Parts of Speech

The Parts of Speech Traditional grammar classifies words based on eight parts of speech: speech : the verb verb,, the noun noun,, th the epronoun pronoun,, the adjective adjective,, the adverb adverb,, the preposition preposition,, the conjunction conjunction,, and the interjection interjection.. Each part of speech explains not what the word is, but how the word is used . In fact, the same word can be a noun in one sentence and a verb or adjective in the next. The next few examples show how a word's part of speech can change from one sentence to the next, and following them is a series of sections on the individual parts of speech, followed by an exercise. Books are made of ink, paper, and glue. In this sentence, "books" is a noun, the subject of the sentence. Deborah waits patiently while Bridget books the tickets. Here "books" is a verb, and its subject is "Bridget." We walk down the street. In this sentence, "walk" is a verb, and its subject is the pronoun "we." The mail carrier stood on the walk walk.. In this example, "walk" is a noun, which is part of a prepositional phrase describing where the mail carrier stood. The town decided to build a new jail new  jail.. Here "jail" is a noun, which is the object of the infinitive phrase "to build." The sheriff told us that if we did not leave town immediately he would  jail us. Here "jail" is part of the compound verb "would jail." They heard high pitched cries in the middle of the night. In this sentence, "cries" is a noun acting as the direct object of the verb "heard." The baby cries all night long and all day long. But here "cries" is a verb that describes the actions of the subject of the sentence, the baby. The next few sections explain each of the parts of speech in detail. When you have finished, you might want to test yourself by trying the exercise. Written by Heather MacFadyen

Parts of Speech Table

This is a summary of the 8 parts of speech*. You can find more detail if you click on each part of speech. part of  speech

function or "job"

example words

example sentences

action or state

(to) be, have, do, like, work, sing, can, must

EnglishClub.com is a web site. Ilike EnglishClub.com.

thing or person

pen, dog, work, music, town, London, teacher, John

This is my dog. dog. He lives in myhouse myhouse.. We live in London. London.

Adjective

describes a noun

a/an, the, 2, some, good, big, red, well, interesting

I have two dogs. My dogs are big. big. I like big dogs.

Adverb

describes a verb, adjective or adverb

quickly, silently, well, badly, very, really

My dog eats quickly. quickly. When he isvery isvery hungry, he eats reallyquickly. reallyquickly.

Pronoun

replaces a noun

I, you, he, she, some

Tara is Indian. She is beautiful.

Preposition

links a noun to another word

to, at, after, on, but

We went to school on Monday.

and, but, when

I like dogs and I like cats. I like cats and dogs. I like dogs but I don't like cats.

oh!, ouch!, hi!, well

Ouch! Ouch! That hurts! Hi! Hi! How are you? Well, Well, I don't know.

Verb

Noun

 joins clauses or Conjunction sentences or words

Interjection

short exclamation, sometimes inserted into a sentence

* Some grammar sources categorize English into 9 or 10 parts of speech. At EnglishClub.com, we use the traditional categorization of 8 of 8 parts of speech. Examples of other categorizations are:

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Verbs may be treated as two different parts of speech: Lexical Verbs (work, like, run) o Auxiliary Verbs (be, have, must ) o Determiners may be treated as a separate part of speech, instead of  being categorized under Adjectives

The history of algebra of algebra began in ancient Egypt and Babylon Babylon,, where people learned to solve linear (ax  (ax = = b) 2

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and quadratic (ax  (ax  + bx = as x  + y  = z , whereby bx = c) equations, as well as indeterminate equations such as x  several unknowns are involved. The ancient Babylonians solved arbitrary quadratic equations by essentially the same procedures taught today. They also could solve some indeterminate e quations. The Alexandrian mathematicians Hero of Alexandria and Diophantus continued the traditions of Egypt and Babylon, but Diophantus's book Arithmetica book  Arithmetica is on a much higher level and gives g ives many surprising solutions to difficult indeterminate equations. This ancient knowledge of solutions of equations in turn found a home early in the Islamic world, where it was known as the "science of re storation and balancing." (The Arabic word for restoration, al-jabru, is the root of the word algebra.) algebra.) In the 9th century, the Arab mathematician al-Khwarizmi wrote one of the first Arabic algebras, a systematic exposé of the basic theory of equations, with both examples and proofs. By the end of t he 9th century, the Egyptian mathematician Abu Kamil had stated and proved the basic laws and identities of algebra 2

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and solved such complicated problems as finding x, finding x, y, and z such that x  that x + + y + y + z = 10, x  10, x  + y  = z , and xz and xz = 2

y  . Ancient civilizations wrote out algebraic expressions using only occasional abbreviations, but by medieval times Islamic mathematicians were able to talk about ar bitrarily high powers of the unknown included  x, and work out the basic algebra of polynomials (without yet using modern symbolism). This included the ability to multiply, divide, and find square roots of polynomials of  polynomials as well as a knowledge of the binomial theorem. The Persian mathematician, astronomer, and poet Omar Khayyam showed how to express roots of cubic of cubic equations by line segments obtained by intersecting conic sections, sections, but he could not find a formula for the roots. A L atin translation of Al-Khwarizmi's Algebra Al-Khwarizmi's Algebra appeared in the 12th century. In the early 13th century, ce ntury, the great Italian mathematician Leonardo Fibonacci achieved a close 3

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approximation to the solution of the cubic equation x  equation x  + 2 x  + cx = cx = d . Because Fibonacci had traveled in Islamic lands, he probably used an Arabic method of successive approximations. Early in the 16th century, the Italian mathematicians Scipione del Ferro, Ferro, Niccolò Tartaglia Tartaglia,, and Gerolamo Cardano solved the general cubic equation in terms o f the constants appearing in the equation. Cardano's pupil, Ludovico Ferrari, soon found an exact solution to equations of the fourth degree (see quartic equation) equation), and as a result, mat hematicians for the next several centuries tried to find a formula for the roots of equations of degree five, or higher. Early in the 19th century, however, the Norwegian mathematician Niels Abel and the French mathematician Evariste Galois proved that no such formula

exists. An important development in algebra in the 16th century was the introduction of symbols for the unknown and for algebraic powers and operations. As a result o f this development, Book III of La of La Descartes,, looks much géometrie (1637), written by the French philosopher and mathematician René Descartes like a modern algebra text. Descartes's De scartes's most significant contribution to mathematics, however, was his discovery of analytic of analytic geometry, geometry, which reduces the solution of geometric problems to the solution of  algebraic ones. His geometry text also contained the essentials of a course on the theory of equations of equations,, including his so-called rule of signs for counting the number of what Descartes c alled the "true" (positive) and "false" (negative) roots of an equation. Work continued through the 18th century o n the theory of equations, but not until 1799 was the proof published, by the German mathematician Carl Friedrich Gauss, Gauss, showing that every polynomial equation has at least one root in the complex plane (see (see Number: Complex Numbers) Numbers). By the time of Gauss, algebra had entered its modern phase. Atte ntion shifted from solving polynomial equations to studying the structure of abstract m athematical systems whose axioms were based on the behavior of mathematical objects, such as complex numbers, numbers, that mathematicians encountered when studying polynomial equations. Two examples of such systems are algebraic groups (see Group) and quaternions,, which share some of the properties of number systems but also depart from them in quaternions important ways. Groups began as systems of permutations and combinations of roots of polynomials, but they became one of the t he chief unifying concepts of 19th-century mathematics. Important contributions to their study were made by the French mathematicians Galois and Augustin Cauchy, Cauchy, the British mathematician Arthur Cayley, and the Norwegian mathematicians Niels Abel and Sophus Lie. Quaternions were discovered by British mathematician and astronomer William Rowan Hamilton, Hamilton , who extended the arithmetic of complex numbers to quaternions while complex numbers are of the form a + bi, quaternions are of the form a + bi + bi + cj + cj + dk. Immediately after Hamilton's discovery, the German mathematician Hermann Grassmann began investigating vectors. Despite its abstract character, Amer ican physicist J. W. Gibbs recognized in vector algebra a system of great utility for physicists, just as Hamilton had recognized the usefulness of  quaternions. The widespread influence of this abstract approach led George Boole to write The Laws of  Thought (1854), Thought (1854), an algebraic treatment of basic logic logic.. Since that time, modern algebra— algebra—also called abstract algebra— algebra—has continued to develop. Important new results have been discovered, and the subject has found applications in all branches of mathematics and in many of the sciences as well. Main page

*The origins of algebra go all the way back to the early Babylonians and Hindus. The Arabs (specifically the person described next) used and formalized algebra, giving it the name by which we now know it. The name is derived from the treatise t reatise written in about the year 830 AD by the Persian Muslim mathematician mathematician Muhammad bin Mūsā alal-Khwārizmī titled (in Arabic

) Al-Kitab al-

Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing"), which provided symbolic operations for the systematic solution of linear and quadratic equations. *Algebra (from Arabic al-jebr meaning al-jebr meaning "reunion of broken parts)

*By the time of Plato of Plato,, Greek mathematics had undergone a drastic change. The Greeks created a geometric algebra where terms were represented by sides of geometric obje cts, usually lines, that [2] had letters associated with them. them. Diophantus (3rd century AD), sometimes called "the father of  algebra", was an Alexandrian Greek mathematician and the author of a series of books called [3]  Arithmetica.. These texts deal with solving algebraic equations.  Arithmetica equations. While the word algebra comes from the Arabic language ( al-jabr "restoration") al-jabr "restoration") and much of its methods from Arabic/Islamic mathematics, mathematics, its roots can be traced to earlier traditions, which had a direct influence on Muhammad ibn Mūsā al-Khwārizmī (c. 780 – 850). 850). He later  wrote The Compendious Book on Calculation by Completion and Balancing , which established [4] algebra as a mathematical discipline that is indep endent of geometry of geometry and arithmetic arithmetic.. [5]

The roots of algebra can be traced to the ancient Babylonians Babylonians,, who developed an advanced arithmetical system with which they were able to do calculations in an algorithmic fashion. The Babylonians developed formulas to calculate solutions for problems typically solved today by using linear equations, equations, quadratic equations, equations, and indeterminate linear equations. equations. By contrast, most Egyptians of this era, as well as Greek  Greek and and Chinese mathematicians in the 1st millennium BC, BC, usually solved such equations by geometric methods, such as those described in the Rhind  the Rhind   Mathematical Papyrus, Papyrus, Euclid's Euclid's Elements  Elements,, and The Nine Chapters on the Mathematical Art . The geometric work of the Greeks, typified in the Elements the Elements,, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations, though this would not be realized until the medieval Muslim [citation needed ] mathematicians.. mathematicians [6]

The Hellenistic mathematicians Hero of Alexandria and Diophantus as well as Indian mathematicians such as Brahmagupta continued the traditions of Egypt and Babylon, though [7] Diophantus' Arithmetica Diophantus' Arithmetica and Brahmagupta's Brahmasphutasiddhanta Brahmagupta's Brahmasphutasiddhanta are on a higher level. level. For  example, the first complete arithmetic solution (including zero and ne gative solutions) to quadratic equations was described by Brahmagupta in his book  Brahmasphutasiddhanta.  Brahmasphutasiddhanta. Later, Arabic and Muslim mathematicians developed algebraic methods to a much higher degree of  sophistication. Although Diophantus and the Bab ylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi was the first to solve equations using general method s. He solved the linear indeterminate equations, quadratic equations, second order indeterminate equations [citation needed ] and equations with multiple variables.

In 1545, the Italian mathematician Girolamo Cardano published Ars published Ars magna -The great art , a 40-chapter masterpiece in which he gave for the first time a method for solving the general quartic equation. equation.

The Greek  Greek mathematician mathematician Diophantus has traditionally been known as the "father of algebra" but in more recent times there is much debate over whether al-Khwarizmi, who founded the [8] discipline of al-jabr  of al-jabr , deserves that title instead. instead. Those who support Diophantus point to the fact that the algebra found in Al-Jabr  in Al-Jabr is is slightly more elementary than the algebra found in [9]  Arithmetica and that Arithmetica that Arithmetica is syncopated while Al-Jabr  while Al-Jabr is is fully rhetorical. rhetorical. Those who support Al-Khwarizmi point to the fact that he introduced the methods of "reduction "reduction"" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like of like terms on opposite sides of the equation) which the term al-jabr originally al-jabr originally [10] [11] referred to, to, and that he gave an exhaustive explanation of solving quadratic equations, equations, supported by geometric proofs, while treating algebra as an independent discipline in its own [12] right. right. His algebra was also no longer concerned "with a series of  problems  problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible  prototypes for equations, which henceforward explicitly constitute the true object of study." study." He also studied an equation for its own sake and "in a generic manner, mann er, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class [13] of problems." problems." The Persian mathematician Omar Khayyam is credited with identifying the foundations of  algebraic geometry and found the general geometric solution of the cubic equation. equation. Another  Persian mathematician, Sharaf al-Dīn al-Tūsī , found algebraic and numerical solutions to various [14] [15] cases of cubic equations. equations. He also developed the concept of a function function.. The Indian [16] mathematicians Mahavira and Bhaskara II, II, the Persian mathematician Al-Karaji Al-Karaji,, and the Chinese mathematician Zhu Shijie, Shijie, solved various cases of cubic, quartic quartic,, quintic and higherorder  polynomial  polynomial equations using numerical methods. In the 13th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra. As the

Islamic world was declining, the European world was ascend ing. And it is here that algebra al gebra was further developed. François Viète’s work at the close of the 16th century marks the start of the classical discipline of algebra. In 1637, René Descartes published Descartes published La  La Géométrie, Géométrie, inventing analytic geometry and introducing modern algebraic notation. Another key event in the further development of algebra al gebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Kowa Seki in the 17th century, followed independently by Gottfried Leibniz ten years later, for the purpose of  solving systems of simultaneous linear equations using matrices matrices.. Gabriel Cramer also Cramer also did some work on matrices and determinants in the 18th century. Permutations were studied by Joseph Lagrange in his 1770 paper  Réflexions  Réflexions sur la résolution algébrique des équations devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents. resolvents. Paolo Ruffini was the first person to develop the theory of  permutation  permutation groups, groups, and like his predecessors, also in the context of solving algebraic equations. Abstract algebra was developed in the 19th century, initially focusing on what is now called [17] Galois theory, theory, and on constructibility issues. issues. The "modern algebra" algebra" has deep nineteenthcentury roots in the work, for example, of Richard of Richard Dedekind and Leopold Kronecker and Kronecker and  profound interconnections with other branches of mathematics such as algebraic number theory [18] and algebraic geometry. geometry. George Peacock was Peacock was the founder of axiomatic thinking in arithmetic and algebra. Augustus De Morgan discovered relation algebra in his Syllabus of a Proposed  System of Logic. Logic. Josiah Willard Gibbs developed an algebra of vectors in three-dimensional space, and Arthur Cayley developed an algebra of matrices (this is a noncommutative [19] algebra). algebra). *The word algebra is a Latin variant of the Arabic Ar abic word al-jabr . This came from the title of o f a book, "Hidab al-jabr wal-muqubala" , written in Baghdad about 825 A.D. by the Arab mathematician Mohammed ibn-Musa al-Khowarizmi. The words jabr  words jabr (JAH-ber) (JAH-ber) and muqubalah (moo-KAH-ba-lah) were used by al-Khowarizmi to designate two basic operations in solving equations. Jabr  equations. Jabr was was to transpose subtracted terms to the other side of  the equation. Muqubalah was to cancel like terms on o pposite sides of the equation. In fact, the title has been translated to mean "science of restoration (or reunion) and opposition" or "science of  transposition and cancellation" and "The Book of Completion and Cancellation" or "The Book of  Restoration and Balancing."  Jabr is  Jabr is used in the step where x - 2 = 12 becomes x = 14. The left-side of the first equation, where x is lessened by 2, is "restored" or "completed" back to x in the second equation. Muqabalah takes us from x + y = y + 7 to x = 7 by "cancelling" or "balancing" the two sides of t he equation. Eventually the muqabalah was left behind, and this type of math became known as algebra in many languages. However, algebra was not invented by any single person or civilization. It is a reasoning skill that is most likely as old as human beings. The concept of algebra began as a reasoning skill to determine unknown quantities.

For example, an early human being (living nearly 7 million years ago) probably ran across the problem of  food being stolen from him by other animals... He may have had 5 berries laying on the ground, but then, suddenly a bird flew by and now only had 2. He probably wondered how many berries the bird ate (an unknown quantity). He could probably reason that 3 berries were missing and thus 3 berries were eaten by the bird. If you perceive algebra in this way, then no one invented algebra because it is a natural instinct encoded in our genetics... it is our ability to reason o ut quantities that produce algebra. However, the e laboration of this reasoning into structured symbolization and manipulation is not credited to any single individual. Many people, throughout the world and throughout the ages, have developed parts of what is now known as ALGEBRA. The word itself -algebra- comes from a book called Kitab al-Jabr wa-l Muqabala (translated: Calculation by Way of Restoration and Confrontation or Calculation by Completion and Balance), written by Persian mathematician Muhammad ibn Mosa al-Khwarizmi (approximately) in the year 820 AD. However, this t his was not the first written recor d of algebraic concepts or manipulation. Ancient Egyptians, Babylonians, Indians, Chinese, and Greeks all have written records of algebra dating far before this date. No one can specify any one time, place, or person solely responsible for the elaboration of algebra as a mathematical discipline. However, it is true t hat the Acient Greeks invented "algebraic met hod" in which you solve a problem by calling a unknown in the question x, then list out all the other expressions containing x. Then you find two equal expressions and form a equation and solve it. *History of numbers. Numbers were probably first used many thousands of years ago in commerce, and initially only whole numbers and perhaps rational numbers were needed. But already in Babylonian times, practical problems of geometry began to require square roots. Nevertheless, for a very long time, and despite some development of algebra, only numbers that co uld somehow in principle be constructed mechanically were ever considered. The invention of fluxions by Isaac Newton in the late 1600s, however, introduced the idea of continuous variables - numbers with a continuous range of  possible sizes. But while this was a convenient and powerful notion, it also involved a new level of  abstraction, and it brought with it considerable confusion about fundamental issues. In fact, it was really only through the development of rigorous mathematical analysis in the late 180 0s that this confusion finally began to clear up. And already by the 1880s Georg Cantor and others had constructed completely discontinuous functions, in which the idea of treating numbers as continuous variables where only the size matters was called into question. But until almost the 1970s, and the emergence of fractal geometry and chaos theory, these functions were largely considered as mathematical curiosities, of no practical relevance. (See also page 1175.) Independent of pure mathematics, however, practical applications of numbers have always had to go beyond the abstract idealization of continuous variables. For whether one does calculations by hand, by mechanical calculator or by electronic computer, one always needs an explicit representation for numbers, typically in terms of a sequence of digits of a certain length. (From the 1930s to 1960s, some work was done on so-called analog computers which used e lectrical voltages to represent continuous variables, but such machines turned out not to be reliable enough for most practical purposes.) From

the earliest days of electronic computing, however, great efforts were made to try to approximate a continuum of numbers as closely as possible. And indeed for studying systems with fairly simple behavior, such approximations can typically be made to work. But as we shall see later in this chapter, with more complex behavior, it is almost inevitable that the approximation breaks down, and there is no choice but to look at the explicit representations of numbers.                

*algebra Diophantus al-Khwarismi Omar Khayyam Leonardo Fibnacci Scipione del Ferro+ Niccolo Tartaglia Gerolamo Cardano Niels Abel Evariste Galois Rene Descartes Carl Frriedrich Gauss Augustin Cauchy+ William Rowan Hamilton Hermann Grassmann George Bhoole

*trigo Thales, Democritus, Pythagoras, Aristotle, Archimedes, Euclid, Erastosthenes, Hipparchus

REAL NUMBERS

In mathematics mathematics,, a real number is a value that represents a quantity along a continuous line. The real numbers include all the rational numbers, numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers such as √2 (1.41421356... the square root of two, two, an irrational algebraic number ) and π (3.14159265..., a transcendental number ). Real numbers can be thought of as  points on an infinitely long line called the number line or real or real line, line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. The real line can be thought of as a part of the complex plane, plane, and correspondingly, complex numbers include real numbers as a special case. These descriptions of the real numbers are not sufficiently rigorous b y the modern standards of   pure mathematics. The discovery of a suitably rigorous definition of the real real numbers — indeed, the realization that a better definition was needed — was one of the most important developments of 19th century mathematics. The c urrently standard axiomatic definition is that [1] real numbers form the unique complete totally ordered field (R ,+,·,