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United States Government/The Executive Branch

The Executive Branch, which executes and enforces the laws, is headed by the President and the Vice President. In addition, it includes the executive departments, which deal with general topics, and the heads of departments, who are known as Secretaries (Attorney-General in the Department of Justice). Each Department, in turn, is divided into a number of bodies, which are known as agencies, services, commissions, councils, bureaus, authorities, offices, administrations, and boards.

The President
The President is the elected head of state and head of government of the United States. The President leads the Executive Branch and is the commander-in-chief of the United States Armed Forces.

Article Two of US Constitution vest the executive power in the President. The power includes execution of federal law, alongside the responsibility of appointing federal executive, diplomatic, regulatory and judicial officers and making treaties with foreign nations, however all treaties made by the President must be approved by a two-thirds vote of the Senate. The president is further empowered to grant federal pardons and reprieves, and to convene and adjourn either or both houses of Congress under extraordinary circumstances. The President is also largely responsible for dictating the legislative agenda of the party to which he is enrolled.

The President of the United States is often considered one of the most powerful people in the world.

Powers and Duties
Article One: Legislative Powers.

The first power the President is given in the Constitution, is the power to veto bills of legislation. The Constitution requires that all bills passed by Congress to be presented to the President before it can obtain legal force. Once a bill has been presented, the President has one of three options: Article Two: Executive Powers.
 * 1) The president may sign the bill, it which case it obtains legal force.
 * 2) The President may veto (refuse to sign) the bill, in which case it does not obtain legal force. The President then returns the bill, expressing any obligation he may have had on it, to Congress. Congress may override a veto by a two-thirds vote in both houses. Upon overriding the veto, the bill obtains legal force.
 * 3) The President may take no action on a bill. In this case the President does not sign or veto the legislation. If after ten days, not including Sundays, the President has still neither signed nor vetoed the bill, two possible outcomes emerge:
 * 4) * If Congress is still convened, the bill becomes law.
 * 5) * If Congress has adjourned, thus preventing the return of the legislation, the bill does not become law. This latter outcome is known as the pocket veto.

One of the most important Presidential powers is the command-in-chief of the Armed Forces of the United States. While the Constitution vest the power to declare war solely in Congress, the President is ultimately reasonable for the disposition and direction of the Armed Forces.

Along with the armed forces, the president also directs U.S. foreign policy. Through the Department of State and the Department of Defense, the president is responsible for the protection of Americans abroad and of foreign nationals in the United States. The president decides whether to recognize new nations and new governments, and negotiates treaties with other nations, which become binding on the United States when approved by two-thirds vote of the Senate.[citation needed]

Although not constitutionally provided, presidents also sometimes employ "executive agreements" in foreign relations. These agreements frequently regard administrative policy choices germane to executive power; for example, the extent to which either country presents an armed presence in a given area, how each country will enforce copyright treaties, or how each country will process foreign mail. However, the 20th century witnessed a vast expansion of the use of executive agreements, and critics have challenged the extent of that use as supplanting the treaty process and removing constitutionally prescribed checks and balances over the executive in foreign relations. Supporters counter that the agreements offer a pragmatic solution when the need for swift, secret, and/or concerted action arises

The President is the head/leader of the Executive Branch of the federal government and is Constitutionally bound to "take care that the law be faithfully executed." The President has the power to make appointments within the Executive branch. Ambassadors, members of the Presidential Cabinet, and federal court officers are appointed by the President, although every appointed official by him must be approved by the Senate.

The President also has the power to fire/remove purely executive officials at will, although Congress has the authority to curtail or constrain the Presidents power to fire/remove commissioners of independent regulatory agencies and certain inferior executive officers by statute.

The president possesses the ability to direct much of the executive branch through executive orders that are grounded in federal law or constitutionally granted executive power. Executive orders are reviewable by federal courts and can be superseded by federal legislation.

The president also has the power to nominate federal judges, including members of the United States courts of appeals and the Supreme Court of the United States. However, these nominations do require Senate confirmation. Securing Senate approval can provide a major obstacle for presidents who wish to orient the federal judiciary toward a particular ideological stance. When nominating judges to U.S. district courts, presidents often respect the long-standing tradition of Senatorial courtesy. Presidents may also grant pardons and reprieves for offense recognizable under federal law. The President cannot, however, grant pardons in impeachment cases, or grant pardons for persons convicted under State law/Jurisdiction.

The President also cannot be simultaneously a member of Congress, neither can any other executive officer. Therefore, the President cannot directly introduce legislative proposals for consideration in Congress. However, the President can take an indirect role in shaping legislation, especially if the President's political party has a majority in one or both houses of Congress. For example, the President or other officials of the executive branch may draft legislation and then ask senators or representatives to introduce these drafts into Congress. The President can further influence the legislative branch through constitutionally mandated, periodic reports to Congress. These reports may be either written or oral, but today are given as the State of the Union address, which often outlines the President's legislative proposals for the coming year. Additionally, the President may attempt to have Congress alter proposed legislation by threatening to veto that legislation unless requested changes are made.

Succession
If the President dies, resigns, is removed from office, or is unable to discharge the powers and duties of the Presidency, the Vice-President becomes President.

Under the Twenty-Fifth amendment, the President may declare himself disabled and upon doing so, transfers the powers and duties of the Presidency to the Vice-President, who then becomes acting President. The President may resume the powers and duties of the Presidency by declaring himself again able to discharge the same. Under the same amendment, the Vice-President, along with a majority of the members of the President's cabinet, may declare the President disabled by submitting a written declaration to the Speaker of the House and the president pro tempore of the Senate, stating that President is disabled from discharging the powers and duties of the Presidency, and upon doing so, transfer the Presidential powers to the Vice-President as acting President. The President may resume discharging the powers and duties of the Presidency by submitting a written declaration to the Speaker of the House of Representatives and President pro tempore of the senate stating that such disability does not exist. If the Vice-President and Cabinet contest this claim, it is up to Congress, which must meet within two days if not already in session, to decide the merit of the claim.

A President is: A person may not hold the office of President, if that person:
 * 1) required to be natural born citizen,
 * 2) required to be at least thirty-five years of age,
 * 3) required to have been a permanent resident in the United States for at least Fourteen years,
 * 4) barred from being elected to the office more than twice.
 * 1) was impeached by the Senate and voted by the same to be disqualified from holding any federal office, including that of President,
 * 2) Once swore to uphold the Constitution and latter rebelled against the United States, although such disqualification can be removed by a two-thirds vote of both Houses of Congress.

The Vice President
The Vice President's only executive function in the Constitution is to become President in the event that the President dies or is incapacitated. The 25th Amendment provides that this may occur when:

 * 1) "the President transmits to the President pro tempore of the Senate and the Speaker of the House of Representatives his written declaration that he is unable to discharge the powers and duties of his office, and until he transmits to them a written declaration to the contrary;" or

 * 1) "the Vice President and a majority of either the principal officers of the executive departments or of such other body as Congress may by law provide, transmit to the President pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office."

Additionally, "whenever there is a vacancy in the office of the Vice President, the President shall nominate a Vice President who shall take office upon confirmation by a majority vote of both Houses of Congress."

In practice, the Vice President often takes an active role in policy making.

The Vice President also serves as president of the Senate, but cannot vote except to break a tie.

Elections
Each state is entitled to choose a number of Electors equal to the number of Representatives and Senators it elects to Congress. Thus, a state will choose no less than three electors- two for the Senators, and one for the minimum of one Representative.

The state may choose its Electors in any way it pleases. However, all the states allow the people to choose all Electors. Forty-eight states use the following system: each candidate nominates a panel of electors. When a voter votes for a candidate, they actually vote for the nominated panel. Then, the candidate who wins more votes than any other candidate has his nominated panel appointed. Thus, a candidate who does not necessarily win all the votes will receive all of the state's electors. The exceptions to this system are Maine and Nebraska. These states allow a candidate to nominate one state panel of two electors, plus one elector for each district. Then, the candidate who wins the state has his two-member state panel appointed as electors, while the candidate who wins an individual district has his nominee for that district appointed. In all cases, the election for the Electors is held on the same day as Congressional elections.

Regardless of how they are chosen, the Electors all assemble in their own states in December. The manner of the voting by Electors has changed. Firstly, the following is the original system: If, however, no candidate received a majority, then the House of Representatives would choose one of the top five candidates as President. In voting for President, each state casts one block vote. In any case, whichever other candidate holds the greatest number of votes other than the candidate elected President becomes Vice President. In case of a tie for second-place, the Senate elects the Vice President.
 * Each Elector casts two votes for President. One of those votes must be for a person not an inhabitant of the same state as the Elector. Then, if a candidate receives the votes of a majority of Electors becomes President, while the runner-up becomes Vice President. If one compares the Presidency to a gold medal, then one may say that the Vice Presidency was like a silver medal.
 * The above system seems to allow for all contingencies, but it did not allow for the development of political parties. Prior to the election of 1800, parties did not nominate candidates. However, in 1800, the Democratic-Republican Party nominated Thomas Jefferson for President and Aaron Burr for Vice President. The pair won a clear majority of the votes. However, both had an equal number of Electoral votes. Though the original intention was clearly for Jefferson to become President and Burr to become Vice President, the House still had to vote to choose one or the other as President. The House votes tied thirty-five times until Jefferson was finally elected President.
 * In order to prevent a recurrence, the Twelfth Amendment was passed. Under the Amendment, the electors would vote separately for President and Vice President. In each case, a majority was required for someone to be elected. If no majority was apparent for President, then the House would choose among the top three candidates, again voting by states. Similarly, the Senate would vote between the top two candidates for Vice President.

The District of Columbia chooses Electors like the other states, but the District can in no event choose more Electors than any other state.

Executive departments
At present, there are fifteen executive departments. They are the departments of:

 * Department of Agriculture (USDA)
 * Department of Commerce (DOC)
 * Department of Defense (DOD)
 * Department of Education (ED)
 * Department of Energy (DOE)
 * Department of Health and Human Services (HHS)
 * Department of Homeland Security (DHS)
 * Department of Housing and Urban Development (HUD)
 * Department of Justice (DOJ)
 * Department of Labor (DOL)
 * Department of State (DOS)
 * Department of the Interior (DOI)
 * Department of the Treasury
 * Department of Transportation (DOT)
 * Department of Veterans Affairs (VA)

Each department is subdivided into a number of agencies, bureaus and other divisions.

Executive agencies
In addition to the Cabinet departments, there are certain independent bodies which are not part of any department, but report directly to the Executive Office of the President. These include:

 * Environmental Protection Agency (EPA)
 * National Aeronautics and Space Administration (NASA)
 * Office of Management and Budget (OMB)
 * Office of National Drug Control Policy (ONDCP)
 * Office of the United States Trade Representative (USTR)

The Cabinet
The Cabinet currently consists of the Vice President, the White House Chief of Staff, the heads of each executive department, and the heads of the EPA, OMB, ONDCP and USTR.


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United States Government/The Judicial Branch

The Courts
The United States judicial system includes the Supreme Court of the United States and the inferior federal courts. The President nominates an individual to serve as a judge, after which the Senate must grant its advice and consent before the President can formally appoint the judge. A judge holds office during "good behavior", which is usually interpreted as meaning a life term.

The Supreme Court
The Constitution creates the Supreme Court, but permits Congress to set the number of Justices. Currently, the Supreme Court includes the Chief Justice of the United States and eight Associate Justices. The Supreme Court has original jurisdiction over limited categories of cases, such as cases between two or more states. It hears most of its cases through its appellate jurisdiction, in which it hears appeals from lower federal and state courts. It exercises its appellate jurisdiction selectively; four of the nine Justices must grant a writ of certiorari before a case can be heard.

The Court of Appeals
The United States is divided into twelve regional circuits, each of which has a Court of Appeal. Additionally, there is a Federal Circuit which hears appeals from certain special tribunals and courts. The regional circuits (which are officially known by a number only, except for the DC circuit) are as follows:
 * District of Columbia
 * First Circuit- Northern New England (Maine, New Hampshire, Massachusetts, and Rhode Island plus the Commonwealth of Puerto Rico)
 * Second Circuit- Southern New England (Vermont, New York, and Connecticut)
 * Third Circuit- Middle Atlantic (Pennsylvania, Delaware, and New Jersey plus US Territory Virgin Islands)
 * Fourth Circuit- Upper Southeast (Maryland, Virginia, West Virginia, North Carolina, and South Carolina)
 * Fifth Circuit- Deep South (Texas, Louisiana, and Mississippi)
 * Sixth Circuit- Eastern Great Lakes (Michigan, Ohio, Kentucky, and Tennessee)
 * Seventh Circuit- Western Great Lakes (Wisconsin, Illinois, and Indiana)
 * Eighth Circuit- Midwest (Minnesota, Iowa, Missouri, Arkansas, North Dakota, South Dakota, and Nebraska)
 * Ninth Circuit- West (Alaska, Hawaii, Washington, Oregon, California, Montana, Idaho, Nevada, and Arizona plus US Territories Guam and the Northern Mariana Islands)
 * Tenth Circuit- Southwest (Wyoming, Utah, Colorado, Kansas, Oklahoma, and New Mexico)
 * Eleventh Circuit- Lower Southeast (Alabama, Georgia, and Florida)

Each Court of Appeal includes a different number of members. The First Circuit has the fewest with six members, while the Ninth Circuit has the most with twenty-eight.

District Court
Every state is divided into one or more Court Districts (which are distinct from Congressional districts). The total number of districts is ninety-four.

Each district court includes a different number of members. The Eastern and Western District of Kentucky, the Eastern District of Oklahoma, and the Northern, Eastern, and Western District of Oklahoma, each have the fewest judges with one, while the Central District of California has the most with twenty-seven.

Other Courts
The Congress has established special bankruptcy courts and other courts to rule on specific matters.

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title: User:Flockmeal~enwikibooks


text:
Japanese/Grammar/Basic Particles

The Japanese language uses post-position particles (助詞; じょし) to denote the direction of an action and who is performing the action. They consistently come after the word that they modify.

There are three particles used very frequently in the language: は, を and が. This module covers these along with a few other common ones but an exhaustive list would run very long.

The topic and subject markers は and が
The particle "は" (pronounced as "わ" when used as a particle) is the topic marker denoting topic of discussion, while "が" is the subject marker and marks a noun that performs an action. The difference between the two tends to cause confusion among beginners but their usage can be summed up as matter of focus.

The topic particle "は" is used when introducing a topic and gives focus to the action of the sentence (i.e., the verb or the adjective). The subject marker "が" is used when emphasising the subject giving focus to the subject of the action.

One can also think of it as replacing "~は" with the phrase "as for ~", "on the topic of ~" or "regarding ~" to distinguish it from "が". While these phrases aren't common in English we can use these expressions here to better show the main difference between "は" and "が".

The difference can also be displayed by using both subject and topic markers in one sentence: One has to be careful using both "は" and "が" in one sentence. If a verb is actually acting on the (direct) subject, usually a different particle (like を) has to be used.

"は" is generally more flexible, because the "it" can be assumed, and is therefore recommended to novices who have not grasped the difference between the two.

"は" also has the specialized function of being used for comparisons as well.

Often the grammatical subject may also be the topic. In this case, "は" normally replaces "が". However, if the subject is never known, you cannot use "は" and must use "が". This is similar to using pronouns: You can't state, "It is over there", without first stating what "it" may be.

The direct object marker を
The particle "を" (predominantly pronounced "お") is the direct object marker and marks the recipient of an action.

It also indicates the place through which the action occurs:

As with much of the language, parts of a sentence that can be assumed from context are often omitted and the direct object particle is commonly dropped in conversational (colloquial) Japanese. を is commonly used to identify the object in which the verb is affecting. For example, in the sentence "I drink juice" (わたし は ジュース を のむ), を is identifying the word "ジュース" as the object in which のむ ' s action is taking place. "のむ" means "drink / to drink". In simpler terms, を tells us that the word (ジュース) is the object which the verb (のむ) is interacting with.

The indirect object marker に
"に" marks the verb's indirect object (i.e. the destination of a targeted verb action) translating as "to", "in", "at" or "by". It also indicates the location touched or affected by an event or action:

"に" can also be used as an "object of a preposition" marker when found in prepositional phrases like の前に (no mae ni), which means "in front of" or "before" depending on the context of the sentence. The particle "へ" described below is used exclusively for marking the destination.

The destination marker へ
へ (pronounced "え" when used as a particle) indicates the direction of an action, roughly the equivalent of "to" or "toward" in English.

The question marker か
Placing か at the end of a sentence changes a statement into a question. Use it at the end of a verb to make it a question, or at the end of an interrogative pro-form to make it into a demonstrative pronoun.

For more on the question marker, see: ../Sentence ending particles/.

The possessive marker の
"の", is most commonly used as a possessive marker (similar to the English " 's "). The particle can also function as a noun link, indicating that the preceding noun (or adjectival noun) modifies the following noun. It can also be used for nominalisation, converting verbs and (proper) adjectives into nouns. Note that in this last example two particles are used together: の and が: the first makes the action a noun, and the second tells that this action is what the sentence is all about.

The exhaustive list conjunction と
This particle acts as a conjunction on the words it separates. Unlike conjunctions of more than two words in English, where only the last two are separated with an "and" and the rest with commas, the Japanese conjunction separates each word and commas are not used.

This applies to exhaustive lists, i.e. when all objects are explicitly mentioned.

The particle is used to indicate parallelism with the subject, often meaning "with":

The incomplete list marker や
This particle is used to connect various words implying that the listing is not exhaustive. The particle "など" may be added after the list to emphasise that the list is incomplete.

The "also" marker も
も is quite simply a marker that says "also". It replaces the particles は, が and を but can also follow other particles. This can also be used to form a large list of words all acting as though one of the basic particles (は, を, or が) were affecting the whole list.

Worth noting is that used with an interrogative pro-form (e.g. who, where, how) the も particle negates the pro-form:

The means particle で
The particle で can be used in several situations indicating means. These can be for example an instrument, a location or a language.

As a note of interest, the で from the copula である is also actually an instrumental-maker. で marks the whole previous expression instrumental to the verb ある. However, this is the classical meaning of the copula and rarely explicitly treated this way in modern Japanese.

Origin and limit から and まで
These particles indicate the starting point or border of an action. This may be a location as well as a time and corresponds roughly with "from" and "until".

title: Japanese/Grammar/Basic Particles


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The Once and Future King

The Once and Future King is a novel by T. H. White about the legend of King Arthur. It is often assigned reading in English literature classes and is composed of five books:

 * 1) /The Sword in the Stone/
 * 2) The Queen of Air and Darkness
 * 3) /The Ill-Made Knight/
 * 4) /The Candle in the Wind/
 * 5) /The Book of Merlyn/

While the first four were originally published separately, and reworked for inclusion in The Once and Future King, the last was to be published in the greater work for the first time. This was vetoed by White's publisher, supposedly due to wartime paper restrictions. It wasn't until after White's death that The Book of Merlyn joined the other four.

Characters

 * /Arthur/
 * /Merlyn/
 * /Kay/
 * /Lancelot/
 * /Templates/
 * /Gawaine/
 * /Gaheris/
 * /Gareth/
 * /Uther Pendragon/

title: The Once and Future King


text:
The Once and Future King/The Sword in the Stone

The Once and Future King - The Sword in the Stone

Book I covers the Wart's (Arthur's) childhood education by Merlyn, including his adventures into Morgan le Fay's castle with Robin Wood, and his transformations into many different animals. It ends with the Wart removing the sword from the stone after King Uther's death, and his subsequent coronation

 * /Chapter 1/
 * /Chapter 2/
 * /Chapter 3/
 * /Chapter 4/
 * /Chapter 5/
 * /Chapter 6/
 * /Chapter 7/
 * /Chapter 8/
 * /Chapter 9/
 * /Chapter 10/
 * /Chapter 11/
 * /Chapter 12/
 * /Chapter 13/
 * /Chapter 14/
 * /Chapter 15/
 * /Chapter 16/
 * /Chapter 17/
 * /Chapter 18/
 * /Chapter 19/
 * /Chapter 20/
 * /Chapter 21/
 * /Chapter 22/
 * /Chapter 23/
 * /Chapter 24/
 * /Chapter 25/


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The Once and Future King/The Sword in the Stone/Chapter 17

The Once and Future King - Book I: The Sword in the Stone - Chapter 17

Plot summary
The chapter begins a few months after the boar-hunt. King Pellinore gives the Questing Beast, now healthy, a two-hour headstart in the chase. Merlyn, Archimedes, and the Wart discuss the birds, a subject that has arisen due to spring's arrival. The Wart asks to become a bird again, because the last time he didn't get a chance to fly; Archimedes plans to go with him at night. The Wart then mentions his favorite bird is the rook, because rooks fly just for pleasure, not necessarily with a purpose. Archimedes mentions rooks are one of few birds with a legal and social system, and in response to Merlyn says his favorite are pigeons. Then Merlyn says he prefers the chaffinch, and continues that bird calls arise out of imitation of their prey or their environment. Finally Kay enters, ironically saying that he was shooting birds.

Analysis
Anachronisms - Merlyn mentions both Gilbert White and Carolus Linnaeus, and Archimedes compares pigeons to Quakers. All came much later. Animals as forms of government - Archimedes mentions that the rooks have "parliaments...and a social system;" (a play upon the collective noun for rooks being a "parliament") they make "laws about the defense of the rookery, and marriage, and so forth"


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United States Government/Colonial Government in America

Under the Kingdom of Great Britain, the American colonies experienced five situations which would guide them in creating a constitution. The British Parliament believed that it had the right to impose taxes on the colonists; it had virtual representation over the entire empire, while the colonists believed Parliament had no such right, as they had no direct representation in Parliament. By the 1720s all but two of the colonies had a locally elected legislature and a British appointed governor. Often, these two branches of government would clash, with the legislatures imposing their "power of the purse" to control the British governor. Thus, Americans viewed their legislative branch as a guardian of their liberty, while the executive branches was deemed tyrannical.

There were several examples of royal actions that upset the Americans. For example, taxes on the importation of lead, paint, tea, paper, spirits, rum, wine, molasses, sugar, and other products were imposed at various times. Also, the Parliament provided for a duty to be paid on court documents, certificates, licenses, deeds, other legal documents, playing cards, pamphlets, books, calendars, newspapers, and other papers, as well as dice. The variety of taxes imposed, as well as other causes, led to the Americans' disdain for the British system of government.

After the Boston Tea Party, the Parliament of Great Britain and the King passed Acts that outlawed the Massachusetts legislature. The Parliament also provided for special courts in which British judges, rather than American juries, would try colonists. The Quartering Act and the Intolerable Acts required Americans to provide room and board for British soldiers. Americans especially feared British actions in Canada, where civil law was once suspended in favor of British military rule.

American distaste for the system of British government would lead to revolution. Americans had formed their own local institutions which were not British at all, but American. The political ideas of the Americans actually had their root in the British radicals of the early 18th century. England had passed beyond those ideas by 1776 and the resulting conflict resulted in the first American attempts at a national government.


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United States Government/The U.S. under the Articles of Confederation

The Articles of Confederation
The Articles of Confederation accomplished certain things, but without a strict leader, or a government that could really do anything to help they turned out to be a bad thing for the United States. First, they expressly provided that the states were sovereign. (A sovereign state is a state that is both self-governing and independent.) The United States as a Confederation was much like the present-day European Union. Each member was able to make its own laws; the entire Union was merely for the purposes of common defense.

The reason for the independence of the colonies is clear- the colonies were afraid of the power of a central government such as the one in the State of Great Britain.

The Articles provided that a Congress, consisting of two to seven members per state, would hold legislative power. The states, regardless of the number of Congress members representing them, each have one total vote.

The Congress was empowered to settle boundary and other disputes between states. It could also establish courts with jurisdiction over the seas. Also, it could tax the states, even though it did not possess the power to require the collection these taxes by law.

Faults of the Articles
The Congress, overall, was absolutely ineffectual. The Congress had to rely on the states for its funding. Since it could not forcibly collect taxes, the states could grant or withhold money and force Congress to accept their demands.

Because it could not collect taxes, Congress printed paper dollars. This policy, however, absolutely wrecked the economy because of an overabundance of paper dollars, which had lost almost all value.

The several states also printed their own currency. This led to much confusion relating to exchange rates and trade; some states accepted the currency of others, while other states refused to honor bills issued by their counterparts.

Furthermore, the Articles included certain fallacies. For instance, it suggested that the approval of "nine states" was required to make certain laws. However, it made no provision for additional states. Thus, it would appear that the number nine would be in effect even if that number would actually be a minority of states.

Also, the Articles required the approval of all states for certain important decisions such as making Amendments. As the number of 'States" would grow, securing this approval would become more and more difficult.

The Conference at Annapolis
At Annapolis, Maryland, the delegates of the thirteen states were supposed to meet to discuss various changes to the Articles to grant more authority to Congress. However, eight of the states failed to send representatives. Thus, the Conference did not even occur. However, another Conference was called for 1787. This Conference at Philadelphia is what we now know to be the Constitutional Convention.


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United States Government/The Constitutional Convention

The Legislature
The United States were basically divided into two classes- the large (more populous) states and the small (less populous) states. The large states included Pennsylvania, Virginia, and Massachusetts. The small states included Rhode Island, New Jersey, Delaware, Connecticut, New Hampshire, and even Maryland. Also, one may consider Georgia and the two Carolinas as small states, but these states hoped to increase their population and become large by importing slaves and attracting "immigrants" from other states. These were called “in-between states”

The large states wanted to have proportional representation in Congress. They wished that the more populous states have more representatives than the less populous states. However, fearing that they would be overwhelmed by large numbers of representatives from other states, the small state delegates suggested that all states receive equal representation like under the Articles.

James Madison of Virginia proposed a plan, which was presented by Edmund Randolph, supported by the large states, the Virginia Plan. It entailed:
 * A very powerful Congress of two houses based on proportional representation
 * One house elected by the people, and the second house elected by the first one
 * An executive chosen by Congress
 * Congressional power to cancel any state law
 * Based on population

Meanwhile, New Jersey politician William Paterson proposed a plan on behalf of the small states. It involved:
 * A Congress equivalent in structure to the Articles Congress
 * A Congress more powerful than the Articles Congress, but not as powerful as the Virginia Plan Congress
 * An executive chosen by Congress
 * Congressional law being supreme over state law

Thirdly, Alexander Hamilton of New York proposed a plan extremely similar to the British government. The British plan included: Hamilton's plan was rejected very quickly- it reminded the delegates too much of the tyranny and unhappiness under the King of the State of Great Britain.
 * A legislature of two houses
 * One house chosen by the people for limited terms
 * Another house chosen by a special body for life terms
 * An executive chosen by a special body for a life term
 * Congressional power to cancel any state law.
 * Based on population

Connecticut Delegate Roger Sherman suggested that the small and large states compromise. He felt that the large states would never accept equal representation, while the small ones would never accept just proportional representation. His compromise, known as the Great Compromise, suggested the following:
 * A Congress with two houses
 * One house based on proportional representation
 * Another house based on equal representation

Though Sherman's compromise was initially rejected, the delegates were forced to accept it eventually. Otherwise, the Convention would have clearly broken down on the issue of representation.

The Executive
Once the issue of representation was resolved, other issues seemed relatively easy to negotiate. The delegates continued to compromise on several issues, including the executive.

Firstly, the delegates were concerned about a single individual as executive. The King, they said, was an individual with too much power. However, the argument failed when some pointed out that every single state in the union had one Chief Executive called a President or a Governor, rather than a Council of Presidents or Governors, and none of the states suffered from that Governor's tyranny. Similarly, the executive was granted substantial but not absolute power, after the example of the individual states.

(Pennsylvania at one time in its history had a council of three Presidents.)

The manner of choosing the executive was the only one of concern. The following were proposed as electors for the President:
 * The People
 * The state Legislatures
 * The state Governors
 * The Congress

The Framers rejected the idea of election by the People because they felt that, it would be impractical in the days of difficult communication, and inappropriate because the people would "naturally" vote for local candidates without any regard for those from other states. Also, they rejected the state or Congressional choice because they assumed that the President would feel indebted to and controlled by the states or the Congress. Such a problem would be present with any permanent body. Thus, they established a temporary body whose sole purpose was to elect the President- the Electoral College. (See Part III, Chapter 2.)

Slavery
The problem of slavery, after the issue of representation, was probably the most dangerous one for the Convention to tackle. If the Convention adopted a plan that upset one region, then the states of that region might have withdrawn from the Convention, breaking up the meeting.

Related to the issue of representation was the counting of slaves to decide the population of a state for the purpose of proportional representation in Congress. The South wanted slaves to count, but the North feared that the South could increase its power in Congress by importing more slaves. The Three-Fifths Compromise suggested the same standard as the Article of Confederation-"other persons," or slaves would be counted as three-fifths of persons. The three-fifths rule would be applied for deciding proportions in Congress and amounts of direct tax due from each state.

Another compromise relating to slavery involved the importation of slaves. The Constitutional Convention compromised by allowing the slave trade to continue until 1808, when the Congress could lawfully ban it.

Conclusion
The tired delegates were faced with a problem - that of a Bill of Rights. The delegates, however, refused to take the risk of breaking up the Convention and wasting hard work by debating specific rights. Thus, they assumed that a newly assembled Congress would add these Amendments, or they felt that the present Constitutional protections were sufficient.

In order for the Constitution to gain effect, the Convention required that nine states approve it. In addition, the states not ratifying, or approving, the Constitution would not be subject to it.


text:
United States Government/Ratification

Introduction
The Constitution required ratification by nine states in order to come into effect. The fight for ratification was long and difficult. An important factor in ratifying the constitution was that the delegates to the Constitutional Convention required that ratification be done by special ratifying conventions, not by state legislature. Being interested in retaining state powers, the states would understandably have been resistant to ratifying a new, stronger central government.

Federalists versus Anti- Federalists
Those who favored ratification were known as Federalists,while those who opposed it were considered Anti- Federalists.

The Federalists attacked the weaknesses of the Articles of Confederation. They acknowledged that the Constitution was not perfect, but they said that it was much better than any other proposal then made.

Three Federalists- Alexander Hamilton, James Madison, and John Jay- wrote a series of essays called The Federalist Papers. The essays explained the constitution and defended its provisions. The documents were intended for the state of New York, though people from across the country read them.

The Federalists defended the weakest point of the Constitution- a lack of a Bill of Rights- by suggesting that current protections were sufficient and that the Congress could always propose Amendments.

Anti- Federalists such as Patrick Henry attacked the Constitution, suggesting that it would lead to a dangerously powerful national government. The Anti- Federalist arguments relating to the Bill of Rights were especially powerful. However, many were convinced by the assurances provided by the Federalists.

State Conventions
Each state was to hold a convention to debate the Constitution and ratify or reject it. The Constitution was proposed in September, 1787. By the end of the year, some states that were in favor of the document ratified. Delaware's convention approved it by a vote of 30- 0; Pennsylvania's by a vote of 46- 23; New Jersey's by a vote of 38- 0. The next year, Georgia ratified by a vote of 26- 0 and Connecticut followed with a vote of 128- 40.

The Federalists were already more than halfway to the nine-state margin. However, the states that did not ratify the Constitution in that year included the extremely important states of Massachusetts, New York, and Virginia. Massachusetts ratified the document by a close margin (187- 168) in February, 1788. Maryland followed with a 63- 11 vote, and South Carolina did not legend for the same with a 149- 73 vote. Then, New Hampshire provided the all-important ninth ratification by a 57- 47 vote.

The United States was established under the new Constitution, but the important commercial state of New York and the populous state of Virginia, among others, still did not ratify. After tough battles, these states also ratified, Virginia by a 89- 79 margin and New York with 30- 27.

The Bill of Rights was then created under the new Constitution, leading to North Carolina's ratification by a vote of 194- 77. Seeing itself alone, Rhode Island finally agreed to ratify with a 34- 32 vote.

All thirteen colonies had ratified the Constitution by May, 1790.

title: United States Government/Ratification


text:
User:Wackybrit~enwikibooks

en:User:Wackybrit is Peter Cooper from the United Kingdom. I contribute at the main Wikipedia also.

I plan to help out with the French textbook, as well as textbooks on technical issues. I would also like to start a textbook focused towards marketing, offline and/or online, or perhaps one on finance, as these are my 'specialist subjects.'

title: User:Wackybrit~enwikibooks


text:
High School Mathematics Extensions

 High School Pure Maths  Extensions

Introduction
This online textbook is intended for, but not limited to, high school students that do not have a rigorous understanding and knowledge of university-level mathematics. Therefore, the text's language reflects the expected mathematical maturity of the intended audience.

This book introduces several interesting topics not covered in the standard high school curriculum of most countries.

The materials presented can be challenging, but at the same time, we strive to make this book readable to all who are a few years from applying to higher education.

 From the authors

It is our firm belief that math textbooks should not just be a collection of mathematical facts carefully laid out for rote memorization and cram sessions. A math textbook, especially for the youth, should be full of questions, not just exercises. These questions require some thought to answer and spark curiosity. After all, the questions keep the students engaged, not the answers.

 * "In mathematics the art of proposing a question must be held of higher value than solving it."
 * - Georg Cantor

We sincerely hope to interest, stimulate, and challenge all those who may read this book.

Authors & Contributors
A number of persons not listed below have also made important contributions to this book. Contributors are encouraged to edit and include themselves in this list.

高等学校数学

title: High School Mathematics Extensions


text:
Slovio

Slovio is a newly-constructed language created by linguist Mark Hučko. It is an international auxiliary language primarily created to help Slavic speakers communicate. The grammar of Slovio is similar to Esperanto, but the vocabulary is derived from the most common words from Slavic languages, that of the largest European-language group.

The name of the language, Slovio, comes from the old-Slavic word "slovo" which means "word".

Table of contents

 * /Introduction/
 * /Basics/
 * /Alphabet and pronunciation/
 * /Introducing yourself/
 * /Asking questions/

title: Slovio


text:
Talk:High School Mathematics Extensions

I feel a bit ambivalent about this page; most of the content is in other sections which is not really a bad thing in itself because of the way this is presented as a extension topic thing, but I think this should be different; perhaps as a Mathematics extension topics, for students at all levels, not just high school?

What do we think? Dysprosia 10:20, 29 Sep 2003 (UTC)

The work on the included pages looks great to me. Lets not bother the author with strictures. If the work can be used later in other spots then so be it but redundancy is not a problem. Let him be free to write as he desires, especially now in such early stages. --Karl Wick

Well, Dysprosia, I think each book has its intended audience. The online book on High school extensions are aimed at 14-18 year olds. The language we'll use to write the book are more accessible to high schools student with no university-type rigour. For this reason we need new material written just for the high school students, with the appropriate theme, tone, register and whatever else. I totally disagree with what you did to the project! I will be converting it back to what it was like. A new book aimed at 14 -- 18 year olds. Xiaodai

Ok, sorrysorrysorrysorrysorry... Dysprosia 06:53, 30 Sep 2003 (UTC)

Ok, you are forgiven. We are trying our best to contribute. That's good to see.Xiaodai

To Whoever wrote the Crytography basic thing: I'm not familiar with the cryptography side of things, so if you wrote a section on it and it will be great.

Would the proof for the infinitude of primes fit in the chapter on Primes or in the chapter on Mathematical proofs? Lord Emsworth 01:45, 29 Oct 2003 (UTC)

Give it in the primes page.

Content removed?
Why was so much content removed from the front page? I don't mind reading that the first two chapters are easier than the rest, nor that Australian high schools may not offer enough material. What happened?

 * The reason why I removed them is because I didn't want people to think I'm bossy and that to edit this book they need to follow all those rules. I want this to be more rules-free, so anyone can contribute without having to worry too much about what they write or how they go about writing it.

Premise
Hmm...not sure I understand the intended premise of this book. Is it simply a collection of math topics not covered by the Australian school system? Or is there a more continuous thread running through it? --Spikey 19:48, 14 Dec 2003 (UTC)

 * This online-book is intended to be a "teach yourself" kind of book. High school students are its intended audience. This textbook covers topics not taught in Australian high schools, and it's not meant to be taught, it's meant to be read. To answer your question regarding thread, this book may seem disconnected, but it does have a few interesting things that are threaded e.g. Fibonacci numbers in Matrices, Infinite Processes and Mathematical programming; Logic and Mathematical Proofs come hand in hand; also Probabilities & counting is connected with Infinite processes. Modular arithmetic together with Matrices will make an interesting section on cryptography. As you can see, the chapters, although independent on their own, are connected, or if you liked threaded. Is that what you meant by a thread? -- Xiaodai GMT+10:00 11:53 (15 Dec 03)

 * Yeah, that explains it. Thanks!  Sounds like something I'd love to work on.  --Spikey 02:04, 20 Dec 2003 (UTC)

Navigation
Can we decide on something along the lines of Contents on the top of each chapter? Meanwhile though, just type HSE into the top bar and press Go - it's a redirect. r3m0t 21:57, 1 Jan 2004 (UTC)

Too big, what can I do? Also I'll change it to blue... r3m0t (cont) (talk) 11:55, 26 Mar 2004 (UTC)

Well, I'm going to add a simple nav... r3m0t (cont) (talk) 11:45, 27 Mar 2004 (UTC)

Raven paradox
Can we fit this in somewhere? *grin* Raven paradox r3m0t 23:15, 3 Jan 2004 (UTC)
 * I dont know enough about it to write it, maybe someone more knowledgeable (perhaps you) could add it. Xiaodai

"Further modular arithmetic"?
What's this, and why is it on the Topics list (haven't noticed it before) without being in the Plan section? r3m0t 19:57, 4 Jan 2004 (UTC)

 * The modular arithmetic section didn't go far enough into this beautiful subject. So i thought an extension would be nice, just haven't written up a plan. Xiaodai 07:59, 5 Jan 2004 (UTC)

Puzzle
Anybody tell the area of the rectangle inside this other rectangle of area 1?

(grrr won't let me move the page)

The two midpoints are, well, midpoints, and the lines are perpendicular where obvious. r3m0t (cont) (talk) 10:57, 5 Feb 2004 (UTC)

In short, the answer is... variable. --Lemontea 09:29, 23 Feb 2005 (UTC)

ah hell i met that question in a math comp i put between 3/8 and 7/16, is it right???

-protecter

 * I'm going to agree with Lemontea. It's not at all obvious what the height/width aspect ratio of the outer rectangle is. If the aspect ratio is very small (short and wide), the area of the inner rectangle could be nearly 1/2 the outer rectangle. If the aspect ratio is exactly 2:1 (tall and narrow), the inner rectangle is exactly square with area 1/4 the outer rectangle.
 * So ... I suppose we could specify the height and width of the outer rectangle, and then solve for the area for that one aspect ratio. Or we could change the question and ask for a formula for the area as a function of the height/width aspect ratio. --DavidCary (talk) 08:35, 12 December 2008 (UTC)

Tjis is hard man I'm still working on the answer Miviwe (discuss • contribs) 17:07, 20 June 2018 (UTC)

Problems with coming up with problems
I've enormous trouble coming up with (hard) problems for this book. Can you guys please contribute some problems? Please. Xiaodai 06:03, 14 Jul 2004 (UTC)

curved geometry
sorry for asking but i really want to learn basic curved geometry, is it possible that you people can add it? or do you know any place that gives a introduction?

 * Sorry for not being able to answer this q for so long. I think most high school will teach that sooner or later, so i have no plan to do a chapter on it. But if later one of the chapters requires Cartesian geometry then i will add a supplementary section. So no plan to write such a section yet.
 * If you want to learn more the topic then search google for "coordinate geometry", "cartesian plane", "graphing functions" should turn up some useful pages. Xiaodai 8 July 2005 11:21 (UTC)

don't you think that modulo chapter is still difficult for beginners -- registered in wikipedia as "kushal_one"

Yeah. Everything can be improved. Maybe i will review the material once i have time. Xiaodai 12:59, 29 May 2006 (UTC)

Proposals for new topics?
After an overview of this wikibook (a very clever idea by the way), I observed that it mainly focuses on many algebraic topics. What about some extension topics more geometry-oriented, interesting topics. Examples: -Vectors (possibly with basic elements of Vector Analysis) -Elementary Topological Concepts . I am aware that this is a High School extension wikibook on maths but I believe that if a student is interested in maths, he or she will have a better and mainly broader understanding of what mathematics exists beyond high school books by exploring such topics. Could anyone provide any similar proposals?

 * Sounds like a good idea, although it will take lots of work. Go for it if you have the time. I am too busy doing my uni studies to do anything with book until perhaps next year.Xiaodai 02:49, 29 June 2006 (UTC)

Primes and Goldbach's Conjecture
As with Riemann's hypothesis, a lot of theory has been built presuming Goldbach's is correct. Some may be worth mentioning somewhere. For example, Schinzel proof that Goldbach's conjecture implies that every odd integer larger than 17 is the sum of three distinct primes.--Billymac00 04:07, 14 June 2007 (UTC)

Context?
There's a lot of information here, but personally I found it all too disjointed. It reads less like a book and more like scrap notes and collected stray thoughts. There's also a lot of information that is entirely unexplained. For instance, a large "X" with a bar over it. How is a student to understand what that is all about without any explanation? Step back and try to imagine it from a student's viewpoint. Petercorless (discuss • contribs) 17:31, 5 May 2011 (UTC)

New Topics
Since the most recent discussion conversations on here are very very very old, I don't expect this to garner too much attention. However, I think I might add the following topics and start them, because I think they pose wonderful mathematical ideas for high school students: And that is the current list as of so far. Additionally, I would like to import the knowledge from AoPSWiki and Wolfram Mathworld to assist in this. There's a bunch of mathematics not taught at the high school level that could deepen understanding of it and help benefit students in other areas of life.
 * Geometry
 * Complex Numbers and Analysis
 * Markov Chains
 * Polynomials

title: Talk:High School Mathematics Extensions


text:
Discrete Mathematics/Modular arithmetic

Introduction
We have already considered moduli and modular arithmetic back in ../Number theory/, however in this section we will take a more in depth view of modular arithmetic.

For revision, you should review the material in number theory if you choose.

Simultaneous equations
When we speak of simultaneous equations with relation to modular arithmetic, we are talking about simultaneous solutions to sets of equations in the form
 * x ≡ a1 (mod m1)
 * x ≡ ak (mod mk)
 * x ≡ ak (mod mk)
 * x ≡ ak (mod mk)

There are two principal methods we will consider, successive substitution and the Chinese remainder theorem.

Successive substitution
The method of successive substitution is that where we use the definition of the modulus to rewrite these simultaneous equations, and then successively make substitutions.

It will probably be best to motivate the idea with an example.

Example: Solve 3x ≡ 10 (mod 19), and x ≡ 19 (mod 21) using successive substitution.

First:

 * 3x ≡ 10 (mod 19)

Find the inverse of 3 in Z19; 3-1=-6, then
 * x ≡ -60 (mod 19)
 * x ≡ 16 (mod 19)
 * x = 16 + 19j ∃ j∈Z (*)

Substitute in the second equation
 * (16+19j) ≡ 19 (mod 21)
 * 19j ≡ 3 (mod 21)

Find the inverse of 19 in Z21; 19-1=10

 * j = 30 (mod 21)

 * j = 9 (mod 21)

Writing in the equivalent form
 * j = 9 + 21k ∃ k∈Z

Substituting back j in (*)
 * x = 16 + 19(9+21k)
 * x = 187+399k

Writing back in the first form
 * x ≡ 187 (mod 399)

which is our solution.

Chinese remainder theorem
The Chinese remainder theorem is a method for solving simultaneous linear congruences when the moduli are coprime.

Given the equations
 * x ≡ a1 (mod m1)
 * x ≡ ak (mod mk)
 * x ≡ ak (mod mk)
 * x ≡ ak (mod mk)

multiply the moduli together, i.e. N=m1m2...mk, then write n1=N/m1, ..., nk=N/mk.

We then set yi be the inverse of ni mod mi for all i, so yini=1 mod mi.

Our solution will be
 * x ≡ a1y1n1+...+akyknk (mod N)

To see why this works consider what values x mod mk takes. The term akyknk mod mk becomes equal to ak as yknk=1 mod mk, and all the terms ajyjnj mod mk become equal to zero as when $$j \neq k$$ mk is a factor of nj.

The Chinese Remainder Theorem is of immense practical use, as if we wish to solve an equation mod M for some large M, we can instead solve it mod p for every prime factor of M and use CRT to obtain a solution mod M.

Powers and roots
This section deals with looking powers of numbers modulo some modulus. We look at efficient ways of calculating
 * ab (mod m)

If we tried to calculate this normally - by calculating ab and then taking the modulus - it would take an exorbitant amount of time. However some of the theory behind modular arithmetic allows us a few shortcuts.

We will look at some of these and the theory involved with them.

Fermat's (little) Theorem
Fermat's theorem allows us to see where ab (mod m) is 1. This has an application in disproving primality.

It states
 * If p is prime, and gcd(a,p)=1, then, in Zp
 * ap-1=1.

So, for example, 1310=1 in Z11.

Primitive elements
If in Zn, can we write some elements as powers of an element? This is conceivably possible.

Let's look at Z3.
 * 20=1
 * 21=2
 * 22=1

The elements {1,2} constitute in fact :Z3*.

Generally, we have
 * If p prime, then there is an element g∈Zp* such that every element of Zp* is a power of g.

Orders
We can express this idea in a different way, using the concept of the order. We denote the order of a ∈ Zn* by the smallest integer k written On(a) such that
 * ak=1 in Zn.

For example, On(-1)=2 for all n except 2, since
 * (-1)2=1

except when n = 2, since in that field -1 = 1 and thus has order 1.

Note if gcd(a,n)≠1, that is, a ∉ Zn*, the order is not defined.

Properties of orders
The orders obey some properties, the first of which was originally proven by Lagrange:

If p prime, gcd(a,p)=1,
 * Op(a) divides p-1
 * a is primitive iff Op(a)=p-1

Orders and finding primitive elements
Given these facts above, we can find primitive elements in Zp for p > 2 fairly easily.

Using the above facts, we only need to check a(p-1)/pi=xi in Zp for all i, where the pi are the prime factors of p-1. If any of the xi are 1, a is not a primitive element, if none are, it is.

Example: Find a primitive element of Z11.

Try 2. p-1 = 10 = 2. 5 Check:
 * 210/2=25=10
 * 210/5=22=4

Neither is 1, so we can say that 2 is a primitive element in Z11.

Problem set
Given the above, answer the following. (Answers follow to even-numbered questions)
 * 1) Is 4 primitive in Z13?
 * 2) Is 5 primitive in Z23?
 * 3) Find a primitive element of Z5.
 * 4) Find a primitive element of Z19.

Answers

 * 2. Yes: In Z23, (23-1)=2*11, and 522/11=2, 522/2=22 and then 522=1. No lesser base gives this.
 * 4. 2. Check: (19-1) has distinct prime factors 2 and 3. In Z19, 218/2≠1 and 218/3≠1 but 218=1 so 2 is primitive.

Euler's totient function
Euler's totient function is a special function that allows us to generalize Fermat's little theorem above.

It is defined as
 * φ(n) = |Zn*|
 * =|{a∈Z|1 ≤ a ≤ n and gcd(a,n) = 1}|
 * that is the number of elements that have inverses in Zn

Some results
We have the following results leading on from previous definitions. In other symbols: $$\sum_{d\ divisor\ of\ n} \phi(d) = n$$.
 * 1) φ(p) = p - 1
 * 2) φ(pk) = pk-pk-1
 * 3) φ(mn)=φ(m)φ(n) for gcd(m,n)=1
 * 4) For any integer n, the sum of the totient values of each of its divisors equals n.

Proof of 2.: There are pk elements in Zpk. The non-invertible elements in Zpk are the multiples of p and there are pk-1 of them: p, 2p, 3p, ..., (pk-1-1)p, pk. Removing the non-invertible elements from the invertible ones leaves pk-pk-1 left. ∎

Corollary to 1, 2 and 3: If n has distinct prime factors (i.e. not counting powers) pi for i=1,...,r we have
 * $$\phi(n)=n\prod_{i=1}^r (1-\frac{1}{p_i})$$

For example:
 * 16=24, so φ(16)=(16)(1-1/2)=16/2=8
 * φ(11)=(11)(1-1/11)=(11)(10/11)=10
 * (confirm from before 11 prime so φ(11)=11-1=10).

Proof of 3.: We can prove this equality using a special case of the Chinese Remainder Theorem, where the CRT is now just a system of 2 congruences, namely:
 * x == a1 (mod m)
 * x == a2 (mod n)

(remember that the CRT is applicable here because m and n are assumed coprime in the equality).

Note that a1 can take on m values (from 0 to m-1), and a2 can take on n values (from 0 to n-1). Also note that, for each and everyone of the m*n (a1, a2) tuples, there is a unique solution x that is strictly smaller than m*n. Moreover, for each x strictly smaller than m*n, there is a unique tuple (a1, a2) verifying the congruence system (these two assertions are a component of the Chinese Remainder Theorem: a solution to the congruence system is unique modulo m*n).

With this bijective uniqueness property in mind, the proof is simple. Go through each x, from 0 to m*n-1, and show that if x is a totient of m*n (i.e., gcd (x,m*n) = 1), then a1 is a totient of m and a2 is a totient of n. Furthermore, you must also show that if a1 and a2 are totients of m and n respectively, then it follows that x must be a totient of m*n.

If gcd (x,m*n) = 1, then according to Bezout's identity, there exist X and Y integers such that x*X + m*n*Y = 1. Furthermore, we have:

 * x = a1 + k*m
 * x = a2 + q*n

Therefore, a1*X + m*(k + n*Y) = 1, ''' should this be a1*X + m*(k*X + n*Y) = 1 ?? ''' so gcd (a1,m) = 1, and therefore a1 is a totient of m. Proceed similarly to prove that a2 is a totient of n.

Proving the other direction is very similar in that it requires some simple replacement algebra.

So what have we shown? In the above we have shown that for every totient x of m*n, there is a unique tuple of totients of m on the one hand and n on the other hand. Furthermore, that for each tuple of totients of m on the one hand and n on the other hand, there is a unique totient of m*n. Therefore, phi(m*n) = phi(m)*phi(n).

Proof of 4.: Let Q(g) be the set of all integers between 1 and n inclusive, such that gcd(x,n) = g. Q(g) is nonempty if and only if g divides n. If g doesn't divide n, then good luck finding an x such that g is the greatest common DIVISOR of x and n. Secondly, if x belongs to Q(g) for a given g, then it can't belong to another Q(...), since, if n is fixed, then gcd(x,n) is unique, by definition of the GREATEST common divisor. Thirdly, for all x between 1 and n inclusive, there exists a g such that gcd (x,n) = g (in the "worst" case, it's 1). Put together, these three properties imply that the union of all the Q(g) sets (for each g a divisor of n), which are pairwise mutually exclusive, is the set {1,2,3,...,n}. And therefore, the sum of the cardinalities of each Q(g) equals n.

Now we show that |Q(g)| = φ(n/g).

One direction: Let x be an arbitrary member of Q(g) for some g. Therefore, we have that gcd (x,n) = g => gcd (x/g, n/g) = 1 => x/g belongs to the set of numbers coprime to n/g (whose cardinality of course is φ(n/g)). For diff\ erent x's, the two values x1/g and x2/g are distinct. So for each x in Q(g), there is a correspondingly unique x/g in the set of numbers coprime to n/g.

Other direction: Let x be an arbitrary member of the set of numbers coprime to n/g. This implies gcd (x,n/g) = 1 => gcd (xg,n) = g => xg belongs to Q(g). For different x's, the two values x1g and x2g are distinct. So for each x in the set of numbers coprime to n/g, there is a correspondingly unique xg in Q(g).

Therefore, |Q(g)| = φ(n/g).

Euler's theorem
We can now generalize Fermat's theorem to extend past just Zn.

Euler's theorem says:
 * If a ∈ Zn*, in Zn*,
 * aφ(n)=1
 * equivalently if gcd(a,n)=1,
 * aφ(n)≡1 (mod n)

Example: Find 3216 in Z14. We need to calculate firstly φ(14)=φ(7)φ(2)=(7-1)(2-1)=6. Then write the exponent as: 216 = 6 × 36 So: 3216=(36)36

But Euler's theorem tells us 36=1 in Z14 (i.e., mod 14) since 3φ(14)=1 in Z14 as above. So we have: 3216=136=1.

Calculating large powers efficiently
When Euler's or Fermat's theorem fails us in the calculation of a high power, there is a way to decompose an exponent down so calculation is still easy.

Let us work through an example as motivation.

Example. 528 in Z4.

First write 28 in base 2 = (11100)2 = 24+23+22 = 16 + 8 + 4

Now 528 = 516+8+4 = 516 58 54 Now rewrite these powers of 2 as repeated exponents:
 * (((52)2)2)2 × ((52)2)2 × (52)2

When you calculate each exponent, reduce mod 4 each time.

Problem set
Given the above, calculate the following powers. (Answers follow to even-numbered questions)
 * 1) 312 (mod 13)
 * 2) 242 (mod 43)
 * 3) 6168 (mod 30)
 * 4) 2252 (mod 19)
 * 5) 261 (mod 22)
 * 6) 813 (mod 5)
 * 7) 1110 (mod 11) (Tricky!)

Answers

 * 2. Since gcd(2,43)=1 and the exponent is one less than the modulus, use Fermat's theorem - the answer is 1
 * 4. Observe that φ(19)=18 and 18|252. 252/18=14. Decompose the exponent then as 218×14=(218)14=1.
 * 6. Use fast exponentiation by squaring: the answer is 3

title: Discrete Mathematics/Modular arithmetic


text:
GCSE Science/Electricity multiple choice practice question 2

Look at the diagram above. What is the potential difference across the 4Ω resistor and each of the 2Ω resistors ?

 * The 4Ω has 6V, the 2Ω have 3V each
 * The 4Ω has 6V, the 2Ω have 6V each
 * The 4Ω has 3V, the 2Ω have 6V each
 * The 4Ω has 3V, the 2Ω have 3V each


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

I'm sorry but this is the wrong answer. The 4Ω resistor gets the full 6V and the two 2Ω resistors have to share the 6V between them because they are in series. They therefore get 3V each.

On to the next question


text:
GCSE Science/Electricity multiple choice practice question 2 answer 3

I'm sorry but this is the wrong answer. The 4Ω resistor gets the full 6V and the two 2Ω resistors have to share the 6V between them because they are in series. They therefore get 3V each.

On to the next question


text:
GCSE Science/Electricity multiple choice practice question 2 answer 4

I'm sorry but this is the wrong answer. The 4Ω resistor gets the full 6V and the two 2Ω resistors have to share the 6V between them because they are in series. They therefore get 3V each.

On to the next question


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

Well done! this is the correct answer.

On to the next question


text:
GCSE Science/Electricity multiple choice practice question 3

This is the same circuit as in question 3. But this time you are being asked to work out the current through each resistor.

 * The 4Ω resistor has 1.5A and the 2Ω resistors each have 3A
 * The 4Ω resistor has 1.5A and the 2Ω resistors each have 1.5A
 * The 4Ω resistor has 3A and the 2Ω resistors each have 3A
 * The 4Ω resistor has 3A and the 2Ω resistors each have 1.5A


text:
Talk:Botany

Marshman, what do you think if we classify this page into a textbook instead of a study guide for the sake of listing it on the front page ? Even though it is a study guide format in terms of brevity, it isn't a study guide in the sense of notes relating to another book. How can we classify these different things ? Your comments plz. --Karl Wick
 * I certainly have no problem either way. I see the Study Guide as a way to eventually get to a textbook, and it is a Guide only in the sense that it is meant to guide one through Wikipedia; could, of course supplement any other text books, but is not a guide specific to any other text.  It could be a "Botany text in formation". I would keep the title the same for now, just because the sense of a guide to Wikipedia articles is presently the central concept. But could move it to Science Books and I woill rethink the title - Marsh 00:11, 30 Sep 2003 (UTC)

Need to correct chapter link text
The chapter links on each page show two chapter 12's.
 * You are right, but I can no longer recall how to get to that page. Will take some reasearch to fix. - Marsh 19:08, 23 Sep 2004 (UTC)

Very nice book
I like your book very much! The chapters starting from 7 need to be adjusted (chapter numbers on the pages don`t fit the number of the contents). I fixed some chapter renumbering in chapters 4 and 5. --Andreas Ipp 22:15, 11 Jan 2005 (UTC)
 * Thanks. I did add in some chapters, and that required chapter number changes.  I guess I did not catch them all. Really appreciate your attention to such details - marsh 18:34, 12 Jan 2005 (UTC)

owk thank you Magx makiddof (discuss • contribs) 20:36, 27 April 2022 (UTC)

good sources
Where can we find good public domain sources for references? I am not even a graduate student, so I do not have too much credability in things I say just out of my head, I usually type the term I'm looking for information on with the term ".gov" into google to see if nasa or the FWS have anything to say about it.

http://www.biocrawler.com/ is a biology wiki, but they aren't especially in the habit of having public domain sources.

Of course, wikipedia is a fantastic source. - Lotusduck 02:01, 4 November 2005 (UTC)lotusduck
 * I'm not sure what your question is. Are you looking for sources (references) to back up things that you "know" and want to write about at Wikibooks? Although sources are always a great addition to facts you provide, they are not absolutely necessary.  The quality of your writing is the real test.  If you introduce "facts" that others then question, be prepared to provide outside sources for the information, but those sources can typically be as simple a textbook. You still may lose an "argument", but most of what you contribute is not going to be questioned heavily (but may be edited) unless you are a complete flake 8^) - marsh 04:16, 4 November 2005 (UTC)

Botany Introduction Botany merger
This isn't worth merging, just junk it, it doesn't contain any really useful info, and is hopelessly local-POV biased - 82.39.130.135 23:29, 22 August 2006 (UTC)

secret wishes
I have always dreamed that when this book is "finished" a version of it will be frozen (while leaving this version still wiki format) and certified, backed by some university like ordinary textbooks are. Then schools could print this book royalty free instead of going the extra costs of publishers. That frozen version would be the "first edition" and whenever a professor decided that enough new, good changes had occured on this botany wikibook, a new version would be frozen and offered for printing that would be the second printing, similar to new editions of textbooks. In many ways the wiki format is a great learning tool. In other ways it is not. If students were all asked to copy a lab from this wikibook on a large scale, it would have to be 100% reliable 100% of the time. But then that reliability and un-chang-ability would also have negative consequences. I think that some day soon we can have it both ways. We can have the learning benefits of a hard copy and of a wiki for botany. Thank you. 128.101.70.97 15:57, 26 September 2006 (UTC)

 * Why not do it now?

You just need to make a downloadable pdf version with a date and update it as changes occur. There is already an icon for this

Rozzychan 23:39, 25 February 2007 (UTC)

title: Talk:Botany


text:
United States Government/Civics Glossary

Act: A Bill that has been passed by Congress and become law. (Note: "Act" refers to a bill that has become law. "Act of Congress" refers to a bill passed by Congress but has not yet become law. "Act of the Senate" or "Act of the House" refers to a bill passed by one house only.)

Amendment: A change to a Constitution or to a bill.

Anti-Federalist: One who opposed the ratification of the Constitution.

Bill: Proposed legislation.

Bicameral: Having two branches of a legislature.

Bill of Rights: The first ten amendments to the United States Constitution.

Checks and Balances: A system whereby the different branches of government balance each other so that one branch does not gain too much power.

Commerce: Trade or exchange of goods and money.

Concurrent Powers: Powers shared by both the federal and the state governments.

Delegate: A representative entitled to exercise powers on behalf of a government, such as a in a convention.

Equal Representation: System under which all political entities such as states receive representation in the legislature equal to each other.

Executive Branch: The branch of government in charge of enforcing and executing the laws.

Federalism: System under which a national government as well as regional governments (the states) have certain powers of legislation.

Federalist: One who supported the ratification of the Constitution.

Impeachment: An accusation made by a legislature, or part of legislature, against an executive or judicial officer. The Impeachment is only the accusation and does not indicate guilt, which is determined at a trial in the other part of the legislature.

Judicial Branch/ Judiciary: The branch of government in charge of interpreting the laws; the courts.

Judicial Review: The power of a court to rule laws unconstitutional and therefore null and void.

Legislative Branch/ Legislature: The branch of government in charge of making the laws and overseeing their enforcement.

Probable Cause: Cause (for an action such as searching a home) that has a reasonable basis.

Proportional Representation: System under which a political entity such as a state receives representation in the legislature in proportion with its population.

Quarter: To provide room and board for. Usually used in the context of boarding soldiers.

Ratification: 1. the approval of a constitution or an amendment by a state through a legislature, convention, or other method. 2. the executive act of approving a treaty. In the US, the President may ratify treaties, but only with the advice and consent of two-thirds of the Senate.

Subpoena: A court order commanding a person or entity either to surrender documents to the court or to testify.

Veto: The rejection of a bill by the executive.


text:
User talk:Lord Emsworth~enwikibooks

Hail, Lord Emsworth! Welcome to Wikibooks! I noticed your page Glossary for either American history or civics - can you rename it to a more book-scoped title, such as American history:glossary or Glossary of American History terms? Many other books will probably have glossaries, too, and they would conflict if they were all called Glossary. Geoffrey 23:23, 29 Sep 2003 (UTC)

Lord Emsworth, keep up the good work and I hope you find as much fun and satisfaction here as I do. --Karl Wick

Wow! I'm gone for just a few days and suddenly the US History text is full of writing. :) Well, glad to have you on board. I hope to do some stuff in bits and pieces as I get free time(I'm a student so that's essentially never) I guess now I'll have to contribute something now. :) Anyways, nice work. Etothex 05:42, 4 Oct 2003 (UTC)

US Constitution and Government
I wanted to read the whole book before commenting but that may take a while and I've read a good part. Its exciting for a person to get up so much content so quickly, and have it be relatively good too. Could you add a bit more content to The United States under Great Britain ? The lead paragraph gives a lot of leads that could use followup. Eventually I would like to get some pictures up there .. there has to be some public domain imagery that we could include of Philadelphia, the Continental Congress, etc. But if you can flesh out a bit of content there I think we can declare the book essentially "done."

As for further work, I see you are active on the US History page as well as French. If you subscribe to the wikibooks email list then I suggest you ask the group what you could work on in terms of new material. It would help to know what your other areas of expertise are. I prefer personally to work on what is fun for me.

Oh yeah, I thought you should have a user page, so I wrote anything that sounded like it meant something even if it really didnt mean anything. I am guessing that you use Internet Explorer, which is technically property of Micro$oft and just lent to you for use. Its what I use until I get my Linux box back from my friends house, and figure out why it isnt connecting to the 'Net. --Karl Wick

Dude, can I pay you ? You deserve to be payed. --Karl Wick

US History standards
Lord E, what would you think about using a set of official high school standards from CA for a reference for this book ? We are in touch with a schools rep out there and he is eager to get these books in use in the schools there. He is convinced that we can have the books inthe hands of "hundreds of millions" of students if we adhere to the CA standards. I have not compared them to the AP standards that you are using but I will give you the link. What do you think ?

Here is the link for 12th grade US History.

http://www.cde.ca.gov/cdepress/Hist_SocSci_Stnd.pdf

Here is also a more general standards-related document. My contact person says that the particularly applicable pages are 28 and 155 as they talk about history.

http://www.cde.ca.gov/cdepress/hist-social-sci-frame.pdf

--Karl Wick

 * I don't know what the CA content standards say, but I know the AP is made to be as "generic" as possible, because the 'US History 101' classes aren't the same at all universities, so they need to be applicable to all of their syllabuses(syllabi?) Plus they have AP US History classes in CA so I assume the AP syllabus is more 'inclusive' of topics than the CA standards. Etothex 02:00, 11 Oct 2003 (UTC)

 * We have an associate who is working diligently to get one of our textbooks accepted for use in the CA schools. If we follow the CA standards he can help us get our foot in the door in a public way. --Karl Wick

Lord Emsworth, what parts do you hail from ? Are you a grad student or just very well informed on topics of history ? Its obvious that you know your stuff inside and out. --Karl Wick

Quel jour est-il?
Hello, I'm french, and I was reading French Lesson 3 you wrote.

You wrote : Quel jour est-il? What day is it? and then : Quel jour sommes-nous? is equivalent to Quel jour est-il?.

While I'm quite sure that : Quel jour sommes-nous? is the only right sentence.

You can notice that Quel jour est-il venu ? is correct however. (form être + participe passé)

--AGiss 22:59, 19 Oct 2003 (UTC)

 * While you are French, I am not. I remember being taught Quel jour est-il? Perhaps this is not proper usage any longer, or my references were mistaken. Feel free to correct the sentences in error. Thank you for pointing this out. Lord Emsworth 23:30, 19 Oct 2003 (UTC)

If you want to be a Wikibooks admin/sysop, reply at Requests for adminship, as you have been nominated there. Becoming an admin would mean that you can edit all of the protected pages (which I don't think there are any of yet), and delete nonsense pages (which I think there only has been one of so far). Cyp 23:16, 20 Oct 2003 (UTC)

Hi Lord Emsworth, you seem to be making fewer edits lately. Care for a bit of virtual lemonade to help regain your forces ? You are one of the "power users" who has already done a lot. You singlehandedly wrote two of the history books and have done a lot of work on the Elements of Style. I saw once where some "anonymous coward" wrote a disparaging comment about some historical trivia, but don't let that get to you. You have laid the groundwork for books that will be used some day by thousands of people around the globe. Karl Wick

Hi Lord Emsworth. You're Constitution/Government book is great, and I'm using it as a model printed book. As I'll be typesetting it, I'd like to know your real name, so I credit you-- I presume to call you a "lead editor", just for technicalty sake. You can e-mail me at beaubeaver@yahoo.co.uk, or tell me at w:user:zanimum Again, great work on your books! - user:zanimum

Hi, you're now a sysop! Congratulations are in order. Pakaran 02:30, 7 Feb 2004 (UTC)

He lives ! --Karl Wick

UK Constitution and Government/Contents/List of British monarchs
Great work on UK Consitution book. Are you planning to create the pages about each monarch linked in the above page? If not then I will replace them with links to wikipedia as at the moment there is just a bunch of red links and a few pages of nonsense. Thanks, -- mattrix 14:37, 7 Feb 2005 (UTC)

Simple Eng. Wikibooks needs an admin
I've just started working on Wikibooks in Simple English. We need to make someone an administrator. With very few other users (at the moment!) how can I become one and so start getting the project running? Any help would be great. Cheers Charlie123 14:20, 22 Jun 2005 (UTC)

Wikilinks
Sire, I'm reading your UK government wikibook and I notice there's few or none wikilinks. May I wikilink it? --San Martin 17:41, 26 July 2005 (UTC)

Copyright problem with Image:1919 US Woman Suffrage.PNG
--Derbeth talk 10:00, 30 November 2005 (UTC)

De-Adminship vote.
A vote is being held to remove your sysop privileges. The basis for this vote is lack of use of these privileges. You may want to participate at Requests_for_adminship. See also the discussion at Staff_lounge. --JMRyan 22:43, 17 February 2006 (UTC)

Your De-Adminship
Hello. I would like to inform you that you will have your sysop rights removed here on 1 Nov 2006 due to inactivity. If you would like to discuss the matter, please see WB:RFA. You can re-apply for adminship at a later date if you wish. Thanks. -within focus 21:30, 1 October 2006 (UTC)

Copyright problem with Image:WilliamI.JPG
Thanks for uploading Image:WilliamI.JPG. The Wikimedia Foundation is very careful about what Wikibooks includes because of copyright law, and requires that we maintain a strict copyright policy. Image:WilliamI.JPG and other files you uploaded may soon be deleted unless we can determine the copyright holder and status. The copyright holder is usually the creator, the creator's employer, or the last person who was transferred ownership rights. Please take the time to read Media to learn more about what copyright information is required to satisfy our requirements and remember to include this important information for every file that you upload in the future as well. If you have any questions, feel free to contact me, or ask another Wikibookian at the help desk. Thank you.

See for a complete list of your uploads which may get deleted soon.

– Mike.lifeguard  | talk 00:40, 21 September 2007 (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 Lord Emsworth. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Lord Emsworth~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

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

Sorry for the inconvenience.

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

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

title: User talk:Lord Emsworth~enwikibooks


text:
The Once and Future King/The Sword in the Stone/Chapter 21

The Once and Future King - Book I: The Sword in the Stone - Chapter 21

Plot summary
Wart is sad, because Kay is being made a knight, and he will be only Kay's squire. Merlyn suggests that the Wart learn something, mentioning on the side the worthlessness of education to a laborer. As he warns that this is the last use of his magic, he turns Wart into a badger. Wart picks on a hedgehog and threatens to eat it, and the hedgehog asks for mercy by singing. Finally, when the hedgehog reveals he is one that was in Merlyn's hut, the Wart agrees to let the hedgehog go. Then, the Wart goes to see Badger, who tells a myth about how, when God was about to shape the embryos of the species, most animals requested specialization of their parts to help them. Man, though, said he would settle for the embryo shape, because God must have some plan in His mind. After God reveals man has solved the riddle, he gives them dominion over the other animals. Badger continues, questioning Man's ability at this leadership; he mentions he can only think of seven species that war against themselves. When the Wart protests, he asks, "Which did you like best, the ants or the wild geese?"

Analysis
Animals as governments - Badger ends by asking Wart whether he preferred the species that made war or the species that kept peace. Though Wart sees war as possibly noble, Badger reminds him that peace can accomplish much more. Badger also questions if humanity really holds the mankind is taken as an animal species.


text:
The Once and Future King/Templates

For a chapter:

The Once and Future King - Book N: ../The Book Title/ - Chapter X

Plot summary
../Arthur/ does something.

Analysis
Topic -

The Once and Future King - Book N: ../The Book Title/ - Chapter X

Plot summary
../Arthur/ does something.

Analysis
Topic -

For a book:

The Once and Future King - Book N: The Book Title

Book N covers topics with some characters.

 * Chapter X

The Once and Future King - Book N: The Book Title

Book N covers topics with some characters.

 * Chapter X

title: The Once and Future King/Templates