Android (operating system)

Context: , Android 11 is most popular on smartphones at 35%, with Android 10 2nd at 24%, then Android 12 at 12%. Usage of Android 10 and newer, i.e. supported versions, is at 71%, the rest of users are not supported with security updates. Android 12 is most popular in a few countries including the United States, but Android 11 is most used in most countries, including India, while in many others, including China, Android 10 is the most popular version.

Question: অ্যানড্রয়েড বা এন্ড্রয়েড নামক ওপেন সোর্স মোবাইল অপারেটিং সিস্টেমটি প্রথম কত সাল থেকে চালু হয় ?


Android (operating system)

Context: , 66% of devices have Vulkan support (47% on newer Vulkan 1.1), the successor to OpenGL. At the same time 91.5% of the devices have support for or higher (in addition, the rest of devices, 8.50%, use version 2.0), with 73.50% using the latest version .

Question: অ্যানড্রয়েড বা এন্ড্রয়েড নামক ওপেন সোর্স মোবাইল অপারেটিং সিস্টেমটি প্রথম কত সাল থেকে চালু হয় ?


Android (operating system)

Context: The success of Android has made it a target for patent and copyright litigation between technology companies, both Android and Android phone manufacturers having been involved in numerous patent lawsuits and other legal challenges.

Question: অ্যানড্রয়েড বা এন্ড্রয়েড নামক ওপেন সোর্স মোবাইল অপারেটিং সিস্টেমটি প্রথম কত সাল থেকে চালু হয় ?


Android (operating system)

Context: In April 2021, the United Supreme Court ruled that Google's use of the Java APIs was within the bounds of fair use, reversing the Federal Circuit Appeals Court ruling and remanding the case for further hearing. The majority opinion began with the assumption that the APIs may be copyrightable, and thus proceeded with a review of the factors that contributed to fair use.

Question: অ্যানড্রয়েড বা এন্ড্রয়েড নামক ওপেন সোর্স মোবাইল অপারেটিং সিস্টেমটি প্রথম কত সাল থেকে চালু হয় ?


Android (operating system)

Context: Turkey's competition authority investigations about default search engine in Android, started in 2017, led to a US$17.4 million fine in September 2018 and a fine of 0.05 percent of Google's revenue per day in November 2019 when Google didn't meet the requirements. In December 2019, Google stopped issuing licenses for new Android phone models sold in Turkey.

Question: অ্যানড্রয়েড বা এন্ড্রয়েড নামক ওপেন সোর্স মোবাইল অপারেটিং সিস্টেমটি প্রথম কত সাল থেকে চালু হয় ?


Android (operating system)

Context: Ouya, a video game console running Android, became one of the most successful Kickstarter campaigns, crowdfunding US$8.5m for its development, and was later followed by other Android-based consoles, such as Nvidia's Shield Portable an Android device in a video game controller form factor.

Question: অ্যানড্রয়েড বা এন্ড্রয়েড নামক ওপেন সোর্স মোবাইল অপারেটিং সিস্টেমটি প্রথম কত সাল থেকে চালু হয় ?


Android (operating system)

Context: At Google I/O in May 2016, Google announced Daydream, a virtual reality platform that relies on a smartphone and provides VR capabilities through a virtual reality headset and controller designed by Google itself. The platform is built into Android starting with Android Nougat, differentiating from standalone support for VR capabilities. The software is available for developers, and was released in 2016.

Question: অ্যানড্রয়েড বা এন্ড্রয়েড নামক ওপেন সোর্স মোবাইল অপারেটিং সিস্টেমটি প্রথম কত সাল থেকে চালু হয় ?


Android (operating system)

Context: The mascot of Android is a green android robot, as related to the software's name. Although it has no official name, the Android team at Google reportedly call it "Bugdroid".

Question: অ্যানড্রয়েড বা এন্ড্রয়েড নামক ওপেন সোর্স মোবাইল অপারেটিং সিস্টেমটি প্রথম কত সাল থেকে চালু হয় ?


Thomas Jefferson

Context: Jefferson was admitted to the Virginia bar in 1767, and lived with his mother at Shadwell. Jefferson also represented Albemarle County as a delegate in the Virginia House of Burgesses from 1769 until 1775. He pursued reforms to slavery, offering legislation in 1769 to allow masters to control emancipation of slaves, taking discretion away from the royal governor and General Court. He persuaded his cousin Richard Bland to spearhead the legislation's passage, but the reaction was strongly negative.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: The British parliament passed the Intolerable Acts in 1774, and Jefferson wrote a resolution calling for a "Day of Fasting and Prayer" in protest, as well as a boycott of all British goods. His resolution was later expanded into "A Summary View of the Rights of British America", in which he argued that people have the right to govern themselves.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: In 1768, Jefferson began constructing his primary residence Monticello (Italian for "Little Mountain") on a hilltop overlooking his plantation. He spent most of his adult life designing Monticello as architect and was quoted as saying, "Architecture is my delight, and putting up, and pulling down, one of my favorite amusements." Construction was done mostly by local masons and carpenters, assisted by Jefferson's slaves.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Martha's father John Wayles died in 1773, and the couple inherited 135 slaves, , and the estate's debts. The debts took Jefferson years to satisfy, contributing to his financial problems.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: After working as secretary of state (1790–93), he returned to Monticello and initiated a remodeling based on the architectural concepts which he had acquired in Europe. The work continued throughout most of his presidency and was completed in 1809.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: He sought out John Adams, an emerging leader of the Congress. They became close friends and Adams supported Jefferson's appointment to the Committee of Five formed to draft a declaration of independence in furtherance of the Lee Resolution passed by the Congress, which declared the United Colonies independent. The committee initially thought that Adams should write the document, but Adams persuaded the committee to choose Jefferson.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson consulted with other committee members over the next seventeen days and drew on his proposed draft of the Virginia Constitution, George Mason's draft of the Virginia Declaration of Rights, and other sources. The other committee members made some changes, and a final draft was presented to Congress on June 28, 1776.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: At the start of the Revolution, Jefferson was a colonel and was named commander of the Albemarle County Militia on September 26, 1775. He was then elected to the Virginia House of Delegates for Albemarle County in September 1776, when finalizing a state constitution was a priority.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: For nearly three years, he assisted with the constitution and was especially proud of his Bill for Establishing Religious Freedom, which forbade state support of religious institutions or enforcement of religious doctrine. The bill failed to pass, as did his legislation to disestablish the Anglican Church, but both were later revived by James Madison.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson was elected governor for one-year terms in 1779 and 1780. He transferred the state capital from Williamsburg to Richmond, and introduced measures for public education, religious freedom, and revision of inheritance laws.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: In April of the same year, his daughter Lucy died at age one. A second daughter of that name was born the following year, but she died at age three.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: In 1782, Jefferson refused a partnership offer by North Carolina Governor Abner Nash, in a profiteering land scheme sale of confiscated Loyalist lands. Unlike other Founders, in the pursuit of land, Jefferson was content with his Monticello estate and the land he owned in the vicinity of Virginia's Shenandoah Valley. Jefferson wanted Monticello to be an intellectual gathering place for his friends James Madison and James Monroe.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: "Notes" was first published in 1785 in French and appeared in English in 1787. Biographer George Tucker considered the work "surprising in the extent of the information which a single individual had been thus able to acquire, as to the physical features of the state", and Merrill D. Peterson described it as an accomplishment for which all Americans should be grateful.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson had Patsy educated at the Pentemont Abbey. In 1786, he met and fell in love with Maria Cosway, an accomplished—and married—Italian-English musician of 27. They saw each other frequently over a period of six weeks. She returned to Great Britain, but they maintained a lifelong correspondence.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: The second major issue was the capital's permanent location. Hamilton favored a capital close to the major commercial centers of the Northeast, while Washington, Jefferson, and other agrarians wanted it located to the south. After lengthy deadlock, the Compromise of 1790 was struck, permanently locating the capital on the Potomac River, and the federal government assumed the war debts of all thirteen states.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Historian Ron Chernow claims that "the theoretical damage of the Kentucky and Virginia Resolutions was deep and lasting, and was a recipe for disunion", contributing to the American Civil War as well as later events. Washington was so appalled by the resolutions that he told Patrick Henry that, if "systematically and pertinaciously pursued", the resolutions would "dissolve the union or produce coercion."

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson had always admired Washington's leadership skills but felt that his Federalist party was leading the country in the wrong direction. Jefferson thought it wise not to attend his funeral in 1799 because of acute differences with Washington while serving as secretary of state, and remained at Monticello.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: The transition proceeded smoothly, marking a watershed in American history. As historian Gordon S. Wood writes, "it was one of the first popular elections in modern history that resulted in the peaceful transfer of power from one 'party' to another."

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson pardoned several of those imprisoned under the Alien and Sedition Acts. Congressional Republicans repealed the Judiciary Act of 1801, which removed nearly all of Adams's "midnight judges" from office. A subsequent appointment battle led to the Supreme Court's landmark decision in "Marbury v. Madison", asserting judicial review over executive branch actions. Jefferson appointed three Supreme Court justices: William Johnson (1804), Henry Brockholst Livingston (1807), and Thomas Todd (1807).

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson took great interest in the Library of Congress, which had been established in 1800. He often recommended books to acquire. In 1802, an act of Congress authorized President Jefferson to name the first Librarian of Congress and gave itself the power to establish library rules and regulations. This act also granted the president and vice president the right to use the library.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Most thought that this was an exceptional opportunity, despite Republican reservations about the Constitutional authority of the federal government to acquire land. Jefferson initially thought that a Constitutional amendment was necessary to purchase and govern the new territory; but he later changed his mind, fearing that this would give cause to oppose the purchase, and he, therefore, urged a speedy debate and ratification. On October 20, 1803, the Senate ratified the purchase treaty by a vote of 24–7.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: The expedition lasted from May 1804 to September 1806 (see Timeline) and obtained a wealth of scientific and geographic knowledge, including knowledge of many Indian tribes.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: In addition to the Corps of Discovery, Jefferson organized three other western expeditions: the William Dunbar and George Hunter expedition on the Ouachita River (1804–1805), the Thomas Freeman and Peter Custis expedition (1806) on the Red River, and the Zebulon Pike Expedition (1806–1807) into the Rocky Mountains and the Southwest. All three produced valuable information about the American frontier.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson's experiences with the American Indians began during his boyhood in Virginia and extended through his political career and into his retirement. He refuted the contemporary notion that Indians were inferior people and maintained that they were equal in body and mind to people of European descent.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson's successful first term occasioned his re-nomination for president by the Republican party, with George Clinton replacing Burr as his running mate. The Federalist party ran Charles Cotesworth Pinckney of South Carolina, John Adams's vice-presidential candidate in the 1800 election. The Jefferson-Clinton ticket won overwhelmingly in the electoral college vote, by 162 to 14, promoting their achievement of a strong economy, lower taxes, and the Louisiana Purchase.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: In Haiti, Jefferson's neutrality had allowed arms to enable the slave independence movement during its Revolution, and blocked attempts to assist Napoleon, who was defeated there in 1803. But he refused official recognition of the country during his second term, in deference to southern complaints about the racial violence against slave-holders; it was eventually extended to Haiti in 1862.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Domestically, Jefferson's grandson James Madison Randolph became the first child born in the White House in 1806.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Following the 1801 electoral deadlock, Jefferson's relationship with his vice president, former New York Senator Aaron Burr, rapidly eroded. Jefferson suspected Burr of seeking the presidency for himself, while Burr was angered by Jefferson's refusal to appoint some of his supporters to federal office. Burr was dropped from the Republican ticket in 1804.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: After leaving office in April 1805, Burr traveled west and conspired with Louisiana Territory governor James Wilkinson, beginning a large-scale recruitment for a military expedition. Other plotters included Ohio Senator John Smith and an Irishman named Harmon Blennerhassett. Burr discussed a number of plots—seizing control of Mexico or Spanish Florida, or forming a secessionist state in New Orleans or the Western U.S. Historians remain unclear as to his true goal.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: In the fall of 1806, Burr launched a military flotilla carrying about 60 men down the Ohio River. Wilkinson renounced the plot, apparently from self-interested motives; he reported Burr's expedition to Jefferson, who immediately ordered Burr's arrest. On February 13, 1807, Burr was captured in Louisiana's Bayou Pierre wilderness and sent to Virginia to be tried for treason.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: When Jefferson died in 1826, James Madison replaced him as rector. Jefferson bequeathed most of his library to the university. Only one other ex-president has founded a university, namely Millard Fillmore who founded the University at Buffalo.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: He also expressed opposition to the idea of a privileged aristocracy made up of large landowning families partial to the King, and instead promoted "the aristocracy of virtue and talent, which nature has wisely provided for the direction of the interests of society, & scattered with equal hand through all its conditions, was deemed essential to a well-ordered republic".

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson gave his insight into people, politics, and events. The work is primarily concerned with the Declaration and reforming the government of Virginia. He used notes, letters, and documents to tell many of the stories within the autobiography. He suggested that this history was so rich that his personal affairs were better overlooked, but he incorporated a self-analysis using the Declaration and other patriotism.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: In the summer of 1824, the Marquis de Lafayette accepted an invitation from President James Monroe to visit the country. Jefferson and Lafayette had not seen each other since 1789. After visits to New York, New England, and Washington, Lafayette arrived at Monticello on November 4.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Shortly after Jefferson had died, attendants found a gold locket on a chain around his neck, where it had rested for more than 40 years, containing a small faded blue ribbon that tied a lock of his wife Martha's brown hair.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: HERE WAS BURIED THOMAS JEFFERSON, AUTHOR OF THE DECLARATION OF AMERICAN INDEPENDENCE, OF THE STATUTE OF VIRGINIA FOR RELIGIOUS FREEDOM, AND FATHER OF THE UNIVERSITY OF VIRGINIA.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson died deeply in debt, unable to pass on his estate freely to his heirs. He gave instructions in his will for disposal of his assets, including the freeing of Sally Hemings's children; but his estate, possessions, and slaves were sold at public auctions starting in 1827. In 1831, Monticello was sold by Martha Jefferson Randolph and the other heirs.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson wrote letters and speeches prolifically, and these show him to be conversant and well read in the philosophical literature of his day and of antiquity. However, since Jefferson did not produce a formal work on philosophy, scholars, in general, do not take him seriously as a philosopher. Nonetheless, Jefferson left behind a "rich philosophical legacy" in his presidential messages, letters to philosophically minded people, and his public papers.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: According to Jefferson's philosophy, citizens have "certain inalienable rights" and "rightful liberty is unobstructed action according to our will, within limits drawn around us by the equal rights of others." A staunch advocate of the jury system to protect people's liberties, he proclaimed in 1801, "I consider [trial by jury] as the only anchor yet imagined by man, by which a government can be held to the principles of its constitution."

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson considered democracy to be the expression of society and promoted national self-determination, cultural uniformity, and education of all males of the commonwealth. He supported public education and a free press as essential components of a democratic nation.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: After resigning as secretary of state in 1795, Jefferson focused on the electoral bases of the Republicans and Federalists. The "Republican" classification for which he advocated included "the entire body of landholders" everywhere and "the body of laborers" without land. Republicans united behind Jefferson as vice president, with the election of 1796 expanding democracy nationwide at grassroots levels. Jefferson promoted Republican candidates for local offices.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson's unorthodox religious beliefs became an important issue in the 1800 presidential election. Federalists attacked him as an atheist. As president, Jefferson countered the accusations by praising religion in his inaugural address and attending services at the Capitol.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson distrusted government banks and opposed public borrowing, which he thought created long-term debt, bred monopolies, and invited dangerous speculation as opposed to productive labor. In one letter to Madison, he argued each generation should curtail all debt within 19 years, and not impose a long-term debt on subsequent generations.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson used agrarian resistance to banks and speculators as the first defining principle of an opposition party, recruiting candidates for Congress on the issue as early as 1792. As president, Jefferson was persuaded by Secretary of Treasury Albert Gallatin to leave the bank intact but sought to restrain its influence.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: During his presidency, Jefferson allowed the diffusion of slavery into the Louisiana Territory hoping to prevent slave uprisings in Virginia and to prevent South Carolina secession. In 1804, in a compromise on the slavery issue, Jefferson and Congress banned domestic slave trafficking for one year into the Louisiana Territory. In 1806 he officially called for anti-slavery legislation terminating the import or export of slaves. Congress passed the law in 1807.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Historian Henry Wiencek said Jefferson "rationalized an abomination to the point where an absolute moral reversal was reached and he made slavery fit into America's national enterprise."

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: On the contrary, historian and biographer John Ferling stated that Jefferson was "zealously committed to slavery's abolition".

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: In July 2017, the TJF announced that archeological excavations at Monticello had revealed what they believe to have been Sally Hemings's quarters, adjacent to Jefferson's bedroom. In 2018, the TJF said that it considered the issue "a settled historical matter." Since the results of the DNA tests were made public, the consensus among most historians has been that Jefferson had a sexual relationship with Sally Hemings and that he was the father of her son Eston Hemings.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson was a farmer, obsessed with new crops, soil conditions, garden designs, and scientific agricultural techniques. His main cash crop was tobacco, but its price was usually low and it was rarely profitable. He tried to achieve self-sufficiency with wheat, vegetables, flax, corn, hogs, sheep, poultry, and cattle to supply his family, slaves, and employees, but he lived perpetually beyond his means and was always in debt.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: In the field of architecture, Jefferson helped popularize the Neo-Palladian style in the United States utilizing designs for the Virginia State Capitol, the University of Virginia, Monticello, and others. Jefferson mastered architecture through self-study, using various books and classical architectural designs of the day. His primary authority was Andrea Palladio's 1570 "The Four Books of Architecture", which outlines the principles of classical design.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: He was interested in birds and wine, and was a noted gourmet; he was also a prolific writer and linguist, and spoke several languages. As a naturalist, he was fascinated by the Natural Bridge geological formation, and in 1774 successfully acquired the Bridge by a grant from George III.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson served as APS president for the next eighteen years, including through both terms of his presidency. He introduced Meriwether Lewis to the society, where various scientists tutored him in preparation for the Lewis and Clark Expedition. He resigned on January 20, 1815, but remained active through correspondence.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson was not an outstanding orator and preferred to communicate through writing or remain silent if possible. Instead of delivering his State of the Union addresses himself, Jefferson wrote the annual messages and sent a representative to read them aloud in Congress. This started a tradition that continued until 1913 when President Woodrow Wilson (1913–1921) chose to deliver his own State of the Union address.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson invented many small practical devices and improved contemporary inventions, including a revolving book-stand and a "Great Clock" powered by the gravitational pull on cannonballs. He improved the pedometer, the polygraph (a device for duplicating writing), and the moldboard plow, an idea he never patented and gave to posterity. Jefferson can also be credited as the creator of the swivel chair, the first of which he created and used to write much of the Declaration of Independence.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: As Minister to France, Jefferson was impressed by the military standardization program known as the "Système Gribeauval", and initiated a program as president to develop interchangeable parts for firearms. For his inventiveness and ingenuity, he received several honorary Doctor of Law degrees.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Noting the huge output of scholarly books on Jefferson in recent years, historian Gordon Wood summarizes the raging debates about Jefferson's stature: "Although many historians and others are embarrassed about his contradictions and have sought to knock him off the democratic pedestal ... his position, though shaky, still seems secure."

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: The Siena Research Institute poll of presidential scholars, begun in 1982, has consistently ranked Jefferson as one of the five best U.S. presidents, and a 2015 Brookings Institution poll of American Political Science Association members ranked him as the fifth greatest president.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: Jefferson has been memorialized with buildings, sculptures, postage, and currency. In the 1920s, Jefferson, together with George Washington, Theodore Roosevelt, and Abraham Lincoln, was chosen by sculptor Gutzon Borglum and approved by President Calvin Coolidge to be depicted in stone at the Mount Rushmore Memorial.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: The Jefferson Memorial was dedicated in Washington, D.C. in 1943, on the 200th anniversary of Jefferson's birth. The interior of the memorial includes a statue of Jefferson by Rudulph Evans and engravings of passages from Jefferson's writings. Most prominent are the words inscribed around the monument near the roof: "I have sworn upon the altar of God eternal hostility against every form of tyranny over the mind of man."

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


Thomas Jefferson

Context: In October 2021, in response to lobbying by activists, the New York City Public Design Commission voted unanimously to remove a statue of the former president from the New York City Council chamber where it had stood for more than a century. The statue was taken down in November 2021.

Question: মার্কিন যুক্তরাষ্ট্রের তৃতীয় রাষ্ট্রপতির নাম কী ?


List of largest cities of Iran

Context: Mashhad, with a population of 3 million, is the second-largest Iranian city and the centre of the province of Razavi Khorasan. Mashhad is one of the holiest Shi'a cities in the world as it is the site of the Imam Reza shrine. It is the centre of tourism in Iran and between 15 and 20 million pilgrims go to Imam Reza's shrine every year.

Question: ইরানের সবচেয়ে জনবহুল শহর কোনটি ?


List of largest cities of Iran

Context: The third most populous city of Iran is Isfahan with a population 1.9 million (as of 2016). This city is one of the most industrial cities of Iran with large industries and hosts several UNESCO World Heritage Sites. The city has a wide variety of historic monuments and is known for its paintings, history and architecture.

Question: ইরানের সবচেয়ে জনবহুল শহর কোনটি ?


List of largest cities of Iran

Context: The fifth most populous Iranian city is Shiraz (population 1.57 million), which is located in the southwest of Iran with a moderate climate and has been a regional trade center for over a thousand years. Shiraz is one of the oldest cities of ancient Persia, known as the city of poets, literature, flowers . It is also considered by many Iranians to be the city of gardens, and also a major center for Iran's electronic industries.

Question: ইরানের সবচেয়ে জনবহুল শহর কোনটি ?


List of largest cities of Iran

Context: The other major cities are Tabriz (population 1.56 million), Qom (population 1.2 million), and Ahvaz (population 1.18 million).

Question: ইরানের সবচেয়ে জনবহুল শহর কোনটি ?


List of largest cities of Iran

Context: The following sorted table, lists the most populous cities in Iran with a population of more than 100,000 according to 2016 Census results announced by Statistical Center of Iran. A city is displayed in bold if it is the province capital.

Question: ইরানের সবচেয়ে জনবহুল শহর কোনটি ?


List of largest cities of Iran

Context: This is a list of the largest cities of the 31 provinces, based on the city's population as of the 2016 census.

Question: ইরানের সবচেয়ে জনবহুল শহর কোনটি ?


Mahabharata

Context: The Mahābhārata ( ; , "", ) is one of the two major Sanskrit epics of ancient India, the other being the "Rāmāyaṇa". It narrates the struggle between two groups of cousins in the Kurukshetra War and the fates of the Kaurava and the Pāṇḍava princes and their successors.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: It also contains philosophical and devotional material, such as a discussion of the four "goals of life" or "puruṣārtha" (12.161). Among the principal works and stories in the "Mahābhārata" are the "Bhagavad Gita", the story of Damayanti, the story of Shakuntala, the story of Pururava and Urvashi, the story of Savitri and Satyavan, the story of Kacha and Devayani, the story of Rishyasringa and an abbreviated version of the "Rāmāyaṇa", often considered as works in their own right.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: The epic is traditionally ascribed to the sage Vyāsa, who is also a major character in the epic. Vyāsa described it as being "itihāsa" (). He also describes the Guru–shishya tradition, which traces all great teachers and their students of the Vedic times.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: The first section of the "Mahābhārata" states that it was Ganesha who wrote down the text to Vyasa's dictation, but this is regarded by scholars as a later interpolation to the epic and the "Critical Edition" doesn't include Ganesha at all.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: The text was described by some early 20th-century Indologists as unstructured and chaotic. Hermann Oldenberg supposed that the original poem must once have carried an immense "tragic force" but dismissed the full text as a "horrible chaos." Moritz Winternitz ("Geschichte der indischen Literatur" 1909) considered that "only unpoetical theologists and clumsy scribes" could have lumped the parts of disparate origin into an unordered whole.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: The "Suparṇākhyāna", a late Vedic period poem considered to be among the "earliest traces of epic poetry in India," is an older, shorter precursor to the expanded legend of Garuda that is included in the "Āstīka Parva", within the "Ādi Parva" of the "Mahābhārata".

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: The Greek writer Dio Chrysostom (c. 40 – c. 120 CE) reported that Homer's poetry was being sung even in India. Many scholars have taken this as evidence for the existence of a "Mahābhārata" at this date, whose episodes Dio or his sources identify with the story of the "Iliad".

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: Several stories within the "Mahābhārata" took on separate identities of their own in Classical Sanskrit literature. For instance, Abhijñānaśākuntala by the renowned Sanskrit poet Kālidāsa (c. 400 CE), believed to have lived in the era of the Gupta dynasty, is based on a story that is the precursor to the "Mahābhārata". Urubhaṅga, a Sanskrit play written by Bhāsa who is believed to have lived before Kālidāsa, is based on the slaying of Duryodhana by the splitting of his thighs by Bhīma.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: The copper-plate inscription of the Maharaja Sharvanatha (533–534 CE) from Khoh (Satna District, Madhya Pradesh) describes the "Mahābhārata" as a "collection of 100,000 verses" ("śata-sahasri saṃhitā").

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: The Mahabharata begins with the following hymn and in fact this praise has been made at the beginning of every Parva:

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: "Om! Having bowed down to Narayana and Nara (Arjuna), the most exalted male being, and also to the goddess Saraswati, must the word Jaya be uttered."

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: Nara-Narayana were two ancient sages who were the portion of Shree Vishnu. Nara was the previous birth of Arjuna and the friend of Narayana, while Narayana was the incarnation of Shree Vishnu and thus the previous birth of Shree Krishna.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: Puranic literature presents genealogical lists associated with the "Mahābhārata" narrative. The evidence of the Puranas is of two kinds. Of the first kind, there is the direct statement that there were 1015 (or 1050) years between the birth of Parikshit (Arjuna's grandson) and the accession of Mahapadma Nanda (400-329  BCE), which would yield an estimate of about 1400  BCE for the Bharata battle. However, this would imply improbably long reigns on average for the kings listed in the genealogies.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: B. B. Lal used the same approach with a more conservative assumption of the average reign to estimate a date of 836 BCE, and correlated this with archaeological evidence from Painted Grey Ware (PGW) sites, the association being strong between PGW artifacts and places mentioned in the epic. John Keay confirms this and also gives 950 BCE for the Bharata battle.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: Another traditional school of astronomers and historians, represented by Vriddha-Garga, Varahamihira (author of the "Brhatsamhita") and Kalhana (author of the "Rajatarangini"), place the Bharata war 653 years after the "Kali Yuga" epoch, corresponding to 2449 BCE.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: The core story of the work is that of a dynastic struggle for the throne of Hastinapura, the kingdom ruled by the Kuru clan. The two collateral branches of the family that participate in the struggle are the Kaurava and the Pandava. Although the Kaurava is the senior branch of the family, Duryodhana, the eldest Kaurava, is younger than Yudhishthira, the eldest Pandava. Both Duryodhana and Yudhishthira claim to be first in line to inherit the throne.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: The struggle culminates in the great battle of Kurukshetra, in which the Pandavas are ultimately victorious. The battle produces complex conflicts of kinship and friendship, instances of family loyalty and duty taking precedence over what is right, as well as the converse.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: The "Mahābhārata" itself ends with the death of Krishna, and the subsequent end of his dynasty and ascent of the Pandava brothers to heaven. It also marks the beginning of the Hindu age of "Kali Yuga", the fourth and final age of humankind, in which great values and noble ideas have crumbled, and people are heading towards the complete dissolution of right action, morality, and virtue.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: Dhritarashtra has a hundred sons, and one daughter—Duhsala—through Gandhari, all born after the birth of Yudhishthira. These are the Kaurava brothers, the eldest being Duryodhana, and the second Dushasana. Other Kaurava brothers were Vikarna and Sukarna. The rivalry and enmity between them and the Pandava brothers, from their youth and into manhood, leads to the Kurukshetra war.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: After the deaths of their mother (Madri) and father (Pandu), the Pandavas and their mother Kunti return to the palace of Hastinapur. Yudhishthira is made Crown Prince by Dhritarashtra, under considerable pressure from his courtiers. Dhritarashtra wanted his son Duryodhana to become king and lets his ambition get in the way of preserving justice.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: After the wedding, the Pandava brothers are invited back to Hastinapura. The Kuru family elders and relatives negotiate and broker a split of the kingdom, with the Pandavas obtaining and demanding only a wild forest inhabited by Takshaka, the king of snakes, and his family. Through hard work, the Pandavas can build a new glorious capital for the territory at Indraprastha.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: Shortly after this, Arjuna elopes with and then marries Krishna's sister, Subhadra. Yudhishthira wishes to establish his position as king; he seeks Krishna's advice. Krishna advises him, and after due preparation and the elimination of some opposition, Yudhishthira carries out the "rājasūya yagna" ceremony; he is thus recognized as pre-eminent among kings.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: Dhritarashtra, Bhishma, and the other elders are aghast at the situation, but Duryodhana is adamant that there is no place for two crown princes in Hastinapura. Against his wishes Dhritarashtra orders for another dice game. The Pandavas are required to go into exile for 12 years, and in the 13th year, they must remain hidden. If they are discovered by the Kauravas in the 13th year of their exile, then they will be forced into exile for another 12 years.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: The Pandavas spend thirteen years in exile; many adventures occur during this time. The Pandavas acquire many divine weapons, given by gods, during this period. They also prepare alliances for a possible future conflict. They spend their final year in disguise in the court of the king Virata, and they are discovered just after the end of the year.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: At the end of their exile, they try to negotiate a return to Indraprastha with Krishna as their emissary. However, this negotiation fails, because Duryodhana objected that they were discovered in the 13th year of their exile and the return of their kingdom was not agreed upon. Then the Pandavas fought the Kauravas, claiming their rights over Indraprastha.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: Before the battle, Arjuna, noticing that the opposing army includes his cousins and relatives, including his grandfather Bhishma and his teacher Drona, has grave doubts about the fight. He falls into despair and refuses to fight. At this time, Krishna reminds him of his duty as a Kshatriya to fight for a righteous cause in the famous Bhagavad Gita section of the epic.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: After "seeing" the carnage, Gandhari, who had lost all her sons, curses Krishna to be a witness to a similar annihilation of his family, for though divine and capable of stopping the war, he had not done so. Krishna accepts the curse, which bears fruit 36 years later.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: Arjuna's grandson Parikshit rules after them and dies bitten by a snake. His furious son, Janamejaya, decides to perform a snake sacrifice ("sarpasattra") to destroy the snakes. It is at this sacrifice that the tale of his ancestors is narrated to him.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: The "Mahābhārata" mentions that Karna, the Pandavas, Draupadi and Dhritarashtra's sons eventually ascended to svarga and "attained the state of the gods", and banded together – "serene and free from anger".

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: A project to translate the full epic into English prose, translated by various hands, began to appear in 2005 from the Clay Sanskrit Library, published by New York University Press. The translation is based not on the "Critical Edition" but on the version known to the commentator Nīlakaṇṭha. Currently available are 15 volumes of the projected 32-volume edition.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: Indian Vedic Scholar Shripad Damodar Satwalekar translated the Critical Edition of Mahabharata into Hindi which was assigned to him by the Government of India. After his death, the task was taken up by Shrutisheel Sharma.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: Many condensed versions, abridgments and novelistic prose retellings of the complete epic have been published in English, including works by Ramesh Menon, William Buck, R. K. Narayan, C. Rajagopalachari, K. M. Munshi, Krishna Dharma, Romesh C. Dutt, Bharadvaja Sarma, John D. Smith and Sharon Maas.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: Between 1919 and 1966, scholars at the Bhandarkar Oriental Research Institute, Pune, compared the various manuscripts of the epic from India and abroad and produced the "Critical Edition" of the "Mahabharata", on 13,000 pages in 19 volumes, over the span of 47 years, followed by the "Harivamsha" in another two volumes and six index volumes. This is the text that is usually used in current Mahābhārata studies for reference. This work is sometimes called the "Pune" or "Poona" edition of the "Mahabharata".

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: Many regional versions of the work developed over time, mostly differing only in minor details, or with verses or subsidiary stories being added. These include the Tamil street theatre, terukkuttu and kattaikkuttu, the plays of which use themes from the Tamil language versions of "Mahabharata", focusing on Draupadi.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: A Kawi version of the "Mahabharata", of which eight of the eighteen "parvas" survive, is found on the Indonesian island of Bali. It has been translated into English by Dr. I. Gusti Putu Phalgunadi.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: In Indian cinema, several film versions of the epic have been made, dating back to 1920. The "Mahābhārata" was also reinterpreted by Shyam Benegal in "Kalyug". Prakash Jha directed 2010 film Raajneeti was partially inspired by the "Mahabharata". A 2013 animated adaptation holds the record for India's most expensive animated film.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: Uncompleted projects on the "Mahābhārata" include one by Rajkumar Santoshi, and a theatrical adaptation planned by Satyajit Ray.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: In the "Bhagavad Gita", Krishna explains to Arjuna his duties as a warrior and prince and elaborates on different Yogic and Vedantic philosophies, with examples and analogies. This has led to the "Gita" often being described as a concise guide to Hindu philosophy and a practical, self-contained guide to life. In more modern times, Swami Vivekananda, Netaji Subhas Chandra Bose, Bal Gangadhar Tilak, Mahatma Gandhi and many others used the text to help inspire the Indian independence movement.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Mahabharata

Context: It has also inspired several works of modern Hindi literature, such as Ramdhari Singh Dinkar's "Rashmirathi", which is a rendition of "Mahabharata" centered around Karna and his conflicts. It was written in 1952, and won the prestigious Jnanpith Award in 1972.

Question: মহাভারতে মোট কতগুলো অধ্যায়ের উল্লেখ রয়েছে ?


Complex number

Context: Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions and .

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule combined with the associative, commutative and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers also form a real vector space of dimension two, with as a standard basis.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: In summary, the complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: A complex number is a number of the form , where and are real numbers, and is an indeterminate satisfying . For example, is a complex number.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The real number is called the "real part" of the complex number ; the real number is called its "imaginary part". To emphasize, the imaginary part does not include a factor ; that is, the imaginary part is , not .

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Formally, the complex numbers are defined as the quotient ring of the polynomial ring in the indeterminate , by the ideal generated by the polynomial (see below).

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: A real number can be regarded as a complex number , whose imaginary part is 0. A purely imaginary number is a complex number , whose real part is zero. As with polynomials, it is common to write for and for . Moreover, when the imaginary part is negative, that is, , it is common to write instead of ; for example, for , can be written instead of .

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Since the multiplication of the indeterminate and a real is commutative in polynomials with real coefficients, the polynomial may be written as This is often expedient for imaginary parts denoted by expressions, for example, when is a radical.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The real part of a complex number is denoted by , formula_3, or formula_4; the imaginary part of a complex number is denoted by , formula_5, or formula_6 For example,

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: In some disciplines, particularly in electromagnetism and electrical engineering, is used instead of as is frequently used to represent electric current. In these cases, complex numbers are written as , or .

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: A complex number can thus be identified with an ordered pair formula_9 of real numbers, which in turn may be interpreted as coordinates of a point in a two-dimensional space. The most immediate space is the Euclidean plane with suitable coordinates, which is then called "complex plane" or "Argand diagram," named after Jean-Robert Argand. Another prominent space on which the coordinates may be projected is the two-dimensional surface of a sphere, which is then called Riemann sphere.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The definition of the complex numbers involving two arbitrary real values immediately suggests the use of Cartesian coordinates in the complex plane. The horizontal ("real") axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical ("imaginary") axis, with increasing values upwards.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: A charted number may be viewed either as the point or as a position vector from the origin to this point. The coordinate values of a complex number can hence be expressed in its "Cartesian", "rectangular", or "algebraic" form.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: An alternative option for coordinates in the complex plane is the polar coordinate system that uses the distance of the point from the origin (), and the angle subtended between the positive real axis and the line segment in a counterclockwise sense. This leads to the polar form

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: If is a real number (that is, if ), then . That is, the absolute value of a real number equals its absolute value as a complex number.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: By Pythagoras' theorem, the absolute value of a complex number is the distance to the origin of the point representing the complex number in the complex plane.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The "argument" of (in many applications referred to as the "phase" ) is the angle of the radius with the positive real axis, and is written as . As with the modulus, the argument can be found from the rectangular form —by applying the inverse tangent to the quotient of imaginary-by-real parts. By using a half-angle identity, a single branch of the arctan suffices to cover the range of the -function, and avoids a more subtle case-by-case analysis

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Together, and give another way of representing complex numbers, the "polar form", as the combination of modulus and argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinates from the polar form is done by the formula called "trigonometric form"

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: In angle notation, often used in electronics to represent a phasor with amplitude and phase , it is written as

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: When visualizing complex functions, both a complex input and output are needed. Because each complex number is represented in two dimensions, visually graphing a complex function would require the perception of a four dimensional space, which is possible only in projections. Because of this, other ways of visualizing complex functions have been designed.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Riemann surfaces and Riemann spheres are oriented complex manifolds of 1 complex dimension (i.e. 2 real dimensions).

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Riemann surfaces can be thought of as deformations of the complex plane; while the horizontal axes represent the real and imaginary inputs, the single vertical axis only represents either the real or imaginary output. However, Riemann surfaces are built in such a way that rotating them 180 degrees shows the imaginary output, and vice versa. Unlike domain coloring, Riemann surfaces can represent multivalued functions like .

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: A Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of stereographic projection). A model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value formula_20 for infinity.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The term "imaginary" for these quantities was coined by René Descartes in 1637, who was at pains to stress their unreal nature

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the complicated identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be simply re-expressed by the following well-known formula which bears his name, de Moivre's formula:

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The idea of a complex number as a point in the complex plane (above) was first described by Danish–Norwegian mathematician Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's "A Treatise of Algebra".

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, -1, formula_31 positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée, Mourey, Warren, Français and his brother, Bellavitis.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The English mathematician G.H. Hardy remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as Norwegian Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Augustin Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others. Important work (including a systematization) in complex multivariate calculus has been started at beginning of the 20th century. Important results have been achieved by Wilhelm Wirtinger in 1927.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Complex numbers have a similar definition of equality to real numbers; two complex numbers and are equal if and only if both their real and imaginary parts are equal, that is, if and . Nonzero complex numbers written in polar form are equal if and only if they have the same magnitude and their arguments differ by an integer multiple of .

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Unlike the real numbers, there is no natural ordering of the complex numbers. In particular, there is no linear ordering on the complex numbers that is compatible with addition and multiplication. Hence, the complex numbers do not have the structure of an ordered field. One explanation for this is that every non-trivial sum of squares in an ordered field is nonzero, and is a non-trivial sum of squares. Thus, complex numbers are naturally thought of as existing on a two-dimensional plane.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The "complex conjugate" of the complex number is given by . It is denoted by either or . This unary operation on complex numbers cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Geometrically, is the "reflection" of about the real axis. Conjugating twice gives the original complex number

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: which makes this operation an involution. The reflection leaves both the real part and the magnitude of unchanged, that is

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The product of a complex number and its conjugate is known as the "absolute square". It is always a non-negative real number and equals the square of the magnitude of each:

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: This property can be used to convert a fraction with a complex denominator to an equivalent fraction with a real denominator by expanding both numerator and denominator of the fraction by the conjugate of the given denominator. This process is sometimes called "rationalization" of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Conjugation is also employed in inversive geometry, a branch of geometry studying reflections more general than ones about a line. In the network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is looked for.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Two complex numbers formula_44 and formula_45 are most easily added by separately adding their real and imaginary parts. That is to say:

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Multiplication of a complex number formula_44 and a real number can be done similarly by multiplying separately and the real and imaginary parts of :

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: In particular, subtraction can be done by negating the subtrahend (that is multiplying it with ) and adding the result to the minuend:

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Using the visualization of complex numbers in the complex plane, addition has the following geometric interpretation: the sum of two complex numbers and , interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices , and the points of the arrows labeled and (provided that they are not on a line). Equivalently, calling these points , , respectively and the fourth point of the parallelogram the triangles and are congruent.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The rules of the distributive property, the commutative properties (of addition and multiplication), and the defining property apply to complex numbers. It follows that

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: This can be used to express a division of an arbitrary complex number by a non-zero complex number as

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers and , because of the trigonometric identities

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by corresponds to a quarter-turn counter-clockwise, which gives back . The picture at the right illustrates the multiplication of

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Since the real and imaginary part of are equal, the argument of that number is 45 degrees, or (in radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) and arctan(1/2), respectively. Thus, the formula

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: holds. As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of .

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: where is the signum function. This can be seen by squaring formula_60 to obtain . Here formula_64 is called the modulus of , and the square root sign indicates the square root with non-negative real part, called the principal square root; also formula_65 where .

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Analytic continuation allows extending this equality for every complex value of , and thus to define the complex exponentiation with base as

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: This can be proved either by comparing the power series expansion of both members or by applying analytic continuation from the restriction of the equation to real arguments.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: formula_74 of the exponential function. For extending this to the complex domain, one can start from Euler's formula. It implies that, if a complex number formula_75 is written in polar form

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: However, because cosine and sine are periodic functions, the addition of an integer multiple of to does not change . For example, , so both and are possible values for the natural logarithm of .

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: If formula_82 is not a non-positive real number (a positive or a non-real number), the resulting principal value of the complex logarithm is obtained with . It is an analytic function outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number formula_83, where the principal value is .

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: It seems natural to extend this formula to complex values of , but there are some difficulties resulting from the fact that the complex logarithm is not really a function, but a multivalued function.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: It follows that if is as above, and if is another complex number, then the "exponentiation" is the multivalued function

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: If, in the preceding formula, is an integer, then the sine and the cosine are independent of . Thus, if the exponent is an integer, then is well defined, and the exponentiation formula simplifies to de Moivre's formula:

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: for . (Here formula_88 is the usual (positive) th root of the positive real number .) Because sine and cosine are periodic, other integer values of do not give other values.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: While the th root of a positive real number is chosen to be the "positive" real number satisfying , there is no natural way of distinguishing one particular complex th root of a complex number. Therefore, the th root is a -valued function of . This implies that, contrary to the case of positive real numbers, one has

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The set formula_8 of complex numbers is a field. Briefly, this means that the following facts hold: first, any two complex numbers can be added and multiplied to yield another complex number. Second, for any complex number , its additive inverse is also a complex number; and third, every nonzero complex number has a reciprocal complex number. Moreover, these operations satisfy a number of laws, for example the law of commutativity of addition and multiplication for any two complex numbers and :

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: These two laws and the other requirements on a field can be proven by the formulas given above, using the fact that the real numbers themselves form a field.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Unlike the reals, formula_8 is not an ordered field, that is to say, it is not possible to define a relation that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so precludes the existence of an ordering on formula_93

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: When the underlying field for a mathematical topic or construct is the field of complex numbers, the topic's name is usually modified to reflect that fact. For example: complex analysis, complex matrix, complex polynomial, and complex Lie algebra.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: There are various proofs of this theorem, by either analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of "odd" degree has at least one real root.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Because of this fact, theorems that hold "for any algebraically closed field" apply to formula_93 For example, any non-empty complex square matrix has at least one (complex) eigenvalue.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Moreover, formula_8 has a nontrivial involutive automorphism (namely the complex conjugation), such that is in for any nonzero in formula_93

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Any field with these properties can be endowed with a topology by taking the sets as a base, where ranges over the field and ranges over . With this topology is isomorphic as a "topological" field to formula_93

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The only connected locally compact topological fields are formula_97 and formula_93 This gives another characterization of formula_8 as a topological field, since formula_8 can be distinguished from formula_97 because the nonzero complex numbers are connected, while the nonzero real numbers are not.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: William Rowan Hamilton introduced the approach to define the set formula_8 of complex numbers as the set formula_121 of of real numbers, in which the following rules for addition and multiplication are imposed:

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Though this low-level construction does accurately describe the structure of the complex numbers, the following equivalent definition reveals the algebraic nature of formula_8 more immediately. This characterization relies on the notion of fields and polynomials. A field is a set endowed with addition, subtraction, multiplication and division operations that behave as is familiar from, say, rational numbers. For example, the distributive law

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: must hold for any three elements , and of a field. The set formula_97 of real numbers does form a field. A polynomial with real coefficients is an expression of the form

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: where the are real numbers. The usual addition and multiplication of polynomials endows the set formula_127 of all such polynomials with a ring structure. This ring is called the polynomial ring over the real numbers.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The formulas for addition and multiplication in the ring formula_130 modulo the relation , correspond to the formulas for addition and multiplication of complex numbers defined as ordered pairs. So the two definitions of the field formula_8 are isomorphic (as fields).

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Accepting that formula_8 is algebraically closed, since it is an algebraic extension of formula_133 in this approach, formula_8 is therefore the algebraic closure of formula_135

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Here the entries and are real numbers. As the sum and product of two such matrices is again of this form, these matrices form a subring of the ring matrices.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: is a ring isomorphism from the field of complex numbers to the ring of these matrices. This isomorphism associates the square of the absolute value of a complex number with the determinant of the corresponding matrix, and the conjugate of a complex number with the transpose of the matrix.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrices by using this correspondence between complex numbers and such matrices. The action of the matrix on a vector corresponds to the multiplication of by . In particular, if the determinant is , there is a real number such that the matrix has the form:

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: In this case, the action of the matrix on vectors and the multiplication by the complex number formula_139 are both the rotation of the angle .

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. This is equivalent to the (ε, δ)-definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view, formula_140, endowed with the metric

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the "exponential function" , also written , is defined as the infinite series

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The series defining the real trigonometric functions sine and cosine, as well as the hyperbolic functions sinh and cosh, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such as tangent, things are slightly more complicated, as the defining series do not converge for all complex values. Therefore, one must define them either in terms of sine, cosine and exponential, or, equivalently, by using the method of analytic continuation.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: for any complex number . It can be shown that any such solution – called complex logarithm of – satisfies

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: where arg is the argument defined above, and ln the (real) natural logarithm. As arg is a multivalued function, unique only up to a multiple of , log is also multivalued. The principal value of log is often taken by restricting the imaginary part to the interval .

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: and is multi-valued, except when is an integer. For , for some natural number , this recovers the non-uniqueness of th roots mentioned above.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see failure of power and logarithm identities. For example, they do not satisfy

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Both sides of the equation are multivalued by the definition of complex exponentiation given here, and the values on the left are a subset of those on the right.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: A function "f": formula_140 → formula_140 is called holomorphic if it satisfies the Cauchy–Riemann equations. For example, any formula_133-linear map formula_140 → formula_140 can be written in the form

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: with complex coefficients and . This map is holomorphic if and only if . The second summand formula_156 is real-differentiable, but does not satisfy the Cauchy–Riemann equations.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Complex analysis shows some features not apparent in real analysis. For example, any two holomorphic functions and that agree on an arbitrarily small open subset of formula_140 necessarily agree everywhere. Meromorphic functions, functions that can locally be written as with a holomorphic function , still share some of the features of holomorphic functions. Other functions have essential singularities, such as at .

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Complex numbers have applications in many scientific areas, including signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. Some of these applications are described below.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Three non-collinear points formula_158 in the plane determine the shape of the triangle formula_159. Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The shape formula_161 of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by an affine transformation), corresponding to the intuitive notion of shape, and describing similarity. Thus each triangle formula_159 is in a similarity class of triangles with the same shape.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The Mandelbrot set is a popular example of a fractal formed on the complex plane. It is defined by plotting every location formula_163 where iterating the sequence formula_164 does not diverge when iterated infinitely. Similarly, Julia sets have the same rules, except where formula_163 remains constant.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Another example is the Gaussian integers; that is, numbers of the form , where and are integers, which can be used to classify sums of squares.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the Riemann zeta function is related to the distribution of prime numbers.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: In differential equations, it is common to first find all complex roots of the characteristic equation of a linear differential equation or equation system and then attempt to solve the system in terms of base functions of the form . Likewise, in difference equations, the complex roots of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the form .

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Eigendecomposition is a useful tool for computing matrix powers and matrix exponentials. However, it often requires the use of complex numbers, even if the matrix is real (for example, a rotation matrix).

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Complex numbers often generalize concepts originally conceived in the real numbers. For example, the conjugate transpose generalizes the transpose, hermitian matrices generalize symmetric matrices, and unitary matrices generalize orthogonal matrices.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: In control theory, systems are often transformed from the time domain to the complex frequency domain using the Laplace transform. The system's zeros and poles are then analyzed in the "complex plane". The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: In the root locus method, it is important whether zeros and poles are in the left or right half planes, that is, have real part greater than or less than zero. If a linear, time-invariant (LTI) system has poles that are

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value of the corresponding is the amplitude and the argument is the phase.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex-valued functions of the form

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: where ω represents the angular frequency and the complex number "A" encodes the phase and amplitude as explained above.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. This approach is called phasor calculus.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: In electrical engineering, the imaginary unit is denoted by , to avoid confusion with , which is generally in use to denote electric current, or, more particularly, , which is generally in use to denote instantaneous electric current.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The complex-valued signal is called the analytic representation of the real-valued, measurable signal .

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time component of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The process of extending the field formula_176 of reals to formula_1 is known as the Cayley–Dickson construction. It can be carried further to higher dimensions, yielding the quaternions formula_178 and octonions formula_179 which (as a real vector space) are of dimension 4 and 8, respectively.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Just as by applying the construction to reals the property of ordering is lost, properties familiar from real and complex numbers vanish with each extension. The quaternions lose commutativity, that is, for some quaternions , and the multiplication of octonions, additionally to not being commutative, fails to be associative: for some octonions .

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Reals, complex numbers, quaternions and octonions are all normed division algebras over formula_176. By Hurwitz's theorem they are the only ones; the sedenions, the next step in the Cayley–Dickson construction, fail to have this structure.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The Cayley–Dickson construction is closely related to the regular representation of formula_181 thought of as an formula_176-algebra (an -vector space with a multiplication), with respect to the basis . This means the following: the formula_176-linear map

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: for some fixed complex number can be represented by a matrix (once a basis has been chosen). With respect to the basis , this matrix is

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: that is, the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of formula_1 in the 2 × 2 real matrices, it is not the only one. Any matrix

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: is also isomorphic to the field formula_181 and gives an alternative complex structure on formula_190 This is generalized by the notion of a linear complex structure.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: Hypercomplex numbers also generalize formula_191 formula_181 formula_193 and formula_194 For example, this notion contains the split-complex numbers, which are elements of the ring formula_195 (as opposed to formula_196 for complex numbers). In this ring, the equation has four solutions.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Complex number

Context: The fields formula_191 formula_203 and their finite field extensions, including formula_181 are called local fields.

Question: গণিতে জটিল সংখ্যা কবে আবিষ্কৃত হয় ?


Metal

Context: A metal (from Greek μέταλλον "métallon", "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typically malleable (they can be hammered into thin sheets) or ductile (can be drawn into wires). A metal may be a chemical element such as iron; an alloy such as stainless steel; or a molecular compound such as polymeric sulfur nitride.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: Metals, as chemical elements, comprise 25% of the Earth's crust and are present in many aspects of modern life. The strength and resilience of some metals has led to their frequent use in, for example, high-rise building and bridge construction, as well as most vehicles, many home appliances, tools, pipes, and railroad tracks. Precious metals were historically used as coinage, but in the modern era, coinage metals have extended to at least 23 of the chemical elements.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: The history of refined metals is thought to begin with the use of copper about 11,000 years ago. Gold, silver, iron (as meteoric iron), lead, and brass were likewise in use before the first known appearance of bronze in the 5th millennium BCE. Subsequent developments include the production of early forms of steel; the discovery of sodium—the first light metal—in 1809; the rise of modern alloy steels; and, since the end of World War II, the development of more sophisticated alloys.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: Metals are shiny and lustrous, at least when freshly prepared, polished, or fractured. Sheets of metal thicker than a few micrometres appear opaque, but gold leaf transmits green light.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: The atoms of metallic substances are typically arranged in one of three common crystal structures, namely body-centered cubic (bcc), face-centered cubic (fcc), and hexagonal close-packed (hcp). In bcc, each atom is positioned at the center of a cube of eight others. In fcc and hcp, each atom is surrounded by twelve others, but the stacking of the layers differs. Some metals adopt different structures depending on the temperature.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: The unit cell for each crystal structure is the smallest group of atoms which has the overall symmetry of the crystal, and from which the entire crystalline lattice can be built up by repetition in three dimensions. In the case of the body-centered cubic crystal structure shown above, the unit cell is made up of the central atom plus one-eight of each of the eight corner atoms.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: The elemental metals have electrical conductivity values of from 6.9 × 103 S/cm for manganese to 6.3 × 105 S/cm for silver. In contrast, a semiconducting metalloid such as boron has an electrical conductivity 1.5 × 10−6 S/cm. With one exception, metallic elements reduce their electrical conductivity when heated. Plutonium increases its electrical conductivity when heated in the temperature range of around −175 to +125 °C.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: Metals are relatively good conductors of heat. The electrons in a metal's electron cloud are highly mobile and easily able to pass on heat-induced vibrational energy.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: Painting, anodizing, or plating metals are good ways to prevent their corrosion. However, a more reactive metal in the electrochemical series must be chosen for coating, especially when chipping of the coating is expected. Water and the two metals form an electrochemical cell and, if the coating is less reactive than the underlying metal, the coating actually "promotes" corrosion.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: An alloy is a substance having metallic properties and which is composed of two or more elements at least one of which is a metal. An alloy may have a variable or fixed composition. For example, gold and silver form an alloy in which the proportions of gold or silver can be freely adjusted; titanium and silicon form an alloy Ti2Si in which the ratio of the two components is fixed (also known as an intermetallic compound).

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: Alloys specially designed for highly demanding applications, such as jet engines, may contain more than ten elements.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: The term "ferrous" is derived from the Latin word meaning "containing iron". This can include pure iron, such as wrought iron, or an alloy such as steel. Ferrous metals are often magnetic, but not exclusively. Non-ferrous metals and alloys lack appreciable amounts of iron.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: While nearly all metals are malleable or ductile, a few—beryllium, chromium, manganese, gallium, and bismuth—are brittle. Arsenic and antimony, if admitted as metals, are brittle. Low values of the ratio of bulk elastic modulus to shear modulus (Pugh's criterion) are indicative of intrinsic brittleness.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: A white metal is any of range of white-coloured metals (or their alloys) with relatively low melting points. Such metals include zinc, cadmium, tin, antimony (here counted as a metal), lead, and bismuth, some of which are quite toxic. In Britain, the fine art trade uses the term "white metal" in auction catalogues to describe foreign silver items which do not carry British Assay Office marks, but which are nonetheless understood to be silver and are priced accordingly.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: A heavy metal is any relatively dense metal or metalloid. More specific definitions have been proposed, but none have obtained widespread acceptance. Some heavy metals have niche uses, or are notably toxic; some are essential in trace amounts. All other metals are light metals.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: In chemistry, the term "base metal" is used informally to refer to a metal that is easily oxidized or corroded, such as reacting easily with dilute hydrochloric acid (HCl) to form a metal chloride and hydrogen. Examples include iron, nickel, lead, and zinc. Copper is considered a base metal as it is oxidized relatively easily, although it does not react with HCl.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: The term noble metal is commonly used in opposition to "base metal". Noble metals are resistant to corrosion or oxidation, unlike most base metals. They tend to be precious metals, often due to perceived rarity. Examples include gold, platinum, silver, rhodium, iridium, and palladium.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: In alchemy and numismatics, the term base metal is contrasted with precious metal, that is, those of high economic value.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: A longtime goal of the alchemists was the transmutation of base metals into precious metals including such coinage metals as silver and gold. Most coins today are made of base metals with low intrinsic value; in the past, coins frequently derived their value primarily from their precious metal content.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: The demand for precious metals is driven not only by their practical use, but also by their role as investments and a store of value. Palladium and platinum, as of fall 2018, were valued at about three quarters the price of gold. Silver is substantially less expensive than these metals, but is often traditionally considered a precious metal in light of its role in coinage and jewelry.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: "This sub-section deals with the formation of periodic table elemental metals since these form the basis of metallic materials, as defined in this article."

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: Metals up to the vicinity of iron (in the periodic table) are largely made via stellar nucleosynthesis. In this process, lighter elements from hydrogen to silicon undergo successive fusion reactions inside stars, releasing light and heat and forming heavier elements with higher atomic numbers.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: Metals condense in planets as a result of stellar evolution and destruction processes. Stars lose much of their mass when it is ejected late in their lifetimes, and sometimes thereafter as a result of a neutron star merger, thereby increasing the abundance of elements heavier than helium in the interstellar medium. When gravitational attraction causes this matter to coalesce and collapse new stars and planets are formed.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: The Earth's crust is made of approximately 25% of metals by weight, of which 80% are light metals such as sodium, magnesium, and aluminum. Nonmetals (~75%) make up the rest of the crust. Despite the overall scarcity of some heavier metals such as copper, they can become concentrated in economically extractable quantities as a result of mountain building, erosion, or other geological processes.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: Metals are often extracted from the Earth by means of mining ores that are rich sources of the requisite elements, such as bauxite. Ore is located by prospecting techniques, followed by the exploration and examination of deposits. Mineral sources are generally divided into surface mines, which are mined by excavation using heavy equipment, and subsurface mines. In some cases, the sale price of the metal(s) involved make it economically feasible to mine lower concentration sources.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: Once the ore is mined, the metals must be extracted, usually by chemical or electrolytic reduction. Pyrometallurgy uses high temperatures to convert ore into raw metals, while hydrometallurgy employs aqueous chemistry for the same purpose. The methods used depend on the metal and their contaminants.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: When a metal ore is an ionic compound of that metal and a non-metal, the ore must usually be smelted—heated with a reducing agent—to extract the pure metal. Many common metals, such as iron, are smelted using carbon as a reducing agent. Some metals, such as aluminum and sodium, have no commercially practical reducing agent, and are extracted using electrolysis instead.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: Metals are good conductors, making them valuable in electrical appliances and for carrying an electric current over a distance with little energy lost. Electrical power grids rely on metal cables to distribute electricity. Home electrical systems, for the most part, are wired with copper wire for its good conducting properties.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?


Metal

Context: The thermal conductivity of metals is useful for containers to heat materials over a flame. Metals are also used for heat sinks to protect sensitive equipment from overheating.

Question: পদার্থবিজ্ঞানে সবথেকে শক্ত ধাতু কোনটি ?