text:
Wikibooks:Mathematics bookshelf

Basic Math

 * Arithmetic
 * Applied Math Basics
 * Beginning Mathematics
 * Introduction to Mathematics
 * Mental Math
 * Prealgebra for Two-Year Colleges
 * Using an Abacus

Elementary

 * Algebra I - A Verbose Approach
 * Basic Algebra

Intermediate

 * Algebra
 * Intermediate Algebra
 * Theory of equations
 * Prealgebra for Two-Year Colleges

Geometry

 * Geometry for Elementary School
 * Geometry
 * Conic Sections
 * High School Mathematics Extensions

Trigonometry

 * Trigonometry

Discrete Mathematics

 * Discrete mathematics
 * Combinatorics
 * Number Theory
 * Fermat's last theorem

Calculus

 * Calculus
 * Calculus optimization methods

Statistics and Probability

 * Probability
 * Statistics
 * Handbook of Descriptive Statistics

Higher Mathematics

 * Problems in Mathematics

Foundations

 * Set Theory
 * Category theory
 * Mathematical Logic
 * Mathematical Proof
 * Finite Model Theory

Analysis

 * Real Analysis
 * Complex Analysis
 * Functional Analysis
 * Residue Theory
 * Measure Theory
 * UMD Analysis Qualifying Exam

Linear Algebra

 * Linear Algebra

Advanced Algebra

 * Abstract Algebra
 * Ring Theory
 * Introduction to Rings and Algebras

Geometry and Topology

 * Algebraic Geometry
 * Crystallography
 * Topology
 * Combinatorial Topology
 * Differential Geometry
 * Amateur's Guide to Proving the Four Color Theorem

Differential Equations

 * Differential Equations
 * Linear Algebra with Differential Equations
 * Partial Differential Equations

Applied Mathematics

 * Applied Mathematics
 * Introduction to Game Theory
 * Numerical Methods
 * Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)

Program Specific Math

 * A-level Mathematics
 * Advanced A-level Mathematics
 * VCE Mathematical Methods

Misc.

 * Solutions To Mathematics Textbooks
 * Programmed Math
 * Vedic Mathematics
 * Applicable Mathematics
 * Fractals
 * The Book of Mathematical Proofs
 * Solved Question Papers - IIT JEE

cy:Silff lyfrau:Mathemateg de:Regal:Mathematik he:מדף:מתמטיקה it:Wikibooks:Ripiano Matematica 数学 Wiskunde 数学书架

title: Wikibooks:Mathematics bookshelf


text:
Wikibooks:Departments

cy:Wicillyfrau:Holl silffoedd llyfrau de:Wikibooks:Bücherregale fr:Wikilivres:Tous les livres he:ויקיספר:כל מדפי הספרים pl:Wikibooks:Księgozbiór pt:predefinição:Biblioteca

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

title: Wikibooks:Departments


text:
Talk:Organic Chemistry/Foundational concepts of organic chemistry

What would be a better title than "foundational concepts" ? I want to communicate that these things are relatively basic and that the person should be familiar with them before moving on. It is pretty much a review of the relevent topics that are covered in general chemistry .. any suggestions please ? --Karl Wick

What about "Fundamental concepts"?


text:
GCSE Science

Welcome to the world's foremost GCSE Science Textbook on the web!

This General Science book is aimed at GCSE students rather than university students.

Contents by Topic
/Introduction/

Biology

 * Cells
 * Plants
 * Photosynthesis and Plant Respiration
 * Digestion
 * Breathing and Respiration
 * The heart and circulation
 * Hormones & The menstrual cycle
 * Disease
 * DNA, Inheritance, Selection, Evolution and Competition
 * Environment
 * Plant Sensitivity and Transport
 * The Nervous System
 * Ecology
 * Genetics
 * Homeostasis

Chemistry

 * Equations and Formulae
 * Atoms and Bonding
 * The Periodic Table
 * Ionic Compounds
 * Simple Molecular Substances
 * Giant Covalent Substances
 * Metals and Non-Metals
 * States of Matter
 * Separation and Purification
 * Acids & Bases, Salts and Electrolysis
 * Transition Metals
 * Periodic Table Groups
 * Rates of Reaction
 * Fuels
 * Earth Science
 * Chemistry Calculations
 * Organic Chemistry

Physics

 * Electricity
 * Forces and Motion
 * Waves
 * Radioactivity
 * Energy
 * Light and the Electromagnetic Spectrum
 * Astronomy
 * Magnetism and Motor Effect
 * Electromagnetic Induction
 * Particle Model

Contents by Modules
The GCSE exam can be modular or combined. It is sensible however to divide the subject into modules despite the exam system used. Although this page is set out using the English system, students from other nations will still find much of the material relevant.

AQA

 * Humans as Organisms
 * Maintenance of Life
 * Metals
 * Earth Materials
 * Energy
 * Electricity

OCR

 * BD1 Supplying the cell
 * BD2 Control in Animals and Plants
 * BD3 Ecology
 * CD1 Rates of Reactions
 * CD2 Energy in Chemistry
 * CD3 Rocks and Metals
 * PD1 Waves in Action
 * PD2 Energy in the Home
 * PD3 Forces and Motion

AQA

 * Environment
 * Inheritance and selection
 * Waves and Radiation
 * Structures and Bonding
 * Forces
 * Patterns of chemical change

Coursework

 * GCSE Science/Coursework marking system
 * GCSE Science/Coursework advice

Common coursework experiments

 * GCSE Science/Osmosis in potato slices coursework
 * GCSE Science/Temperature dependence of enzymes coursework
 * GCSE Science/Stretch of an elastic band coursework
 * GCSE Science/Voltage/current characteristics of a bulb coursework
 * GCSE Science/Rates of reaction coursework
 * Carbon dioxide production of yeast as affected by temperature

title: GCSE Science


text:
GCSE Science/Electricity multiple choice

Questions List

 * Question 1
 * Question 2
 * Question 3
 * Question 4
 * Question 5
 * Question 6
 * Question 7
 * Question 8

Question 5

A student rubs a polythene strip with fur and suspends it from a clamp stand. She then rubs another polythene strip with fur and brings it up to the first strip. The two strips repel each other.

Look at the following four statements:

 * A) One of the strips is neutral, the other charged.
 * B) The strips are both negatively charged.
 * C) The strips are both positively charged.
 * D) The strips are both neutral.

Which two of the above statements are false?

 * A and B

 * A and C

 * A and D

 * B and C

 * B and D

 * C and D

Question 6

A student decided to silver plate his/her mother's forks. S/he set up the apparatus above. S/he set the power pack and variable resistor so that the ammeter read 0.1 A. S/he allowed the fork to remain in the electrolysis apparatus for 10 min. Once s/he removed the fork s/he tested it by trying to scratch the silver off. S/he found that the layer of silver was too thin and decided that s/he would have to take steps to make the layer a lot thicker.

Which one of the following steps would not produce a thicker layer of silver ?

 * Increasing the voltage of the power pack

 * Increasing the resistance of the variable resistance

 * Increasing the time spent in the apparatus

 * Increasing the concentration of the silver nitrate solution

Question 7
What is the maximum power that an appliance should have if it is connect to a 230V supply by a 5A cable ?

 * 46W

 * 1.15 kW

 * 115W

 * 4600W

Question 8

The graph above shows how the potential difference across a conductor varied with the current flowing through it. Which of the following statements is true ?

 * The resistance of the conductor increases as the current increases

 * The resistance of the conductor decreases as the current increases

 * The resistance of the conductor stays constant as the current increases

 * It is not possible to say anything about the resistance of the conductor based on the information given

Question 9
Two students, Jane and John are working together to make an electromagnet. John coils some plastic coated wire around a pencil and attaches the two ends to a 1.5V cell. He finds the electromagnet that he has made will deflect a compass placed near it but will not pick up a paperclip.

Jane suggest that using an iron nail rather than a pencil will make the magnet stronger. Is she correct, and why?

 * She is correct, the iron core will make the magnet stronger, because iron conducts electricity.

 * She is correct, the iron core will make the magnet stronger because iron is a magnetic material.

 * She is incorrect, the iron core will not make the magnet stronger, because iron conducts electricity.

 * She is incorrect, the iron core will not make the magnet stronger because iron is a magnetic material.

Question 10
Jane and John have made an electromagnet using a soft iron core with 10 windings of insulated copper wire and a voltage of 1.5V. They found that they could pick up 3 paper clips with this magnet if they were careful. Which of the following changes would not result in more paper clips being picked up?

 * Increasing the voltage (and therefore the current).
 * Using 15 turns of wire instead of 10.
 * Reverse the connections to the battery so that the current flows in the opposite direction.
 * Add a resistor in series with the electromagnet.

Question 11
A photocopying machine creates a picture by making use of static electricity. Put the following statements about how the process works in the correct order.

 * 1) Paper is given an opposite charge to the toner and pressed onto the plate. This causes the toner to stick to the paper rendering a copy of the picture.
 * 2) A plate is given a charge, an image is projected onto the plate. Where the image was dark, a dark area forms on the plate.
 * 3) Particles of toner are sprayed over the plate and are attracted to the charged areas, but not the neutral areas.
 * 4) Light shining on the plate gives the charges enough energy to leak away. The dark areas of the plate stay charged.

 * 1 2 3 4
 * 3 2 1 4
 * 4 2 3 1
 * 1 3 2 4
 * 1 3 4 2
 * 2 1 4 3
 * 2 4 3 1
 * 2 4 1 3

(Don't see your order in among the options? Check again because the correct order is certainly in there).

Question 12
A wire is connected to a power pack and placed in the magnetic field of two bar magnets. When the power pack is switched on the wire jumps upwards.

What effect will increasing the current in the wire have ?

 * There will be no effect because the magnets haven't been changed.
 * The wire still moves upwards, but this time the greater current causes it to move more slowly.
 * The wire will now move downwards.
 * The wire will move upwards more quickly than before.

Question 13
This is the same setup as in question 12.

What can be done to make the wire move downwards ? (Choose two methods from the list)

 * increase the current

 * reverse the current

 * make the wire shorter

 * reverse the magnetic field

External Question papers

 * http://www.aqa.org.uk/qual/gcse/sci_da_m_assess.html Requires adobe acrobat viewer.


text:
GCSE Science/Electricity multiple choice practice question 1

A kettle is fitted with a 13A fuse which one of the following statements is correct.

 * The fuse's main purpose is to protect a person should the live wire touch the metal case
 * The current in the live wire is kept at 13A by the fuse
 * The fuse is a special sort of resistor. It keeps the current below 13A by resisting any extra current
 * If the current goes above 13A the fuse will melt, and so break the circuit


text:
User:Alex756~enwikibooks

You can find out more about me at: Alex756 @ Wikipedia

title: User:Alex756~enwikibooks


text:
Spanish/Exercises/Comparisons

^Lesson 5^

Llena los blancos. Fill in the blank
1. Mario es __________ alto como Luigi.

2. El Sr. Perez tiene __________ perros como Sr. Gonzales.

3. Serena es __________ fuerte que Venus.

4. Mi carro cuesta __________ dinero como tu carro.

5. Los niños juegan __________ horas que los mayores.

6. México es __________ hermoso como España.

7. El abuelo tiene __________ pelo como su hijo.

8. Los gatos son __________ atrevidos como los tigres.

9. Argentina tiene __________ personas como Colombia.

Soluciones a los ejercicios Solutions to exercices

^Lesson 5^

title: Spanish/Exercises/Comparisons


text:
User talk:Formulax~enwikibooks

Heya :) I've moved some of the stuff that you moved to Alg II to Discrete mathematics, notably the permutations and combinations stuff. Just thought you'd like to know...thanks :) Dysprosia 09:48, 25 Sep 2003 (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 Formulax. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Formulax~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 23:12, 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) 04:59, 19 April 2015 (UTC)

title: User talk:Formulax~enwikibooks


text:
GCSE Science/Electricity multiple choice practice question 1 answer 2

Sorry this is the wrong answer

The current is determined by the voltage and the resistance of the kettle element. The fuse can't contol the current.


text:
GCSE Science/Electricity multiple choice practice question 1 answer 3

Sorry this is the wrong answer.

Although the circuit symbol for a fuse looks a bit like a resistor, inside a fuse is just a piece of wire. It can't keep the current down to below 13A in this way.


text:
GCSE Science/Electricity multiple choice practice question 1 answer 4

Well done! This is the correct answer!

Next question


text:
GCSE Science/Magnetic effects of a current

When a current flows down a wire it creates a magnetic field. To see this, place a small plotting compass near the wire and turn the current on. The compass needle will deflect. The shape of the magnetic field around a wire is circular. Look at the diagram on the right. The wire is coming straight out of the screen so you only see it's cross section (the red circle) The plotting compasses show how the field wraps around the wire.

Creating an electromagnet
Solenoids are made by coiling wire. The magnetic field of a solenoid looks like the field of an ordinary bar magnet.

A solenoid makes a pretty weak bar magnet on its own, but if a piece of iron is put inside the solenoid, the field becomes much much stronger.

Try the following experiment:
 * Wrap a piece of plastic coated wire around a pencil.Remove the pencil then connect to a power supply set at 1V DC current. Sprinkle iron filings around the coil of wire. Put a plotting compass near the wire. Try to pick up a paper clip.
 * Wrap a piece of plastic coated wire around a large nail. Repeat all of the above experiments

You will find that the solenoid with the iron core is much much stronger than the one with the air core.

Q1) Why is it important to use plastic-coated wire?

Making the field stronger
To make the field stronger we can:
 * Use a soft iron core
 * Increase the amount of current flowing in the wire
 * Increase the number of turns on the solenoid.

Reversing the field
To make the north pole and south pole swap positions we can:
 * Reverse the direction of the current by swapping the positive and negative leads on the power pack.
 * Wind the wire the other way round. {So if you wound it clockwise, take it off and wind it counterclockwise.

Speaking of north and south poles, there is a little trick to help find out which end is the north pole and which is the south.

Look down the solenoid and work out which way the current is flowing. Remember, current flows from the positive to the negative terminal on the power pack. If the current is flowing clockwise, the end will be a south pole. If it is going counterclockwise, the end will be a north pole. The easy way to remember this is to put arrows on the end of a capital N and S like this:

If you do not understand how to determine the pole using the clockwise and anti-clockwise way you can use the right hand grip rule


text:
Talk:Geometry

Prerequisites for geometry readers
I love math but I know the frustration of trying to learn math by myself when the source is obtuse, ambiguous and incomplete. Wiki's geometry readers might not have teachers or tutors to help them. We should specify particular prerequisites for such readers. We should build only on material already presented in previous more elementary wikibooks.

 * In this case that means Geometry for elementary school which is great. Some arithmetic and simple algebra is needed, but unfortunately I haven’t found a good wikibook source that isn't too advanced.

--Cretaceous9 (talk) 14:54, 5 January 2009 (UTC)
 * Each new paragraph should be tested to be sure it is built on these basic assumed skills plus what was taught in the previous paragraphs. It is easier for a well prepared reader to deal with a presentation that is too basic in the first few chapters than for someone new to geometry to bridge that gap when advanced material is introduced too early.

Aligning with California Standards
Okay I like the concept aligning with California standards and creating a High School Level textbook. Nothing has been done in awhile. I am thinking about taking this up. My only qualification is that I teach this subject in a CA highschool.

A great looking intro page. I can't wait 'till it's finished as I really want to learn about advanced geometries. Theresa knott 22:25, 26 Sep 2003 (UTC)

Who has a right to dictate the syllabus?
However, why is the book set up so the contents are set in stone? Why did I only have that chapter available to edit for where I wanted to define R^3? I am asking not just because it seems hard to find out, but because it might help me as a student but also tutor who may one day more formally teach math. If there is some sort of 'standard curricula' in academia I need to know who thinks they are qualified to make it up and why.--Dchmelik (talk) 23:55, 26 September 2009 (UTC)

Book 1 / Book 2 Structure
This book is gaining a similar structural problem to what the Trigonometry book had - advanced material being added that is beyond a high school level. The solution there was to split the ToC into two parts, a book 1 and a book 2. I think the same could and should be done for this book. JamesCrook (talk) 11:35, 13 November 2010 (UTC)

Categorisation of pages, please
Please fix the links so that each chapter of this book is not categorized as a whole book of its own! --Daniel 09:47, 3 May 2004 (UTC)

More illustration, please
Seconded.DroEsperanto 12:24, 21 June 2006 (UTC)
 * A book about Geometry with few or none ilustrations...

Appendix for definitions and postulates, please
I think there should be an appendix listing out all the major postulates and definitions, grouping them by topic (e.g., SSS and SAS would go in "Triangle Congruency," whereas the Corresponding Angles Postulate and that parallel-transitivity property would go in "parallel lines."DroEsperanto 12:24, 21 June 2006 (UTC)

Appendix for Theorems, please
I think there should also be an appendix listing all of the important theorems, also grouped by categories. I think there should also either be a list of the theorems with their proofs or, since that might be seen as helping students cheat, doing what my old geometry textbook did: having a two-column proof chart with only some of the boxes filled in, and having the student fill statements for a given reason or reasons for a given statement to have the complete the proof.DroEsperanto 12:24, 21 June 2006 (UTC)

Vocab, please
I think there should be bolded vocabulary words throughout the text, and a glossary. These don't need to be the precise words used by California (we should try to keep this pretty even and not tailored to one state!), but whatever comes up.--HereToHelp (talk) 02:45, 22 October 2006 (UTC)

I'm adding information as I go
I'm starting to add information on some of the subjects listed in geometry. --Isipeoria 01:08, 23 December 2005 (UTC)

Possibility of students working on this
I am a teacher in a small high school and would like to use this text next semester and have my students help edit and illustrate as we work through the subject. I am new to Wikiland and would appreciate any comments or suggestions. Is anyone working on this subject intensely right now? --Forrestbrinker 18:41, 2 December 2006 (UTC)Forrestbrinker

What styles should new terms be in?
I want to know how the style of the book is outlined. Should new terms or definitions be bold or italicized or underlined. Musical Inquisit (discuss • contribs) 22:30, 8 July 2019 (UTC)

Chapter Order
Hello, I'm currently editing "Parallel Lines, Quadrilaterals, and Circles" and noticed that it comes before polygons. I think polygons should come before it and before triangles since a discussion of either requires a knowledge of what a polygon is.

Exercises Too Complex
The exercises at the end of chapter 2 require concepts that have not yet been introduced. These should be simpler exercises that clearly explain the basic proof concepts introduced in chapter 2. I'm going to change these exercises shortly, if nobody objects.

Euclid Old Fashioned
The five postulates of Euclidean geometry? Which set of postulates are you using? Euclid's? That's has been shown not to stand up to rigorous standards a long time ago...

Isn't it a bit old-fashioned to teach 2D Euclidean geometry along the lines of Euclid's elements?

I agree. And I believe that Euclid's fifth "postulate" was derived from his other four in the 19th century.DroEsperanto 12:24, 21 June 2006 (UTC)

Euclid's fifth postulate
Euclid's Fifth was shown to be independent of (i.e., not derivable from) the other four axioms by the work of Lobachevsky, Bolyai, and Riemann, and secretly by Gauss, in the 19th century, a result reconfirmed by many others in the years since. Although Euclid did not have the insight we have today, Euclidean Geometry is every bit as good a tool for helping build a working understanding of deduction as it has been for the past 2300 years. It is the responsibilty of the instructors and modern mathematicians to point out and amend the program as needed.

Seriously try Euclid
The original The Elements of Euclid, and the best. I wish I had this when I was learning geometry at school. It is crystal clear and free of "pedagogical" techniques. It takes time to gain any mastery over the content, but it is better then any textbook I have had to suffer through in 16 years of schooling (up to Bachelors degree). As a bonus it's in the public domain. --41.243.101.252 (talk) 14:39, 19 August 2008 (UTC)

More modern than Euclid
I think I agree with the poster that says 'seriously try Euclid,' except I wrote the chapter 11 that defines R^3, etc., so I think there is value in a geometry text using modern set theory--surely it is necessary now that Topology exists or just that some teachers and even students think it is worth it to them.--Dchmelik (talk) 23:55, 26 September 2009 (UTC)

Geometry ..
.. зиждетца .. как бы на костях. Без суставов лишних! Ну и конечно .. от ненужных соблазнов и в стороны - не имеющие какой-либо ?? толщины !! .. вооот .. А далее - от брассываем и мясожирыблоньикровьигной .. бррр .. Нет мяса! Нет "буквы" "м" ! Нет "метра" длины. Только "цифра". Теперь - всё хорошо! Всё отлично!! Всё по-вы-брос-или !! ))) ..

А дальше всё - элементарно! Числим! Вся "Геометрия" - одна "теорема Пифпафпуфгора"-а .. Плоская гора. Без объёма! Потомушто длина, плющщадь, мясса и объём - весь этот ненужный хлам как бы "жизни", но! - реально грязного мозгоуродства .. Ура! Вперёд! Даёшь семиПифагоровы числа! 13! 21! 10! Нискоко!! )))

P.S. Vыccocov..or. "Шизофрения" всё это. Нельзя отдалять себя от жизни. Жизнь всегда больше нашей Галактики и других. Што есть "на самом деле" её жизнь ? Ну и конешно в "окружении" всех иных. Не "тел". Тело живёт своей жизнью. "И у опарыша есть тело" - говорил великий Черчиль ) .. "Я - да и не видел?!" - возмущался великий Дарвин. И плюнув на трибуну ! уходил .. ) !! .. О которых мы ничего не знаем! Ну и я пойду .. 176.59.208.115 (discuss) 06:32, 2 June 2020 (UTC)

title: Talk:Geometry


text:
United States Government

Part I- Before the Constitution

 * 1) Colonial Government in America
 * 2) The United States under the Articles of Confederation
 * 3) /The Constitutional Convention/
 * 4) /Ratification/

Part II- The Constitution

 * 1) The Three Branches of Government
 * 2) /The Federal System/
 * 3) /General Provisions/
 * 4) /The Bill of Rights/
 * 5) /The Later Amendments/

Part III- Present Government Structure

 * 1) /The Legislative Branch/
 * 2) /The Executive Branch/
 * 3) /The Judicial Branch/

Part IV- American Government: Theory and Analysis

 * 1) Criticism of the Legislative Branch
 * 2) Criticism of the Executive Branch
 * 3) Criticism of the Judicial Branch
 * 4) /Federalism and State Authority/
 * 5) /Constitutional Issues/
 * 6) /Civic Society/
 * 7) /Foreign Policy/
 * 8) /Domestic Issues/

Part V- Appendices

 * 1) /Civics Glossary/
 * 2) /The Declaration of Independence/
 * 3) /The Articles of Confederation/
 * 4) /The Constitution and Amendments/
 * 5) /The Annotated Constitution of the United States/
 * 6) List of Authors

title: United States Government


text:
United States Government/The Three Branches

Introduction
The United States Constitution divides government into three separate and distinct branches: the Executive, Legislative and Judicial branches. The concept of separate branches with distinct powers is known as "separation of powers." That doctrine arose from the writings of several European philosophers. The Englishman John Locke first pioneered the idea, but he only suggested a separation between the executive and legislative. The Frenchman Charles-Louis de Secondat, Baron de Montesquieu, added the judicial branch.

Each branch is theoretically equal to each of the others. The branches check each others powers and use a system known as  checks and balances. Thus, no branch can gain too much power and influence, thus reducing the opportunity for tyrannical government.

The Preamble to the American Constitution sets out these aims in the general statement:

"'We the people of the United States, in order to form a more perfect union, establish justice, insure domestic tranquility, provide for the common defense, promote the general welfare, and secure the blessings of liberty to ourselves and our posterity, do ordain and establish this constitution for the United States of America'."

The Legislative
The Congress is the Legislative Branch. Its main function is to make laws. It also oversees the execution of these laws, and checks various executive and judicial powers. The Congress is bicameral - it is composed of two houses. One house is the House of Representatives and the other is the Senate.

The House of Representatives is currently composed of four hundred and thirty-five members. Each of the fifty states is allocated one or more representatives based on its population which is calculated on a decennial basis (once in ten years). Each state is guaranteed at least one representative. A state that is allocated more than one representative divides itself, as state procedures dictate, into a number of districts equal to the number of representatives to which it is entitled. The people of each district vote to elect one representative to Congress (States that have only one representative allocated choose at-large representatives - the state votes as one entire district). The District of Columbia and a number of U.S. territories have been permitted to elect delegates to the House of Representatives. These delegates may participate in debates, and sit and vote in committee, but are not allowed to vote in the full House. Every House member faces re-election in an even-numbered year and is elected to a two-year term. The House is presided over by a Speaker, who is directly elected by the members of the House.

The Senate is the upper house of the legislative branch of the United States and possesses one hundred members which is considerably less than the four hundred and thirty-five members of the House of Representatives. Each state chooses two senators, regardless of that state's population. The Constitution originally dictated that a state's senators were to be chosen by the state's legislature; after the Seventeenth Amendment was ratified in 1913, senators were elected directly by the state's population. In contrast to the House's two-year terms, Senators are elected to a six-year stint in office. In addition, only one-third of the Senate stands for election during an even year. These differences between the two houses were deliberately put into place by the Founding Fathers; the Senate was intended to be a more stable, austere body, whereas the House would be more responsive to the people's will.

The Vice-President is President of the Senate, but he/she only votes if there is a tie. The Senate also chooses a President Pro Tempore to preside in the Vice-President's absence (though, in practice, most of the time, senators from the majority take turns presiding for short periods).

The Senate and the House are both required to approve legislation before it becomes a law. The two houses are equal in legislative power, but revenue bills (bills relating to taxation) may only originate in the House. However, as with any other bill, the Senate's approval is still required, and the Senate may amend such bills.

The Senate holds additional powers relating to treaties and the appointments of executive and judicial officials. This power is known as "advice and consent." The Senate's advice and consent is required for the President to appoint judges and many executive officers, and also to ratify treaties. To grant advice and consent on treaties, two-thirds of the Senators must concur (agree).

While most votes require a simple majority to pass, it sometimes takes three-fifths of senators to bring a bill to a vote. This is because Senate rules hold that a bill cannot be voted on as long as it is being debated--and there is no limit on how long a senator may debate a bill. Senators sometimes use this rule to filibuster a bill--that is, continue debating a bill endlessly so that it cannot be voted on. The only way to end a filibuster is for three-fifths of all Senators to vote for a cloture resolution, which ends all debate and brings the bill up for voting. Use of the filibuster tends to be controversial. Whichever party is in the majority tends to call its use "obstructionism," while the other side sees it as an important check on the majority.

The House has the sole power to impeach federal executive and judicial officers. According to the Constitution, officers may be impeached for "treason, bribery, or other high crimes and misdemeanors.” The Senate has the sole power to try all such impeachments, a two-thirds vote being required for conviction. The Constitution requires that any individual convicted by the Senate to be removed from office. The Senate also has the power to bar that individual from further federal office. The Senate may not impose any further punishment, although the parties are still subject to trial in the courts. As the Vice-President (being next-in-line to the Presidency) would have an obvious conflict of interest in presiding at a trial of the President, in such cases, the Chief Justice presides. Interestingly, no similar provision prevents the Vice-President from presiding at his or her own trial.

The Executive
The President, Vice President, and other executive officials make up the Executive Branch. The main function of this branch is to execute the laws created by Congress. The President and the Vice-President are chosen by the Electoral College, a body of people elected for the purpose of electing the President. One may wait to consider the Electoral College in further detail.

The President appoints several Secretaries to head executive departments. An executive department is a body covering a broad topic of law- examples include the Department of Agriculture and the Department of Justice. The several secretaries (in the case of the Justice Department, the Attorney General) serve as advisors to the President and also as the chief officers of their own departments. This group of advisors is collectively known as the President's cabinet. The President nominates these Secretaries, as well as other important federal officials, and the Senate advise and consents to them.

The Judicial
The Supreme Court and the lower courts compose the Judicial Branch. The judiciary must interpret the laws of the United States. In the course of such interpretations, the courts may find that a law violates the constitution. If so, the court declares the law unconstitutional. Thus, the judiciary also has a role in determining the law of the land.

The judges of federal courts are nominated by the President and advised and consented to by the Senate. The number of judges and the exact structure of the courts is set by law, and not by the Constitution.

The Legislative Process: How A Bill Becomes A Law
After both houses of Congress pass a bill, perhaps observing the different rules and procedures in each house, but with the exact same final text, the bill is submitted to the President. Immediately, a ten-day clock for the president to act in starts to tick. Sundays are excluded in this calculation.

Once he receives the bill, the President has many options. The outcome of the process depends on the route taken by him.
 * Signature- If the President signs a bill, it immediately takes effect as law.
 * Veto- The President may, if he pleases, return a bill to the house in which it originated. That house may then consider the President's objections. If it wishes to pass the law in any event, it must repass the bill with a two-thirds majority. If the same occurs, then the other house considers the bill, perhaps repassing it with the same two-thirds majority. If both houses pass the bill with the requisite majority, then the bill becomes law despite the President's veto. If one or both houses fail to provide an adequate majority, then the bill fails
 * No Action- The President may decide to take no action at all on a bill. If, for 10 days, the President has neither signed nor returned the bill, the bill becomes law without a signature.
 * Pocket veto- The pocket veto is an absolute veto- Congress may not override it by any majority (though it may choose to repass the bill). The pocket veto occurs near the end of a Congressional session. If the bill is with the President, and he has not signed it or returned it, and the ten-day limit has not expired, and Congress adjourns for the year (that is, it decides to meet no longer for the rest of the year), then the President may pocket veto the bill.

Checks and Balances
In order to prevent any branch of government from becoming too powerful, the Framers of the Constitution created a system of checks and balances. Each branch of government has checks on the others, while it is itself also checked. The complex system can be outlined as follows:

Checks of the Legislative

Checks on the Executive Checks on the Judicial Internal Checks
 * Power to override vetoes
 * Power to confirm the President's appointment of a Vice President when a vacancy in the Vice Presidency occurs
 * Power to tax and allocate tax revenues for executive activities
 * Oversight and investigation of executive activities
 * (Senate) Power to approve treaties
 * (Senate) Power to approve Presidential nominees
 * (House) Impeachment, or accusation of a federal official for bribery, treason, or another high crime
 * (Senate) Trial of Impeachment
 * Power to set size and structure of courts
 * (Senate) Power to approve Presidential nominees for judgeships
 * (House) Impeachment
 * (Senate) Trial of Impeachment
 * Approval of both houses required for passage of a law

Checks of the Executive

Checks on the Legislative Checks on the Judicial
 * Power to veto
 * Power to pocket veto
 * Power to call a special session of Congress when Congress is not already meeting
 * Vice President presides over Senate meetings as President of the Senate
 * Congress cannot reduce the salary of the President while he continues in office
 * Power to nominate judges
 * Power to fully or partially pardon convicted criminals

Checks of the Judicial

Checks on the Legislative Checks on the Executive
 * Power to declare bills unconstitutional
 * Congress cannot reduce the salary of a judge while he continues in office
 * The Chief Justice presides over an Impeachment Trial of a President
 * Power to declare executive acts unconstitutional


text:
Algebra

Welcome to the Wikibook of

Preface
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures.

You've been asked to buy peanuts for you and your friends at a football game. You've collected $12.50. One bag is $2.75. You want to know how many bags you can buy. This is an algebra problem! Related problems are how much money will be left over, and what should you buy or in what proportions should you return the extra money to your friends.

Algebra helps us to predict things that we don't yet know and to determine the relationships between the things we do. Algebra is a powerful and rich branch of mathematics that is useful in everyday life as well as business, engineering, and other technical fields.

Contents

 * /Cover/
 * /Authors/
 * For Contributors

Preface
Chapter 0: An Introduction to Mathematics

Unit 1: Numbers, Variables and Relationships
How can we use numbers and variables to find out unknown information?

Chapter 1: Elementary Arithmetic Chapter 2: An Introduction to Algebra Chapter 3: Solving Equations Chapter 4: Inequalities

Unit 2: An Introduction To Graphing
Math is a method of solving problems. You take information you know, and by manipulating it using mathematical principles, you can find information you don't know. Functions are the mathematical framework for solving problems. They have parameters, rules, and ways of being solved. This section will introduce you to functions and how to use them.

Chapter 5: The Cartesian Plane Chapter 6: Graphing Linear Functions Chapter 7: Systems and Matrices Chapter 8: Piecewise Functions

Unit 3: Polynomials
Chapter 9: Quadratic Functions Chapter 10: Higher-Degree Polynomials

Unit 4: Other Functions
Chapter 11: Properties of Functions Chapter 12: Exponents and Logarithms Chapter 13: Rational and Radical Functions Chapter 14: Trigonometry

Unit 5: Algebra in Discrete Mathematics
Chapter 15: Sequences and Series Chapter 16: Probability and Statistics

Unit 6: Analytic Geometry
Chapter 17: Conic Sections Chapter 18: Parametric Equations and Polar Coordinates Chapter 19: Vectors

Unit 7: Advanced Algebra
Now that you have completed your mathematical journey, you may be wondering where to head over next. This unit provides six optional chapters that can be done in any order you please. The six chapters are meant to be independent from one another, and any of the chapters can be skipped entirely to put more focus on the topics that are of greater interest to you.

Complex Numbers
 * 1) /Complex Numbers/
 * 2) /More about the Quadratic Formula/
 * 3) More about Polynomials.
 * 4) Advanced Polynomials
 * 5) /Algebraic Manipulation/
 * 6) /Polynomial Interpolation/
 * 7) /Symettric Polynomials/
 * 8) /Chebyshev Polynomials/
 * 9) Continued Fractions
 * 10) /Diophantine Equations/
 * 11) Theory of Equations and Inequalities
 * 12) Limits of Functions
 * 13) Group Theory
 * 14) /Symmetries/
 * 15) /Binary Operators/
 * 16) /Groups/
 * 17) Lagrange's Theorem

Resources

 * /Resources/
 * /Bibliography/
 * /Glossary/
 * /Index/

Algebra Algèbre Algebra Wiskunde/Algebra ពិជគណិត 代數

title: Algebra


text:
Algebra/Solving equations

Quadratic equations
Up to now you have only dealt with equations and expressions involving just x; in this section we'll move onto solving things which have $$x^2$$ in them.

All quadratic equations can be arranged in the form $$ax^2+bx+c=0\ (a \ne 0)$$, and a,b,c are all constants. Now let's look at some examples:

Examples: Rearrange the following equations in the form $$ax^2+bx+c=0 \qquad$$:

(1)\qquad x(x-3)=3-5x; \qquad

$$

Solution for (1):

 * $$ x^2 - 3x = 3 - 5x \qquad$$
 * $$ x^2 + 2x = 3     \qquad$$
 * $$ x^2 + 2x - 3 =0  \qquad$$

 * Note that in the first step you distributed the x on the left side of the equation. The second step was obtained by adding a 5x to both sides of the equation and subsequently subtracting a 3 from both sides of the equation.

(2)\qquad 2x+1=(x^2+2)\sqrt{3} $$

Solution for (2):

\begin{matrix} 2x+1&=&x^2\sqrt{3}+2\sqrt{3}\\ -x^2\sqrt{3}+2x+1-2\sqrt{3}&=&0\\ x^2\sqrt{3}-2x-1+2\sqrt{3}&=&0 \end{matrix} $$

 * Note that in the last step, both sides are multiplied by -1, to make the term $$-x^2\sqrt{3}$$ positive, so that the solving of the equation would be easier.

Factorization
Factorization is the most common way to solve quadratic equations. Let us consider again the first example above: $$ x(x-3)=3-5x $$ We have already simplified the equation into

x^2+2x-3=0 $$

Now, we want to factorize the equation - that is to say, get it into a form such as:

(x+\text{something})(x+\text{something else})=0 $$

Look at the number term c. In this example, it is -3. Now, if we are lucky, the numbers "something" and "something else" will turn out to be nice whole numbers, so let's think of two numbers that will multiply together to give -3. Either 3 and -1, or -3 and 1. But we also need to get the x term correct (here, b=2). In fact, we need our two factors of c to add together to make b. And (3)+(-1)=2. So, we have found our 'somethings': they are 3 and -1. Let's fill them in.

(x+3)(x-1)=0 $$ Just to check, we can multiply out the brackets to check we have what we started with:

\begin{matrix} (x+3)(x-1)=0\\ x^2 +3x -1x -3=0\\ x^2 +2x -3=0 \end{matrix} $$

Now, we know that in an equation the left side is always equal to the right side. And in this case the right side of the equation is 0, so from that we can conclude the term $$(x+3)(x-1)$$ must equal to zero as well. And that means that either $$(x+3)$$ or $$(x-1)$$ must equal zero. (Not convinced? Remember (x+3) and (x-1) are just numbers. Can you find two non-zero numbers which multiply to make zero?)

Let's write that algebraically:

\begin{matrix} x^2 +2x -3=0\\ x+3=0 \qquad \mbox{or} \qquad x-1=0\\ x=-3 \qquad \mbox{or} \qquad x=1 \end{matrix} $$

Thus, there are two different solutions to the same equation! This is the case for all quadratic equations. We say that this quadratic equation has two distinct and real roots.

With practice, you will often be able to write down the equation in factorised form almost immediately. Here is another example, in this case the x easily factorises out:

\begin{matrix} 2x^2&=9x\\ 2x^2-9x&=0\\ x(2x-9)&=0\\ x=0&\qquad \mbox{or} \qquad x=\frac{9}{2} \end{matrix} $$

Completing the square
Sometimes the roots (solutions) of a quadratic equation cannot be easily obtained by factorisation. In such cases, we have to solve the equation by completing the square, or using the quadratic formula (see below).

In order to complete the square, we need to rewrite the given equation in the form $$(x+a)^2=b$$. Now here is an example:

\begin{matrix} x^2+8x+9&=&0\\ x^2+8x&=&-9\\ x^2+8x+4^2&=&4^2-9\\ (x+4)^2&=&7\\ &x+4=\sqrt{7}& \quad \mbox{or} \quad x+4=-\sqrt{7}\\ &x=-4+\sqrt{7}& \quad \mbox{or} \quad x=-4-\sqrt{7}\\ &x=-1.35& \quad \mbox{or} \quad x=-6.65 \end{matrix} $$

In general, we get

x^2+kx+\left({k \over 2}\right)^2=\left(x+{k \over 2}\right)^2 $$

Note that when we reach the stage of taking the square root of both sides of the equation, we might have a negative left-hand side. In this case, the roots will be complex. If you have not yet learned about complex numbers, it is possible to simply state that the equation "has no real roots".

Quadratic Formula
The quadratic formula is a special generalization of completing the square that allows the two roots of a quadratic equation to be obtained by simple substitution. It can be used to solve any quadratic equation and is very quick to work out on a calculator.

 * $$ax^2 + bx = -c \Rightarrow x^2 + \frac {b}{a}x = \frac {-c}{a}$$

Complete the square:

 * $$x^2 + \frac {b}{a}x + \left(\frac {b}{2a}\right)^2 = \frac {-c}{a} + \left(\frac {b}{2a}\right)^2$$

Simplify:

Which equals 4y to the 19th power.

 * $$\left(x + \frac {b}{2a} \right )^2 = \frac {-c}{a} + \left(\frac {b}{2a}\right)^2$$

 * $$x + \frac {b}{2a} = \pm \sqrt {\frac {b^2}{4a^2} + \frac {-c}{a}}$$

 * $$x + \frac {b}{2a} = \pm \sqrt {\frac {b^2}{4a^2} + \frac {-4ac}{4a^2}}$$

 * $$x = \frac {-b}{2a} \pm \sqrt {\frac {b^2-4ac}{4a^2}}$$

 * $$x = \frac {-b}{2a} \pm \frac {\sqrt {b^2-4ac}}{2a}$$

 * $$x = \frac {-b \pm \sqrt {b^2-4ac}}{2a}$$

which is the desired form of the quadratic formula.

Hence, given that a quadratic is in the form $$ax^2+bx+c=0$$, the two roots are:
 * $$x={-b \pm \sqrt{b^2-4ac} \over 2a}$$

The quantity $${b^2-4ac}$$ in the equation, known as the discriminant, is an indication of the solubility and nature of the roots:
 * discriminant is positive—soluble over R, real roots
 * discriminant is zero—soluble over R, real repeated (single) roots
 * discriminant is negative—insoluble over R (yet soluble over C), no real roots

Weda's Theorem
If the quadratic equation $$ax^2+bx+c=0\ (a\ne0)$$ has two real roots $$x_1$$ and $$x_2$$, then

$$ \left\{ \begin{matrix} x_1+x_2&=&-\cfrac{b}{a}\\ x_1x_2&=&\cfrac{c}{a} \end{matrix} \right. $$

This is because $$x_1=\frac{-b+\sqrt{b^2-4ac}}{2a}$$ and $$x_2=\frac{-b-\sqrt{b^2-4ac}}{2a}$$. By simply adding or multiplying the two roots we will get the above two equations. This is called Weda's Theorem.

Using Weda's Theorem we can find the second root of a given quadratic equation without solving the equation.

Example: Given that one of the real roots of the equation $$4x^2-13x+10=0$$ is 2, find the other root without solving the equation.

Solution:

$$ \begin{matrix} x_1 \cdot x_2 &=& \cfrac c a \\ \Rightarrow x_1 \cdot 2 &=& \cfrac{10}{4}\\ x_1&=&\cfrac{5}{4} \end{matrix} $$

We can also determine the signs of two roots by applying the following rules:
 * 1) the equation has two positive roots if $$\Delta\ge0,\ \frac{c}{a}>0,\ \mbox{and}\ \frac{b}{a}<0$$;
 * 2) the equation has two negative roots if $$\Delta\ge0,\ \frac{c}{a}>0,\ \mbox{and}\ \frac{b}{a}>0$$;
 * 3) the equation has two roots with different signs if $$\ \frac{c}{a}<0$$

($$\Delta$$ represents the discriminant of the equation.)

Another problem involving Weda's Theorem:

Example: For the equation $$2x^2+ax+1=0$$, given that the sum of squares of roots is $$7\frac{1}{4}$$, find the value of $$a$$.

Solution:

$$ \begin{matrix} (x_1+x_2)^2-2x_1x_2&=&x_1^2+x_2^2\\ \left(-\cfrac{a}{2}\right)^2-2\cdot\cfrac{1}{2}&=&7\cfrac{1}{4}\\ \cfrac{a^2}{4}&=&8\cfrac{1}{4}\\ a^2&=&33\\ a&=&\pm\sqrt{33} \end{matrix} $$

Pythagorean Theorem ____________________

a^+b^=c^

Solving simultaneous linear and nonlinear equations
In previous chapters you have already learned how to solve simultaneous linear equations. Now we will learn how to solve a system of simultaneous linear and non-linear equations with two unknowns. It is usually done by substitution method.

Example: Solve the following simultaneous equations:

$$ \left\{ \begin{matrix} 2x^2+y^2-5xy&=8 & \mbox{-- (1)}\\ y-x&=2 & \mbox{-- (2)} \end{matrix} \right. $$

Solution:

$$ \begin{matrix} \mbox{Rearrange (2):} & y=x+2 & \mbox{-- (3)}\\ \mbox{Substitute(3) into (1):} & 2x^2+(x+2)^2-5x(x+2)=8\\ & 2x^2+x^2+4x+4-5x^2-10x-8=0\\ & x^2+3x+2=0\\ & (x+2)(x+1)=0\\ & x=-2 \quad \mbox{or} \quad -1\\ \mbox{Substitute x=-2 back into (3):} & y=-2+2=0\\ \mbox{Substitute x=-1 back into (3):} & y=-1+2=1 \end{matrix} $$

∴ x=-1 and y=1, or x=-2 and y=0.

Memecahkan masalah perhitungan aljabar Matemática elementar/Equações algébricas

title: Algebra/Solving equations


text:
User:TakuyaMurata

See w:en:User:TakuyaMurata for who I am. I've been contributing to the following books:
 * Functional Analysis
 * Problems in Mathematics
 * An Introduction to Analysis

title: User:TakuyaMurata


text:
United States Government/The Federal System

Introduction
The United States is exactly that--a Union of states. Each state has its own individual powers. However, that does not mean that the states have power to legislate on all matters. The Constitution of the United States spells out the powers of the federal government and of the "several states." The Union government (known as the federal government) has its own fields of legislation, and if federal legislation conflicts with the state laws, the federal legislation prevails. If this occurs, the state must defer to the federal government. The alternative, that any state may at any time leave the Union and thus be free from Union interference in the state's internal affairs, was tried during the American Civil War.

Types of Federalism
There are two types of federal systems. The first, dual federalism, holds that the Union and the state are equal; under this view of federalism, the Union government only has the powers expressly granted to it, while the states retain all other powers. The second view, cooperative federalism, states that the federal government is definitively superior to the state government, and the federal government should stretch its powers as far as possible. The US is a Union that does not completely fit either definition. The type of federalism in effect really depends on who is in power at the time. In any case, the Constitution prevents stretching federalism too far in either extreme.

All powers retained by the states are known as reserved powers. Those specifically granted only to the Union government are known as enumerated powers. Finally, matters over which both the Union and the state governments have control are known as concurrent powers.

The Tenth Amendment provides that the Union government has only the powers expressly designated to it by the Constitution, and the states control all other matters.

Enumerated powers relate to the following:
 * Regulation of interstate and foreign commerce
 * Immigration and Citizenship
 * Bankruptcy
 * Weights and Measures
 * Currency
 * Copyrights and Patents (Special rights granted to authors and inventors relating to the writing or discovery)
 * Crimes at sea
 * Military
 * Declaration of War
 * Control of the District of Columbia
 * Federal Courts
 * "Necessary and Proper"- The necessary and proper clause allows the Union government to make laws necessary to utilize other powers. For instance, this clause allows the government to jail persons who violate laws created under the aforestated powers.

Reserved Powers
As has been mentioned, the states have control over all matters not controlled by the Union government. Some of these matters include:
 * Education
 * Police protection
 * Licensing
 * Control local governments
 * Regulate individual and corporate activities in order to protect and enhance public health, welfare, safety, morals, and convenience.

Concurrent Powers
Several powers belong concurrently to the Union and the state governments. These include:
 * Transportation
 * Taxation
 * Make and enforce laws
 * Establish courts
 * Borrow
 * Spend money for general welfare
 * Take private property for public purposes, with just compensation.
 * Charter banks and corporations


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United States Government/General Provisions

Introduction
The Constitution provides for many things other than the Union and member state government operation.

Supreme Law
The Constitution calls itself, the United States laws, and treaties, the Supreme Law of the land. No state law may conflict with the Supreme Law. If they do, the conflicting parts are ruled void. However the supremacy of federal law over state law only applies if the federal government is acting in pursuance to its constitutionally granted powers.

Also, the Constitution is itself superior to federal laws and treaties. If either of these, or any Executive order made by the President, or a state law, or other regulation similar to law, conflicts with the constitution, the Courts may declare it unconstitutional, and the law or regulation becomes void. Note that this is not explicitly provided for by the Constitution. Chief Justice John Marshall originally used judicial review- the power to declare laws unconstitutional- in the early 1800's when he decided the case Marbury v. Madison.

Oaths
The Union and member state officers take an oath to support and defend the Constitution. Only the Union President's oath is explicitly spelled out- the form of the other oaths is left to the appropriate government. The Constitution, while mentioning oaths, specifically prohibits the United States or a member state from requiring a religious oath or observance of a certain religion in order to qualify for any office.

Amendment
The Constitution is not an unchangeable document. Changes to it are known as amendments. By tradition, an amendment does not strike out words and insert others. In order to leave the original Constitution whole, Amendments are added as separate Articles at the end of the Constitution.

There are two ways in which an amendment can be proposed, and two ways in which it can be ratified, or approved. First, the Amendment can be proposed by Congress. For this to occur, two-thirds of the House of Representatives and two-thirds of the Senate must vote for the Amendment. Second, an Amendment can be proposed by a Constitutional Convention. If two-thirds of the member states make an application to Congress, Congress must call this Convention, which then proposes Amendments. However, a Constitutional Convention has never been called.

Regardless of the way in which the Amendment is proposed, it must be ratified by three-fourths of the member states. The first manner in which ratification by a member state may occur is through the legislature. Secondly, a state convention can be called to ratify an amendment. The second method has only been used once. Note that each member state cannot decide which method it wishes to use. Whoever proposed the Amendment- the Congress or a Union Convention- will decide if legislatures or state conventions will ratify.

There have been twenty-seven Constitutional Amendments to date. The first ten guarantee certain already existing rights to the people and member states- they are therefore known as the Bill of Rights.


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United States Government/The Bill of Rights

First Amendment
"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances."

The First Amendment is the best-known amendment to most Americans. It provides for and guarantees certain fundamental and basic human rights, such as:

1. Freedom of religion. The amendment states: "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof." This means that Congress cannot make one religion official, or require people to worship in a certain way. It also forbids Congress, Federal and State courts, and State legislatures from prohibiting any one religion from being practiced.

2. Freedom of speech and of the press. The amendment also guarantees that Americans have freedom of speech, and forbids both the Federal Government and State Governments from punishing a person for expressing their views. It also calls for the freedom of the press and bars the Government from interfering with/censoring the press. Therefor the government cannot control the press (i.e. controlling what the press publishes or punishing the same for publishing a particular article or airing on television or radio a particular segment/video.)

3. Freedom of assembly and petition. The amendment guarantees that the people have the right to assemble in a peaceful manner and to consult for their common good. This has also be interpreted by the courts to also guarantee freedom of association. It also states that Americans may petition the Federal Government for a redress of grievances.

Second Amendment
"A well regulated militia, being necessary to the security of a free state, the right of the people to keep and bear arms, shall not be infringed."

The second amendment guarantees that individual Americans have the right to keep and bear firearms. The amendment is subject to many controversies about whether it applies to only people in the armed forces or everyday citizens. The Supreme Court has ruled however in District of Columbia vs Heller (2008) that the amendment applies to everyday citizens. A majority of gun laws and regulation in the USA are at State level.

Third Amendment
"No soldier shall, in time of peace be quartered in any house, without the consent of the owner, nor in time of war, but in a manner to be prescribed by law."

Soldiers, according to the Third, may not during peacetime be quartered or housed in a residence without the consent of the owner. Furthermore, even in wartime, a law is required to force people to quarter soldiers.

The Supreme Court and others have also attributed a right to personal privacy within this amendment.

Fourth Amendment
The fourth amendment of the United States Constitution reads as follows: The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized.

The fourth amendment can be broken into two distinct parts. The first part provides protection against unreasonable searches and seizures, although unreasonable has been defined in a myriad of different ways. However, the framers did not qualify how a violation of this amendment was to be punished. Case law has afforded the exclusionary rule to ensure that evidence improperly collected is excluded from trial. (See Wolf v. Colorado, Mapp v. Ohio, and United States v. Leon for more information on the progression of the exclusionary rule.) This gives law enforcement officers the incentive to respect the amendment.

The second part of the amendment provides for the proper issue of warrants. However, it is important to note that it does not require that warrants be obtained prior to a search or seizure, only that searches and seizures are reasonable. When warrants are issued, their must be probable cause. Probable cause is tested using the "totality of circumstances" test as defined in Illinois v. Gates. (See Spinelli v. United States and Illinois v. Gates for the progression on probable cause tests. Also see Maryland v. Garison and Richards v. Wisconsin for more on search warrants.)

Interesting Search and Seizure Cases
 * http://www.wdnweb.com/2011/02/17/city-beefs-up-code/
 * Katz v. United States: use of electronic eavesdropping by law enforcement
 * California v. Greenwood: examination of curbside garbage
 * Florida v. Riley: use of aerial surveillance by law enforcement
 * United States v. Karo: use of electronic tracking by law enforcement
 * Kyllo v. United States: use of thermal imaging by law enforcement

Fifth Amendment
"No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a grand jury, except in cases arising in the land or naval forces, or in the militia, when in actual service in time of war or public danger; nor shall any person be subject for the same offense to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation."

While the Fourth protects one against the police, the Fifth guarantees one's rights relating to actual charges of criminal activity. Firstly, the Fifth requires that a Grand Jury (a special body of lay-people) indict, or accuse, a person of a serious crime before he may be tried for it. The sole exception relates to the military, where Grand Juries are not used.

2: the Amendment prohibits "double jeopardy." Once a person has been tried and acquitted (found not guilty), the prosecutor cannot retry him, even if new evidence is discovered. However, the prohibition against double jeopardy does not extend to mistrials. If a jury cannot reach a verdict and the judge declares a mistrial, then the case may be retried.

3:  the Amendment provides that a person cannot be compelled to testify against themselves. A person may "plead the Fifth" to avoid self-incrimination. However, the government may grant immunity, or formal protection from prosecution, to a person. Then, the person cannot be punished for what crimes he might have committed, and can be forced to give evidence.

4: the Fifth requires due process to be used. While due process is not defined in the Constitution, one may conclude that it involves fair trials with impartial juries and judges.

5: the Fifth prohibits government from taking away a person's property unless it provides fair compensation.

Sixth Amendment
In all criminal prosecutions, the accused shall enjoy the right to a speedy and public trial, by an impartial jury of the State and district wherein the crime shall have been committed, which district shall have been previously ascertained by law, and to be informed of the nature and cause of the accusation; to be confronted with the witnesses against him; to have compulsory process for obtaining witnesses in his favor, and to have the Assistance of Counsel for his defence.

The sixth amendment, like the fifth amendment, also guarantees and lays out a person's rights when that person had been accused of committing a crime. Firstly, a person is entitled to a "speedy and public trial by an impartial jury." This means a person cannot be arrested/detained for an unreasonable amount of time before that person goes to trial, and the trial cannot be private, unless the person accused waivers this right and ask that the trial be private. The Supreme Court has also ruled that a trial can be held in private, if by having excess publicity would serve to undermine the accused person's right to due process. Also an accused person has the right to trial by jury, and the trial must held in the State or district in which the crime was alleged to be committed.

Secondly, an accused person has the right to be informed of the "nature and cause of the accusation" against him. Therefore, an indictment must allege all the ingredients of the crime to such a degree of precision that it would allow the accused person to assert double jeopardy if the same charges are brought up in subsequent prosecution.

Thirdly, an accused person has the right to confront/cross-examine the wittiness against him, this also includes the right to cross-examine physical evidence that the prosecution will use against the accused person during the trial. An accused person also has the right to call witness to testify in his favor. If the witness refuses to testify, the court may, at the accused's request, order the witness to do so. However, in some cases the court may refuse to permit a defense witness to testify. For example, if a defense lawyer fails to notify the prosecution of the identity of a witness to gain a tactical advantage, that witness may be precluded from testifying

Fifthly, an accused person has the right to be represented by counsel (i.e. a lawyer) and if the accused person does not have a lawyer, and cannot afford one, the court must, upon the accused's request, appoint a lawyer to represent him. If the accused person is non compos mentis and incapable adequately of making his own defense, the court must, with or without the accused's request, appoint a lawyer to represent him.

Seventh Amendment
"In suits at common law, where the value in controversy shall exceed twenty dollars, the right of trial by jury shall be preserved, and no fact tried by a jury, shall be otherwise reexamined in any court of the United States, than according to the rules of the common law.

The seventh amendment guarantees the right to trial by jury in common law (i.e. civil) suits.

Eighth Amendment
"Excessive bail shall not be required, nor excessive fines imposed, nor cruel and unusual punishments inflicted."

The eighth amendment prohibits excessive bail from being required. This means that bail must appropriately match the crime in which the arrested/detained person is accused of committing. Therefor a person arrested for littering cannot be put on bail for 50,000 dollars, as that would be excessive for that respective offense. However bail may be denied if a person is arrested on murder charges. It also prohibits excessive fines from being imposed as punishment for crime and again must appropriately match the respective crime. Lastly, it forbids cruel and unusual punishments.

Ninth Amendment
The enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people.

The ninth amendment states that Americans have more rights than what are listed in the Bill of Rights and that all because a right is not listed in the same, does not mean Americans do not posses that right.

Tenth Amendment
"The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people.

The Tenth Amendment prevents the Union government from regulating a matter it is not constitutionally entitled to regulate. The Union government, therefore, cannot infringe upon a state's reserved powers.