Parts of Speech

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The Parts of Speech

The Parts of Speech

Traditional grammar classifies words based on eight

Traditional grammar classifies words based on eight parts of speechparts of speech: the: theverbverb,,thethenounnoun,,ththeepronounpronoun,,thetheadjectiveadjective,,

the

theadverbadverb,,thetheprepositionpreposition,,thetheconjunctionconjunction,,and theand theinterjectioninterjection..

Each

Eachpart of speechpart of speech explains not what the wordexplains not what the wordisis, but how the word, but how the wordis used is used . In fact, the same word can be a noun in. In fact, the same word can be a noun in

one

onesentencesentenceand a verb or adjective in the next. The next few examples show how a word's part of speech can changeand 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

from one sentence to the next, and following them is a series of sections on the individual parts of speech, followed by an

exercise.

exercise.

Books

Booksare made of ink, paper, and glue.are made of ink, paper, and glue.

In this sentence, "books" is a noun, the

In this sentence, "books" is a noun, the subjectsubjectof the sentence.of the sentence.

Deborah waits patiently while Bridget

Deborah waits patiently while Bridget booksbooksthe tickets.the tickets.

Here "books" is a verb, and its subject is "Bridget."

Here "books" is a verb, and its subject is "Bridget."

We

Wewalkwalkdown the street.down the street.

In this sentence, "walk" is a verb, and its subject is the pronoun "we."

In this sentence, "walk" is a verb, and its subject is the pronoun "we."

The mail carrier stood on the

The mail carrier stood on the walkwalk..

In this example, "walk" is a noun, which is part of a

In this example, "walk" is a noun, which is part of a prepositional phraseprepositional phrasedescribing where the mail carrier stood.describing where the mail carrier stood.

The town decided to build a new

The town decided to build a new jail jail..

Here "jail" is a noun, which is the

Here "jail" is a noun, which is the objectobjectof theof theinfinitive phraseinfinitive phrase"to build.""to build."

The sheriff told us that if we did not leave town immediately he would

The sheriff told us that if we did not leave town immediately he would jail jailus.us.

Here "jail" is part of the

Here "jail" is part of thecompound verbcompound verb"would jail.""would jail."

They heard high pitched

They heard high pitched criescriesin the middle of the night.in the middle of the night.

In this sentence, "cries" is a noun acting as the

In this sentence, "cries" is a noun acting as the direct objectdirect objectof the verb "heard."of the verb "heard."

The baby

The babycriescriesall night long and all day long.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.

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

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.

yourself by trying the exercise.

Written by Heather MacFadyen

Written by Heather MacFadyen

Parts of Speech Table

Parts of Speech Table

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

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

click on each part of speech.

click on each part of speech.

part of 

part of 

speech

speech

function or

function or

"job"

"job"

example

example

words

words

example sentences

example sentences

Verb

Verb

action or state

action or state

(to) be, have,

(to) be, have,

do, like,

do, like,

work, sing,

work, sing,

can, must

can, must

EnglishClub.com

EnglishClub.com is

is a web site.

a web site.

IIlike

like EnglishClub.com.

EnglishClub.com.

Noun

Noun

thing or person

thing or person

pen, dog,

pen, dog,

work, music,

work, music,

town,

town,

London,

London,

teacher, John

teacher, John

This is my

This is my dog

dog. He lives in

. He lives in

my

myhouse

house. We live in

. We live in London

London..

Adjective

Adjective

describes a

describes a

noun

noun

a/an, the, 2,

a/an, the, 2,

some, good,

some, good,

big, red, well,

big, red, well,

interesting

interesting

I have

I have two

two dogs. My dogs

dogs. My dogs

are

are big

big. I like

. I like big

big dogs.

dogs.

Adverb

Adverb

describes a

describes a

verb, adjective

verb, adjective

or adverb

or adverb

quickly,

quickly,

silently, well,

silently, well,

badly, very,

badly, very,

really

really

My dog eats

My dog eats quickly

quickly. When he

. When he

is

isvery

very hungry, he

hungry, he

eats

eats really

reallyquickly.

quickly.

Pronoun

Pronoun

replaces a noun

replaces a noun

I, you, he,

I, you, he,

she, some

she, some

Tara is Indian. She

Tara is Indian.

She is beautiful.

is beautiful.

Preposition

Preposition

links a noun to

links a noun to

another word

another word

to, at, after,

to, at, after,

on, but

on, but

We went to

We went

to school

school on

on Monday.

Monday.

Conjunction

Conjunction

 joins clauses or

 joins clauses or

sentences or

sentences or

words

words

and, but,

and, but,

when

when

I like dogs

I like dogs and

and I like cats. I

I like cats. I

like cats

like cats and

and dogs. I like

dogs. I like

dogs

dogs but

but I don't like cats.

I don't like cats.

Interjection

Interjection

short

short

exclamation,

exclamation,

sometimes

sometimes

inserted into a

inserted into a

sentence

sentence

oh!, ouch!,

oh!, ouch!,

hi!, well

hi!, well

Ouch

Ouch! That hurts!

! That hurts! Hi

Hi! How are

! How are

you?

you? Well

Well, I don't know.

, I don't know.

* Some grammar sources categorize English into

* Some grammar sources categorize English into 9

9 or

or 10

10 parts of speech. At

parts of speech. At

EnglishClub.com, we use the traditional

EnglishClub.com, we use the traditional categorization of 

categorization of 8

8 parts of speech.

parts of speech.

Examples of other categorizations are:

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Verbs may be treated as two different parts

Verbs may be treated as two different parts of speech:

of speech:

o

o

Lexical Verbs

Lexical Verbs ((work, like, run

work, like, run))

o

o

Auxiliary Verbs ((be, have, must 

Auxiliary Verbs

be, have, must ))

Determiners may be treated as a

Determiners

may be treated as a separate part of speech, instead of 

separate part of speech, instead of 

being categorized under Adjectives

being categorized under Adjectives

The history of 

The history of algebraalgebrabegan in ancient Egypt andbegan in ancient Egypt and BabylonBabylon,, where people learned to solve linear (where people learned to solve linear (ax ax == bb)) and quadratic (

and quadratic (ax ax 22++ bx bx == cc) equations, as well as) equations, as well as indeterminate equationsindeterminate equations such assuch as x  x 22++ y y 22== zz22, whereby, whereby several unknowns are involved. The ancient Babylonians solved arbitrary

several unknowns are involved. The ancient Babylonians solved arbitrary quadratic equationsquadratic equationsbyby essentially the same procedures taught today. They also could solve some indeterminate e

essentially the same procedures taught today. They also could solve some indeterminate e quations.quations. The Alexandrian mathematicians Hero of Alexandria and

The Alexandrian mathematicians Hero of Alexandria and DiophantusDiophantuscontinued the traditions of Egyptcontinued the traditions of Egypt and Babylon, but Diophantus's book

and Babylon, but Diophantus's book Arithmetica Arithmetica is on a much higher level and gis on a much higher level and gives many surprisingives many surprising solutions to difficult indeterminate equations. This ancient knowledge of solutions of equations in turn solutions to difficult indeterminate equations. This ancient knowledge of solutions of equations in turn found a home early in the Islamic world, where

found a home early in the Islamic world, where it was known as the "science of reit was known as the "science of re storation andstoration and balancing." (The Arabic word for restoration,

balancing." (The Arabic word for restoration, al-jabru,al-jabru, is the root of the wordis the root of the word algebra.algebra.) In the 9th) In the 9th century, the Arab mathematician

century, the Arab mathematician al-Khwarizmial-Khwarizmiwrote one of the first Arabic algebras, a systematicwrote one of the first Arabic algebras, a systematic exposé of the basic theory of

exposé of the basic theory of equations, with both examples and proofs. By the end of tequations, with both examples and proofs. By the end of t he 9th century,he 9th century, the Egyptian mathematician Abu Kamil had stated and proved the

the Egyptian mathematician Abu Kamil had stated and proved the basic laws and identities of algebrabasic laws and identities of algebra and solved such complicated problems

and solved such complicated problems as findingas finding x, y, x, y, andand zz such thatsuch that x  x ++ y y ++ zz = 10,= 10, x  x 22++ y y 22== zz22, and, and xz xz == y 

y 22..

Ancient civilizations wrote out algebraic expressions using only occasional abbreviations, but by Ancient civilizations wrote out algebraic expressions using only occasional abbreviations, but by medieval times Islamic mathematicians were able to talk about ar

medieval times Islamic mathematicians were able to talk about ar bitrarily high powers of the unknownbitrarily high powers of the unknown  x,

 x, and work out the basic algebra of and work out the basic algebra of polynomials (without yet using modern symbolism). This includpolynomials (without yet using modern symbolism). This includeded the ability to multiply, divide, and find square roots of 

the ability to multiply, divide, and find square roots of polynomialspolynomialsas well as a knowledge of theas well as a knowledge of the binomial theorem. The Persian mathematician, astronomer, and poet

binomial theorem. The Persian mathematician, astronomer, and poet Omar KhayyamOmar Khayyamshowed how toshowed how to express roots of 

express roots of cubic equationscubic equationsby line segments obtained by intersectingby line segments obtained by intersecting conic sectionsconic sections,, but he couldbut he could not find a formula for the roots. A L

not find a formula for the roots. A L atin translation of Al-Khwarizmi'satin translation of Al-Khwarizmi's Algebra Algebra appeared in the 12thappeared in the 12th century. In the early 13th ce

century. In the early 13th century, the great Italian mathematician Leonardontury, the great Italian mathematician Leonardo FibonacciFibonacciachieved a closeachieved a close approximation to the solution of the

approximation to the solution of the cubic equationcubic equation x  x 33+ 2+ 2 x  x 22++ cx cx == d d . Because Fibonacci had traveled in. Because Fibonacci had traveled in Islamic lands, he probably used an Arabic method of successive approximations.

Islamic lands, he probably used an Arabic method of successive approximations. Early in the 16th century, the

Early in the 16th century, the Italian mathematiciansItalian mathematicians Scipione del FerroScipione del Ferro,, NiccolòNiccolò TartagliaTartaglia,, andand GerolamoGerolamo

Cardano

Cardanosolved the general cubic equation in terms osolved the general cubic equation in terms o f the constants appearing in the equation.f the constants appearing in the equation. Cardano's pupil, Ludovico Ferrari, soon found an exact solution to equations of the

Cardano's pupil, Ludovico Ferrari, soon found an exact solution to equations of the fourth degree (seefourth degree (see quartic equation

quartic equation)), and as a result, mat, and as a result, mat hematicians for the next several centuries tried hematicians for the next several centuries tried to find a formulato find a formula for the roots of equations of degree five, or higher. Early in the 19th century, however, the Norwegian for the roots of equations of degree five, or higher. Early in the 19th century, however, the Norwegian mathematician

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exists. exists.

An important development in algebra in the 16th

An important development in algebra in the 16th century was the introduction of symbols for thecentury was the introduction of symbols for the unknown and for algebraic powers and operations. As a result o

unknown and for algebraic powers and operations. As a result o f this development, Book III of f this development, Book III of LaLa géometrie

géometrie (1637), written by the French philosopher and mathematician René(1637), written by the French philosopher and mathematician René DescartesDescartes,, looks muchlooks much like a modern algebra text. De

like a modern algebra text. Descartes's most significant contribution to mathematics, however, was hisscartes's most significant contribution to mathematics, however, was his discovery of 

discovery of analytic geometryanalytic geometry,, which reduces the solution of geometric problems to which reduces the solution of geometric problems to the solution of the solution of  algebraic ones. His geometry text

algebraic ones. His geometry text also contained the essentials of a course on the also contained the essentials of a course on the theory of theory of equationsequations,, including his so-called

including his so-called rule of signsrule of signs for counting the number of what Descartes cfor counting the number of what Descartes c alled the "true"alled the "true" (positive) and "false" (negative) roots of an equation. Work continued through the 18th century o (positive) and "false" (negative) roots of an equation. Work continued through the 18th century o n then the theory of equations, but not until 1799 was

theory of equations, but not until 1799 was the proof published, by the German mathematicianthe proof published, by the German mathematician CarlCarl

Friedrich Gauss

Friedrich Gauss,, showing that every polynomial equation has at least one root in the complex showing that every polynomial equation has at least one root in the complex plane (plane (seesee Number:

Number: Complex NumbersComplex Numbers)).. By the time of Gauss,

By the time of Gauss, algebra had entered its modern phase. Attealgebra had entered its modern phase. Atte ntion shifted from solvingntion shifted from solving polynomialpolynomial

equations

equationsto studying the structure of abstract mto studying the structure of abstract m athematical systems whose axioms were based on theathematical systems whose axioms were based on the behavior of mathematical objects, such as

behavior of mathematical objects, such as complex numberscomplex numbers,, that mathematicians encountered whenthat mathematicians encountered when studying polynomial equations. Two examples of such systems are

studying polynomial equations. Two examples of such systems are algebraic groupsalgebraic groups((seesee Group) andGroup) and quaternions

quaternions,, which share some of the properties of which share some of the properties of number systems but also depart from them innumber systems but also depart from them in important ways. Groups began as systems of permutations and combinations of roots of polynomials, important ways. Groups began as systems of permutations and combinations of roots of polynomials, but they became one of t

but they became one of the chief unifying concepts of 19th-century mathematics. Importanthe chief unifying concepts of 19th-century mathematics. Important contributions to their study were made

contributions to their study were made by the French mathematicians Galois andby the French mathematicians Galois and Augustin CauchyAugustin Cauchy,, thethe British mathematician Arthur Cayley, and the Norwegian mathematicians Niels Abel and Sophus Lie. British mathematician Arthur Cayley, and the Norwegian mathematicians Niels Abel and Sophus Lie. Quaternions

Quaternionswere discovered by British mathematician and astronomerwere discovered by British mathematician and astronomer William Rowan HamiltonWilliam Rowan Hamilton,, whowho extended the arithmetic of complex

extended the arithmetic of complex numbers to quaternions while complex numbers are of the formnumbers to quaternions while complex numbers are of the form aa ++ bi,

bi, quaternions are of the formquaternions are of the form aa ++ bi bi ++ cj cj ++ dk.dk.

Immediately after Hamilton's discovery, the German mathematician

Immediately after Hamilton's discovery, the German mathematician Hermann GrassmannHermann Grassmannbeganbegan investigating vectors. Despite its abstract character, Amer

investigating vectors. Despite its abstract character, Amer ican physicist J. W. Gibbs recognized inican physicist J. W. Gibbs recognized in vectorvector

algebra

algebraa system of great utility for physicists, just a system of great utility for physicists, just as Hamilton had recognized the usefulness of as Hamilton had recognized the usefulness of  quaternions. The widespread influence of this abstract approach led

quaternions. The widespread influence of this abstract approach led George BooleGeorge Booleto writeto write The Laws of The Laws of  Thought 

Thought (1854), an algebraic treatment of basic(1854), an algebraic treatment of basic logiclogic.. Since that time, modern algebraSince that time, modern algebra——also calledalso called abstract algebra

abstract algebra——has continued to develop. Important new results have been discovered, has continued to develop. Important new results have been discovered, and theand the subject has found applications in all branches of mathematics and in many of the

subject has found applications in all branches of mathematics and in many of the sciences as well.sciences as well. MainMain

page

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*The origins of algebra go all the way

*The origins of algebra go all the way back to the early Babylonians and Hindus. The Arabs (specificallyback to the early Babylonians and Hindus. The Arabs (specifically the person described next) used and formalized

the person described next) used and formalized algebra, giving it the name by which we algebra, giving it the name by which we now know it.now know it. The name is derived from the t

The name is derived from the treatise written in about the year reatise written in about the year 830 AD by the Persian Muslim830 AD by the Persian Muslim mathematic

mathematician Muhammad bin Mūsā alian Muhammad bin Mūsā al--Khwārizmī titled (in ArabicKhwārizmī titled (in Arabic ) Al-Kitab al-) Al-Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on

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

*AlgebraAlgebra (from(from ArabicArabical-jebr al-jebr meaning "reunion of broken parts)meaning "reunion of broken parts)

*By the time of 

*By the time of Plato

Plato

,, Greek mathematics

Greek mathematics

had undergone a drastic change. The

had undergone a drastic change. The Greeks

Greeks

created a

created a

geometric algebra

geometric algebra

where terms were represented by sides of geometric obje

where terms were represented by sides of geometric obje cts, usually lines, that

cts, usually lines, that

had letters associated with them

had letters associated with them..

[2][2]

Diophantus

Diophantus

(3rd century AD), sometimes called "the father of 

(3rd century AD), sometimes called "the father of 

algebra", was an

algebra", was an Alexandrian

Alexandrian

Greek mathematician

Greek mathematician

and the author of a series of books called

and the author of a

series of books called

 Arithmetica

 Arithmetica

.. These texts deal with solving

These texts deal with solving algebraic equations

algebraic equations

..

[3][3]

While the word

While the word algebra

algebra comes from the

comes from the Arabic language

Arabic language

((

al-jabr 

al-jabr "restoration") and much

"restoration") and much

of its methods from

of its methods from Arabic/Islamic mathematics

Arabic/Islamic mathematics

,, its roots can be traced to earlier traditions,

its roots can be traced to earlier traditions,

which had a direct influence on

which had a direct influence on

Muhammad ibn Mūsā al

Muhammad ibn Mūsā al

-

-

Khwārizmī 

Khwārizmī 

(c. 780

(c. 780

 – 

 – 

850). He later 

850). He later 

wrote

wrote The Compendious Book on Calculation by Completion and Balancing 

The Compendious Book on Calculation by Completion and Balancing 

,, which established

which established

algebra as a mathematical discipline that is indep

algebra as a mathematical discipline that is independent of 

endent of geometry

geometry

and

and arithmetic

arithmetic

..

[4][4]

The roots of algebra can be traced to the ancient

The roots of algebra can be traced to the ancient Babylonians

Babylonians

,,

[5][5]

who developed an advanced

who developed an advanced

arithmetical system with which they were able to do

arithmetical system with which they were able to do calculations in an

calculations in an algorithmic

algorithmic

fashion. The

fashion. The

Babylonians developed formulas to calculate solutions for problems typically solved

Babylonians developed formulas to calculate solutions for problems typically solved today by

today by

using

using linear equations

linear equations

,, quadratic equations

quadratic equations

,, and

and indeterminate linear equations

indeterminate linear equations

.. By contrast, most

By contrast, most

Egyptians

Egyptians

of this era, as well as

of this era, as well as Greek 

Greek 

and

and Chinese

Chinese

mathematicians in the

mathematicians in the 1st millennium BC

1st millennium BC

,,

usually solved such equations by geometric methods,

usually solved such equations by geometric methods, such as those described in the

such as those described in the Rhind 

 Rhind 

 Mathematical Papyrus

 Mathematical Papyrus

,, Euclid's

Euclid's

 Elements

 Elements

,, and

and The Nine Chapters on the Mathematical Art 

The Nine Chapters on the Mathematical Art 

.. The

The

geometric work of the Greeks, typified in the

geometric work of the Greeks, typified in the Elements

 Elements, provided the framework for generalizing

, provided the framework for generalizing

formulae beyond the solution of particular problems into

formulae beyond the solution of particular problems into more general systems of stating and

more general systems of stating and

solving equations, though this would not be realized until the

solving equations, though this would not be realized until the medieval Muslim

medieval Muslim

mathematicians

mathematicians

..

[[citation needed citation needed ]]

The

The Hellenistic

Hellenistic

mathematicians

mathematicians Hero of Alexandria

Hero of Alexandria

and

and Diophantus

Diophantus

[6][6]

as well as

as well as Indian

Indian

mathematicians

mathematicians

such as

such as Brahmagupta

Brahmagupta

continued the traditions of Egypt and Babylon, though

continued the traditions of Egypt and Babylon, though

Diophantus'

Diophantus' Arithmetica

 Arithmetica

and Brahmagupta's

and Brahmagupta's Brahmasphutasiddhanta

 Brahmasphutasiddhanta

are on a higher level

are on a higher level..

[7][7]

For 

For 

example, the first complete arithmetic solution (including zero and ne

example, the first complete arithmetic solution (including zero and ne gative solutions) to

gative solutions) to

quadratic equations

quadratic equations

was described by Brahmagupta in his book 

was described by Brahmagupta in his book  Brahmasphutasiddhanta

 Brahmasphutasiddhanta. Later,

. Later,

Arabic and Muslim mathematicians developed algebraic methods to a much higher degree of 

Arabic and Muslim mathematicians developed algebraic methods to a much higher degree of 

sophistication. Although Diophantus and the Bab

sophistication. Although Diophantus and the Babylonians used mostly special

ylonians used mostly special ad hoc

ad hoc methods to

methods to

solve equations, Al-Khwarizmi was the first to solve equations using general method

solve equations, Al-Khwarizmi was the first to solve equations using general method s. He solved

s. He solved

the linear indeterminate equations, quadratic equations,

the linear indeterminate equations, quadratic equations, second order indeterminate equations

second order indeterminate equations

and equations with multiple variables.

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In 1545, the Italian mathematician

In 1545, the Italian mathematician Girolamo CardanoGirolamo Cardanopublishedpublished Ars magna Ars magna--The great art The great art , a 40-chapter, a 40-chapter masterpiece in which he gave for the first time a method for solving the general

masterpiece in which he gave for the first time a method for solving the general quartic equationquartic equation..

The

The Greek 

Greek 

mathematician

mathematician Diophantus

Diophantus

has traditionally been known as the

has traditionally been known as the "father of algebra" but

"father of algebra" but

in more recent times there is much debate

in more recent times there is much debate over whether al-Khwarizmi, who founded the

over whether al-Khwarizmi, who founded the

discipline of 

discipline of al-jabr 

al-jabr , deserves that title instead

, deserves that title instead..

[8][8]

Those who support Diophantus point to

Those who support Diophantus point to the fact

the fact

that the algebra found in

that the algebra found in Al-Jabr 

 Al-Jabr is slightly more elementary than the algebra found

is slightly more elementary than the algebra found in

in

 Arithmetica

 Arithmetica and that

and that Arithmetica

 Arithmetica is syncopated while

is syncopated while Al-Jabr 

 Al-Jabr is fully rhetorical

is fully rhetorical..

[9][9]

Those who

Those who

support Al-Khwarizmi point to the fact that he introduced the

support Al-Khwarizmi point to the fact that he introduced the methods of "

methods of "reduction

reduction

"" and

and

"balancing" (the transposition of subtracted terms to the other side of

"balancing" (the transposition of subtracted terms to the other side of an equation, that is, the

an equation, that is, the

cancellation of 

cancellation of like terms

like terms

on opposite sides of the equation) which

on opposite sides of the equation) which the term

the term al-jabr 

al-jabr originally

originally

referred to

referred to,,

[10][10]

and that he gave an exhaustive explanation of solving quadratic equations

and that he gave an exhaustive explanation of solving quadratic equations,,

[11][11]

supported by geometric proofs, while treating algebra as an

supported by geometric proofs, while treating algebra as an independent discipline in its own

independent discipline in its own

right

right..

[12][12]

His algebra was also no longer concerned

His algebra was also no longer concerned "with a series of 

"with a series of  problems

 problems

to be resolved, but

to be resolved, but

an

an exposition

exposition

which starts with primitive terms in which the combinations must give all possible

which starts with primitive terms in which the combinations must give all possible

 prototypes for equations, which henceforward explicitly constitute the true object of st

 prototypes for equations, which henceforward explicitly constitute the true object of study." He

udy." He

also studied an equation for its own

also studied an equation for its own sake and "in a generic mann

sake and "in a generic manner, insofar as it does not simply

er, insofar as it does not simply

emerge in the course of solving a

emerge in the course of solving a problem, but is specifically called on to

problem, but is specifically called on to define an infinite class

define an infinite class

of problems.

of problems.""

[13][13]

The Persian mathematician

The Persian mathematician Omar Khayyam

Omar Khayyam

is credited with identifying the foundations of 

is credited with identifying the foundations of 

algebraic geometry

algebraic geometry

and found the general geometric solution of the

and found the general geometric solution of the cubic equation

cubic equation

.. Another 

Another 

Persian mathematician,

Persian mathematician, Sharaf al-

Sharaf al-

Dīn al

Dīn al

-

-

Tūsī 

Tūsī 

,, found algebraic and numerical solutions to various

found algebraic and numerical solutions to various

cases of cubic equations

cases of cubic equations..

[14][14]

He also developed the concept of a

He also developed the concept of a function

function

..

[15][15]

The Indian

The Indian

mathematicians

mathematicians Mahavira

Mahavira

and

and Bhaskara II

Bhaskara II

,, the Persian mathematician

the Persian mathematician Al-Karaji

Al-Karaji

,,

[16][16]

and the

and the

Chinese mathematician

Chinese mathematician Zhu Shijie

Zhu Shijie

,, solved various cases of cubic,

solved various cases of cubic, quartic

quartic

,, quintic

quintic

and higher-

and

higher-order 

order  polynomial

 polynomial

equations using numerical methods. In the 13th century, the solution of a cubic

equations using numerical methods. In the 13th century, the solution of a cubic

equation by

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Islamic world was declining, the European world was ascend

Islamic world was declining, the European world was ascend ing. And it is here that al

ing. And it is here that algebra was

gebra was

further developed.

further developed.

François Viète

François Viète

’’s work at the close of the 16th century marks the start of the classical discipline

s work at the close of the 16th century marks the start of the classical discipline

of algebra. In 1637,

of algebra. In 1637, René Descartes

René Descartes

 published La Géométrie

 published

 La Géométrie

,, inventing

inventing analytic geometry

analytic geometry

and

and

introducing modern algebraic notation. Another key

introducing modern algebraic notation. Another key event in the further development of al

event in the further development of algebra

gebra

was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th

was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th

century. The idea of a

century. The idea of a determinant

determinant

was developed by Japanese mathematician

was developed by

Japanese mathematician

Kowa Seki

Kowa Seki

in the

in the

17th century, followed independently by

17th century, followed independently by Gottfried Leibniz

Gottfried Leibniz

ten years later, for the purpose of 

ten years later, for the purpose of 

solving systems of simultaneous linear equations using

solving systems of simultaneous linear equations using matrices

matrices

.. Gabriel Cramer 

Gabriel Cramer 

also did some

also did some

work on matrices and determinants in the

work on matrices and determinants in the 18th century. Permutations were studied by

18th century. Permutations were studied by Joseph

Joseph

Lagrange

Lagrange

in his 1770 paper 

in his 1770 paper  Réflexions sur la résolution algébrique des équations

 Réflexions sur la résolution algébrique des équations devoted to

devoted to

solutions of algebraic equations, in which he introduced

solutions of algebraic equations, in which he introduced Lagrange resolvents

Lagrange resolvents

.. Paolo Ruffini

Paolo Ruffini

was

was

the first person to develop the theory of 

the first person to develop the theory of  permutation groups

 permutation groups

,, and like his predecessors, also in the

and like his predecessors, also in the

context of solving algebraic equations.

context of solving algebraic equations.

Abstract algebra

Abstract algebra

was developed in the 19th century, initially focusing on what is now

was developed in the

19th century, initially focusing on what is now called

called

Galois theory

Galois theory

,, and on

and on constructibility

constructibility

issues

issues..

[17][17]

The

The ""modern algebra

modern algebra

"" has deep nineteenth-

has deep

nineteenth-century roots in the work, for example, of 

century roots in the work, for example, of Richard Dedekind

Richard Dedekind

and

and Leopold Kronecker 

Leopold Kronecker 

and

and

 profound interconnections with other branches of mathematics such as

 profound interconnections with other branches of mathematics such as algebraic number theory

algebraic number theory

and

and algebraic geometry

algebraic geometry

..

[18][18]

George Peacock 

George Peacock 

was the founder of axiomatic thinking in arithmetic

was the founder of axiomatic thinking in arithmetic

and algebra.

and algebra. Augustus De Morgan

Augustus De Morgan

discovered

discovered relation algebra

relation algebra

in his

in his Syllabus of a Proposed 

Syllabus of a Proposed 

System of Logic

System of Logic.. Josiah Willard Gibbs

Josiah Willard Gibbs

developed an algebra of vectors in

developed an algebra of vectors in three-dimensional

three-dimensional

space, and

space, and Arthur Cayley

Arthur Cayley

developed an algebra of

developed an algebra of matrices (this is a noncommutative

matrices (this is a noncommutative

algebra)

algebra)..

[19][19]

*The word

*The word algebraalgebra is a Latin variant of the Aris a Latin variant of the Arabic wordabic word al-jabr al-jabr . This came from the title o. This came from the title of a book,f a book, ""Hidab al-jabr Hidab al-jabr wal-muqubala" wal-muqubala" , written in Baghdad about 825 , written in Baghdad about 825 A.D. by the Arab mathematicianA.D. by the Arab mathematician Mohammed ibn-Musa al-Khowarizmi.

Mohammed ibn-Musa al-Khowarizmi. The words

The words jabr  jabr (JAH-ber) and(JAH-ber) and muqubalahmuqubalah (moo-KAH-ba-lah) were used by al-Khowarizmi to designate(moo-KAH-ba-lah) were used by al-Khowarizmi to designate two basic operations in solving equations.

two basic operations in solving equations. Jabr  Jabr was to transpose subtracted terms to the other side of was to transpose subtracted terms to the other side of  the equation.

the equation. MuqubalahMuqubalah was to cancel like terms on owas to cancel like terms on opposite sides of the equation. In fact, the title haspposite sides of the equation. In fact, the title has been translated to mean "science of

been translated to mean "science of restoration (or reunion) and opposition" or "science of restoration (or reunion) and opposition" or "science of  transposition and cancellation" and "The Book of Completion and Cancellation" or "The Book of  transposition and cancellation" and "The Book of Completion and Cancellation" or "The Book of  Restoration and Balancing."

Restoration and Balancing."  Jabr 

 Jabr is used in the step where x - 2 = 12 becomes x = 14. The left-side of the first equation, where x isis 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

lessened by 2, is "restored" or "completed" back to x in the second equation."completed" back to x in the second equation. Muqabalah

Muqabalah takes us from x + y takes us from x + y = y + 7 to x = y + 7 to x = 7 by "cancelling" or "balancing" the two sides of t= 7 by "cancelling" or "balancing" the two sides of t hehe equation.

equation. Eventually the

Eventually the muqabalahmuqabalah was left behind, and this type of math was left behind, and this type of math became known as algebra in manybecame known as algebra in many languages.

languages.

However, algebra was not invented by

However, algebra was not invented by any single person or civilization. It is a reasoning skill that is mostany single person or civilization. It is a reasoning skill that is most likely as old as human beings. The concept of algebra began

likely as old as human beings. The concept of algebra began as a reasoning skill to determine unknownas a reasoning skill to determine unknown quantities.

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For example, an early

For example, an early human being (living nearly 7 million years ago) probably ran across the problem of human being (living nearly 7 million years ago) probably ran across the problem of  food being stolen from him by other

food being stolen from him by other animals... He may have had 5 berries laying on the ground, butanimals... He may have had 5 berries laying on the ground, but then, suddenly a bird flew by and now only had 2.

then, suddenly a bird flew by and now only had 2. He probably wondered how many berries the bird ateHe probably wondered how many berries the bird ate (an unknown quantity). He could probably reason that 3 berries were

(an unknown quantity). He could probably reason that 3 berries were missing and thus 3 berries weremissing and thus 3 berries were eaten by the bird.

eaten by the bird.

If you perceive algebra in this way,

If you perceive algebra in this way, then no one invented algebra because it then no one invented algebra because it is a natural instinct encodedis a natural instinct encoded in our genetics... it is our ability to reason o

in our genetics... it is our ability to reason o ut quantities that produce algebra. However, the eut quantities that produce algebra. However, the e laborationlaboration of this reasoning into structured symbolization and manipulation is not credited to any single individual. of this reasoning into structured symbolization and manipulation is not credited to any single individual. Many people, throughout the world and throughout the ages,

Many people, throughout the world and throughout the ages, have developed parts of what is nowhave developed parts of what is now known as ALGEBRA. The word itself -algebra-

known as ALGEBRA. The word itself -algebra- comes from a book called Kitab al-Jabr wa-l Muqabalacomes from a book called Kitab al-Jabr wa-l Muqabala (translated: Calculation by Way of Restoration and Confrontation or Calculation by Completion and (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 Balance), written by Persian mathematician Muhammad ibn Mosa al-Khwarizmi (approximately) in the year 820 AD. However, t

year 820 AD. However, this was not the first written recorhis was not the first written recor d of algebraic concepts or manipulation.d of algebraic concepts or manipulation. Ancient Egyptians, Babylonians, Indians, Chinese, and Greeks all have written records of

Ancient Egyptians, Babylonians, Indians, Chinese, and Greeks all have written records of algebra datingalgebra dating far before this date. No one can

far before this date. No one can specify any one time, place, or specify any one time, place, or person solely responsible for theperson solely responsible for the elaboration of algebra as a mathematical discipline.

elaboration of algebra as a mathematical discipline. However, it is true t

However, it is true t hat the Acient Greeks invented "algebraic methat the Acient Greeks invented "algebraic met hod" in which you solve a problem byhod" in which you solve a problem by calling a unknown in the question x, then

calling a unknown in the question x, then list out all the other expressions containing x. Then you list out all the other expressions containing x. Then you findfind two equal expressions and form a equation and solve it.

two equal expressions and form a equation and solve it. *

*History of numbers.History of numbers. Numbers were probably first used many thousands of years ago in commerce, Numbers were probably first used many thousands of years ago in commerce, andand initially only whole numbers and perhaps rational numbers were needed. But already in

initially only whole numbers and perhaps rational numbers were needed. But already in BabylonianBabylonian times, practical problems of geometry began to

times, practical problems of geometry began to require square roots. Nevertheless, for a very require square roots. Nevertheless, for a very long time,long time, and despite some development of algebra, only numbers that co

and despite some development of algebra, only numbers that co uld somehow in principle beuld somehow in principle be constructed mechanically were ever considered. The invention of fluxions by

constructed mechanically were ever considered. The invention of fluxions by Isaac NewtonIsaac Newtonin the latein the late 1600s, however, introduced the idea of

1600s, however, introduced the idea of continuous variables - numbers with a continuous range of continuous variables - numbers with a continuous range of  possible sizes. But while this was a convenient and powerful notion, it

possible sizes. But while this was a convenient and powerful notion, it also involved a new level of also involved a new level of  abstraction, and it brought with it considerable confusion about fundamental issues. In fact, it was abstraction, and it brought with it considerable confusion about fundamental issues. In fact, it was reallyreally only through the development of rigorous mathematical analysis in the late 180

only through the development of rigorous mathematical analysis in the late 180 0s that this confusion0s that this confusion finally began to clear up. And already by the

finally began to clear up. And already by the 1880s1880s Georg CantorGeorg Cantorand others had constructed completelyand others had constructed completely discontinuous functions, in which the idea of treating numbers as continuous variables where only the 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

size matters was called into question. But until almost the 1970s, and the emergence of fractal geometry and chaos theory, these

geometry and chaos theory, these functions were largely considered as mathematical curiosities, of nofunctions were largely considered as mathematical curiosities, of no practical relevance. (See also page 1175.)

practical relevance. (See also page 1175.)

Independent of pure mathematics, however, practical applications of numbers have always had Independent of pure mathematics, however, practical applications of numbers have always had to goto go beyond the abstract idealization of continuous variables. For whether one does calculations by hand, by beyond the abstract idealization of continuous variables. For whether one does calculations by hand, by mechanical calculator or by electronic computer, one

mechanical calculator or by electronic computer, one always needs an explicit representation foralways needs an explicit representation for numbers, typically in terms of a sequence of

numbers, typically in terms of a sequence of digits of a certain length. (From the digits of a certain length. (From the 1930s to 1960s, some1930s to 1960s, some work was done on so-called analog computers which used e

work was done on so-called analog computers which used e lectrical voltages to represent continuouslectrical voltages to represent continuous variables, but such machines turned out not to be

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the earliest days of electronic computing, however, great efforts were made to try to approximate a 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 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

behavior, such approximations can typically be made to work. But as we shall see later in as we shall see later in this chapter,this chapter, with more complex behavior, it is almost

with more complex behavior, it is almost inevitable that the approximation breaks down, and there is noinevitable that the approximation breaks down, and there is no choice but to look at the

choice but to look at the explicit representations of numbers.explicit representations of numbers.

  *algebra*algebra   DiophantusDiophantus   al-Khwarismial-Khwarismi 

 Omar KhayyamOmar Khayyam 

 Leonardo FibnacciLeonardo Fibnacci 

 Scipione del Ferro+Scipione del Ferro+ 

 Niccolo TartagliaNiccolo Tartaglia 

 Gerolamo CardanoGerolamo Cardano 

 Niels AbelNiels Abel 

 Evariste GaloisEvariste Galois 

 Rene DescartesRene Descartes 

 Carl Frriedrich GaussCarl Frriedrich Gauss 

 Augustin Cauchy+Augustin Cauchy+ 

 William Rowan HamiltonWilliam Rowan Hamilton 

 Hermann GrassmannHermann Grassmann 

 George BhooleGeorge Bhoole

*trigo *trigo

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

REAL NUMBERS REAL NUMBERS

In

In mathematics

mathematics

,, aa

real numberreal number

is a value that represents a quantity along a continuous line. The

is a value that represents a quantity along a continuous line. The

real numbers include all the

real numbers include all the rational numbers

rational numbers

,, such as the

such as the integer 

integer 

−5 and the

−5 and the

fraction

fraction

4/3, and all

4/3, and all

the

the irrational numbers

irrational numbers

such as √2 (1.41421356... the

such as √2 (1.41421356... the

square root of two

square root of two

,, an irrational

an irrational algebraic

algebraic

number 

number 

)) and

and

π

π

(3.14159265..., a

(3.14159265..., a transcendental number 

transcendental number 

)). Real numbers can be thought of as

. Real numbers can be thought of as

 points on an infinitely long

 points on an infinitely long line

line

called the

called the number line

number line

or 

or real line

real line

,, where the points

where the points

corresponding to

corresponding to integers

integers

are equally spaced. Any real number can be determined by a possibly

are equally spaced. Any real number can be determined by a possibly

infinite

infinite decimal representation

decimal representation

such as that of 8.632,

such as that of 8.632, where each consecutive digit is measured in

where each consecutive digit is measured in

units one tenth the size of the previous one. The

units one tenth the size of the previous one. The real line

real line

can be thought of as a part of the

can be thought of as a part of the

complex plane

complex plane

,, and correspondingly,

and correspondingly, complex numbers

complex numbers

include real numbers as a special case.

include real numbers as a special case.

These descriptions of the real numbers are not sufficiently rigorous b

These descriptions of the real numbers are not sufficiently rigorous b y the modern standards of 

y the modern standards of 

 pure mathematics. The discovery of a suitably rigorous definition of the r

 pure mathematics. The discovery of a suitably rigorous definition of the real numbers

eal numbers

 — 

 — 

indeed,

indeed,

the realization that a better definition was needed

the realization that a better definition was needed

 — 

 — 

was one of the most important

was one of the most important

developments of 19th century mathematics. The c

developments of 19th century mathematics. The currently standard axiomatic definition is that

urrently standard axiomatic definition is that

real numbers form the unique

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Whereas popular constructive definitions of real numbers include declaring

Whereas popular constructive definitions of real numbers include declaring them as

them as equivalence

equivalence

classes

classes

of 

of Cauchy sequences

Cauchy sequences

of rational numbers, Dedekind cuts

of rational numbers,

Dedekind cuts

,, or certain infinite "decimal

or certain infinite "decimal

representations", together with precise interpretations for the arithmetic operations and the order 

representations", together with precise interpretations for the arithmetic operations and the order 

relation. These definitions are equivalent in the realm

relation. These definitions are equivalent in the realm of 

of classical mathematics

classical mathematics

..

A real number may be either 

A real number may be either rational

rational

or 

or irrational

irrational

;; either 

either algebraic

algebraic

or 

or transcendental

transcendental

;; and either 

and either 

 positive

 positive

,, negative

negative

,, or 

or zero

zero

.. Real numbers are used to measure

Real numbers are used to measure continuous

continuous

quantities. They may in

quantities. They may in

theory be expressed by

theory be expressed by decimal representations

decimal representations

that have an infinite sequence of digits to the

that have an infinite sequence of digits to the

right of the decimal point; these are often represented in the same form as 324.823122147… The

right of the decimal point; these are often represented in the same form as 324.823122147… The

ellipsis

ellipsis

(three dots) indicate that there would still be more digits to come.

(three dots) indicate that there would still be more digits to come.

More formally, real numbers have the two b

More formally, real numbers have the two basic properties of being an

asic properties of being an ordered field

ordered field

,, and having

and having

the

the least upper bound

least upper bound

 property. The first says that real numbers comprise a

 property. The first says that real numbers comprise a field

field

,, with addition

with addition

and multiplication as well as division by nonzero numbers, which can be

and multiplication as well as division by nonzero numbers, which can be totally ordered

totally ordered

on a

on a

number line in a way co

number line in a way compatible with addition and multiplication. The seco

mpatible with addition and multiplication. The second says that if a

nd says that if a

nonempty set of real numbers has an

nonempty set of real numbers has an upper bound

upper bound

,, then it has a

then it has a least upper bound

least upper bound

.. The second

The second

condition distinguishes the real numbers from the rational numbers: for example, the

condition distinguishes the real numbers from the rational numbers: for example, the set of 

set of 

rational numbers whose square is less than 2 is a s

rational numbers whose square is less than 2 is a s et with an upper bound (e.g.

et with an upper bound (e.g. 1.5) but no least

1.5) but no least

upper bound: hence the

upper bound: hence the rational numbers do not satisfy the least upper bou

rational numbers do not satisfy the least upper bou nd property.

nd property.

Natural Numbers Natural Numbers

 1, 2, 3, 4, 5, 6 …

1, 2, 3, 4, 5, 6 …

numbers you count with

numbers you count with

positive (not zero) whole numbers

positive (not zero) whole numbers

Whole numbers Whole numbers

 0, 1, 2, 3, 4, 5, …

0, 1, 2, 3, 4, 5, …

the natural numbers, and also zero.

the natural numbers, and also zero.

No negatives; no fractions

No negatives; no fractions

Integers Integers

 …

-3,-2,-

-3,-2,-

1, 0, 1, 2, 3, …

1, 0, 1, 2, 3, …

Whole numbers and their opposites

Whole numbers and their opposites

Absolute value Absolute value

Distance a number is from zero

Distance a number is from zero

 Can’t be negative

Can’t be negative

Figure

Updating...

References

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