position and therefore does not affect the possible choices. With 7 slots and 7 options, the answer is simply $$7!$$.
 * $$(7!)=5040$$

Review Of The Counting Principle
So, we use the Counting Principle to determine the different unique ways we can do something, such as a sandwich or a selection of classes. Sometimes, these events will affect each other, such as when you can't choose the same number twice for your garage door code, so they are dependent events. However, other times, one event has no effect on the next event, such as when you have different cheeses and breads to choose for your sandwich, so they are independent events. The Counting Principle is a fundamental mathematical idea and an essential part of probability.

Counting Rules
Rule 1: If any one of $$k$$ mutually exclusive and exhaustive events can occur on each of $$n$$ trials, there are $$k^n$$ different sequences that may result from a set of such trials.
 * Example: Number of possible sequences resulted from flipping a coin three times is $$k^n=2^3=8$$.

Rule 2: If $$k_1,\dots,k_n$$ are the numbers of distinct events that can occur on trials $$1,\dots,n$$ in a series, the number of different sequences of $$n$$ events that can occur is $$k_1\times\cdots\times k_n$$.
 * Example: Number of possible sequences resulted from flipping a coin and roll a six-faced die is $$(k_1)(k_2)=(2)(6)=12$$.

Rule 3: The number of different ways that $$n$$ distinguishable things may be arranged in order is $$n!=(1)(2)(3)\times\cdots\times(n-1)(n)$$, in which $$0!\overset{\text{def}}=1$$. An arrangement in order is called a permutation. The total number of permutations of $$n$$ objects is $$n!$$ (the symbol $$n!$$ Is called n-factorial).
 * Example: Number of possible ways to arrange 10 items in order is $$10!=10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1=3,628,800$$

Rule 4: The number of ways of selecting and arranging $$k$$ distinguishable objects from among $$n$$ distinct objects is: $$\frac{n!}{(n-k)!}$$, or as seen on calculators, [nPr].
 * Example: Number of possible ways to pick 3 things from 10 items, and arrange them in order is $$\frac{10!}{(10-3)!}=\frac{10!}{7!}=720$$.

Rule 5: The total number of ways of selecting $$k$$ distinct combinations of $$n$$ distinguishable objects, irrespective of order (i.e. order is NOT important), is: $$\binom{n}{k}=\frac{n!}{k!(n-k)!}$$ or as seen on calculators, [nCr].
 * Example: Number of possible ways to pick 3 items from 10 items irrespective of order is $$\binom{10}{3}=\frac{10!}{3!(7!)}=\frac{720}{6}=120$$.

title: Probability/The Counting Principle