Multiplication Mastery

Multiplication in Mathematics

Multiplication is one of the four elementary mathematical operations of arithmetic, alongside addition, subtraction, and division.
The result of a multiplication operation is called a product, and it involves the multiplication of two numbers known as the multiplier and the multiplicand.
Multiplication is often denoted by the cross symbol ×, by the mid-line dot operator ⋅, by juxtaposition, or, on computers, by an asterisk *.
The commutative property of multiplication states that the order of the factors does not affect the result of the multiplication.
Systematic generalizations of multiplication define the multiplication of integers, rational numbers, and real numbers.
Multiplication can be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have some given lengths.
The inverse operation of multiplication is division, and the division of a number other than 0 by itself equals 1.
Multiplication is defined for other types of numbers, such as complex numbers and matrices.
Multiplication tables of memorized or consulted products of small numbers are often used in pencil and paper multiplication.
Historical civilizations, such as the Egyptians, Babylonians, and Chinese, had their own methods of multiplication.
Modern methods of multiplication are based on the Hindu-Arabic numeral system and have greatly reduced the need for multiplication by hand.
The product of measurements depends on the types of measurements, and the general theory is given by dimensional analysis.Multiplication: Definition, Properties, Axioms, and Applications
Multiplication is an operation in mathematics that is used to find the product of two or more numbers.
The product of an infinite sequence is defined as the limit of the product of the first n terms as n grows without bound.
Exponentiation is repeated multiplication, where the exponent indicates how many times the base appears in the expression.
For real and complex numbers, multiplication is commutative, associative, and distributive over addition.
Peano arithmetic proposes axioms for arithmetic based on natural numbers, including two axioms for multiplication.
The axioms for integers define them as equivalence classes of ordered pairs of natural numbers, with the multiplication axiom defined accordingly.
The product of non-negative integers can be defined with set theory using cardinal numbers or the Peano axioms.
Groups are sets that satisfy the axioms of closure, associativity, identity element, and inverses under multiplication.
The set of non-zero rational numbers forms an abelian group under multiplication with identity 1.
Invertible square matrices over a field form a non-abelian group under matrix multiplication.
The integers under multiplication do not form a group, except for 1 and -1, which have inverses.
Multiplication has been generalized to more abstract types of numbers and non-numerical entities, such as matrices and quaternions.

A Brief History of the United Kingdom

The United Kingdom was formed in 1707 with the political union of England and Scotland, creating Great Britain. In 1800, Ireland was added to create the United Kingdom of Great Britain and Ireland.
The 19th century saw political and economic reforms, including the expansion of the franchise and the creation of the welfare state. The Victorian era was marked by prosperity and British dominance in the world economy.
The First World War was a total war in which the UK, France, Russia, and the US were allied against Germany. The resulting League of Nations was a favorite project in Interwar Britain.
In the Second World War, the Soviet Union and the US joined the UK as the main Allied powers. The UK was no longer a military or economic superpower, and it granted independence to almost all its possessions.
Since the 1990s, neoliberalism led to the privatization of nationalized industries and significant deregulation of business affairs. London's status as a world financial hub grew continuously.
Large-scale devolution movements in Northern Ireland, Scotland, and Wales have decentralized political decision-making.
The UK joined the European Economic Community in 1973, thereby weakening economic ties with its Commonwealth. However, the Brexit referendum in 2016 committed the UK to leave the European Union, which it did in 2020.
The Stuart line died with Queen Anne in 1714, and the Elector of Hanover became king as George I. The Hanoverian kings enhanced the stability of the constitutional system in Britain and helped bring peace to northern Europe.
Robert Walpole is regarded as the first Prime Minister, inventing the role from 1719-1742. He built up the First British Empire, strengthening the colonies in the Caribbean and North America.
Hypocrisy became a major topic in English political history in the early 18th century, with the Toleration Act 1689 allowing for certain rights for religious minorities. This campaign of moderation versus zealotry peaked in 1709 during the impeachment trial of high church preacher Henry Sacheverell.
Britain was involved in 137 wars or rebellions from 1700 to 1850, maintaining a relatively large and expensive Royal Navy, along with a small standing army. The rising costs of warfare forced a shift in the sources of government financing.
The Seven Years' War, which began in 1756, was the first war waged on a global scale, fought in Europe, India, North America, the Caribbean, the Philippines, and coastal Africa. Britain emerged as the world's dominant colonial power.
During the 1760s and 1770s, relations between the Thirteen Colonies and Britain became increasingly strained, primarily because of growing anger against Parliament's repeated attempts to tax American colonists without their consent. The American Revolutionary War began in 1775, and in 1776, the Patriots declaredBritish History: From the American Revolution to Whig Reforms
The American Revolution led to Britain forming an alliance with France, equalizing military and naval power, and putting Britain at risk of invasion.
The Treaty of Paris in 1783 ended the war and recognized the independence of the United States, marking the transition from the "first" to "second" British empires.
Adam Smith's Wealth of Nations argued that colonies were redundant and free trade should replace mercantilist policies, which was confirmed by the growth of trade between the US and Britain.
The British East India Company shifted from trade to territory in India, gradually increasing the size of territories under its control.
James Cook discovered the eastern coast of Australia in 1770, leading to the establishment of a penal settlement in 1788.
Britain initially remained neutral during the French Revolution but declared war in 1793 after Louis XVI was beheaded.
During the Napoleonic Wars, Britain's naval supremacy and mastery of the seas allowed it to build up considerable economic strength through trade to its rapidly expanding new empire.
The War of 1812 with the US was little noticed in Britain, where all attention was focused on the struggle with France.
The postwar era was a time of economic depression, poor harvests, growing inflation, and high unemployment among returning soldiers.
Repression techniques included the suspension of Habeas Corpus, the Gagging Acts, and the Six Acts, but many were repealed by the end of the 1820s.
George IV let his ministers take full charge of government affairs, playing a far lesser role than his father, George III, and William IV ruled during several reforms, including the Reform Act 1832.
The Whig Party supported moral reforms, especially the reform of the electoral system, the abolition of slavery, and Catholic emancipation, and made Lord Grey prime minister 1830–1834, passing the Reform Act 1832.Britain in the 19th Century: Politics, Empire, and Society
The 19th century saw the rise of the industrial revolution in Britain, leading to economic and political power.
The period was marked by several movements demanding democracy, including Chartism and the People's Charter.
The Victorian era, which began in 1837, saw Britain reach the zenith of its economic and political power, with new technologies and a growing British empire.
The era was preceded by the Georgian era and succeeded by the Edwardian period.
The mid-Victorian era, from 1850 to 1870, was characterized as Britain's "Golden Years" with peace, prosperity, and social reforms.
The working class movement for democracy, such as Chartism, eventually gave way to trade unions and cooperative societies.
Britain's foreign policy was one of free trade imperialism, and it exerted major influence on many countries outside Europe, especially in Latin America and Asia.
The American Civil War (1861-1865) was a significant event for Britain, which personally disliked American republicanism and favored the Confederacy, but eventually remained neutral.
The 19th century saw a major expansion of Britain's colonial empire in Asia, Africa, and the Pacific, defended by the world's dominant navy and a small professional army.
The Boer War (1899-1902) demonstrated that Britain was not loved around the world and needed new friends, leading to a military alliance with Japan and a close relationship with the United States.
Ireland was a major issue for British politics, with Home Rule Bills introduced by Liberal Prime Minister Gladstone failing of passage and a significant unionist minority opposing Home Rule.
Queen Victoria played a small role in politics but became the symbolic icon of the nation, the empire, and proper, restrained behavior. Lord Palmerston and Benjamin Disraeli were two dominant figures in British politics during the 19th century.British Prime Ministers: Disraeli, Gladstone, Salisbury, and the Edwardian Era
Benjamin Disraeli was a Conservative Prime Minister who emphasized national leadership, imperial expansion, and one-nation conservatism.
Disraeli's policies were opposed by Liberal leader William Gladstone who denounced territorial aggrandizement and imperialism.
Gladstone was a moralist and orator who championed social reforms and home rule for Ireland, but his opposition to imperialism and support for Irish autonomy split the Liberal party.
Lord Salisbury was a Conservative Prime Minister who was a strong and effective leader in foreign affairs, but his negative and reactionary policies at home were criticized by some historians.
The Edwardian Era was characterized by displays of wealth, optimism, and technological advancements, but it was also marked by political crises and instability.
The Liberals were in power from 1906 to 1915 and passed welfare reforms, weakened the veto power of the Lords, and blocked woman suffrage.
The outbreak of World War I in 1914 put domestic issues on hold and led to the formation of a coalition government.
The cabinet's reasons for declaring war focused on a deep commitment to France and avoidance of splitting the Liberal Party, but the public reason given was to safeguard Belgium's neutrality.
The main British contribution to the war effort was financial, as loans and grants helped smaller allies afford the war.
The stalemate on the Western Front required an endless supply of men and munitions, leading to the imposition of conscription in Britain in 1916.
H. H. Asquith was replaced by David Lloyd George, who set up a new small war cabinet and moved towards prime ministerial control.
The war forced Britain to use up its financial reserves and borrow large sums from New York banks, but the economy still grew about 14% from 1914 to 1918.