### 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

### 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:

### 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))

oo

### 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

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

*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.

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

### 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.

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

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

### 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