A. I Kypour AJII
ESPAMtIECKHE
IIPON3BOJlbHbIX H3gRTe.1bCTBO «Hayrca»
YPABHEHNSI
CTEl1EHEA
A.U.
Kurosh
ALGE
B RAIC
EQUATIONS OF ARBITRARY
DEGREES Translated
from
by V.
Kisin
the Russian
t erst punhsned I/li Revised from the l975 Russian edition
HQ OHZABQCKO.u A3slKC ! English
translation, Mir Publishers, 1977
t 'ontents
Preface
7
troduction
9
Complex Numbers
10
In 1.
Evolution. Quadratic Equations
te
2.
Cubic Equations
19
3.
Solution of Equations in Terms of Radicals and the Existence of Roots of Equations
22
The Number of Real Roots
24
Approximate Solution of Equations
27
6.
Fields
30
7.
Conclusion
35
Bibliography
36
IIUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
Prefact
the author's lecture to high Mathematics Olympiad at a review
of the results
and
algebraic equations with due >f its readers. No proofs are have required copying almost ,her algebra. Despite such an eke for light reading. Even a ir the reader's concentration, initions and statements, check , application of the methods
This
booklet
is a revision
of
school students taking part in thi Moscow State University. It gives methods of the general theory of regard for the level of knowledge i included
in the text since this would
half of a university textbook on hi~ approach, this booklet does not m: popular mathematics book calls fc thorough consideration of all the del of calculations in all the exatnples described to his own examples, etc.
IIUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUURRRRRRRRRRRR
Introduction
A secondary school course of algebra is diversified but equations its focus. Let us restrict ourselves to equations with one tnown, and recall what is taught in secondary. school. Any pupil can solve jirst degree equations: if an equation ax+
are unl
b =0
~Jven,in which a 4 0, then its single root is X=
is
a
Furthermore, a pupil knows the formula l'or solving quadratic
eqt ax'+
bx+
c= 0
ere a/0:
wh
5+
x =
rb'
4ac-
2a
III
i< l
tations. For example, the third-degree cubic! equation ax
+ bx
+ ex=0
ich has one root x = 0, and. after factoring out x, is transformed a a quadratic equation ax
+bx+c=O n
wh inti
legree quartic! equation ay +by
A fourth-d
+c=0
atic, may also be reduced to a quadratic equation = x, calculating the roots of the resulting quadratic then extracting their square roots. phasize once again that these are only very special c and quartic equations. Secondary school algebra lods of solving arbitrary equations of these degrees,
called biquadr
by setting y~ equation and Let us em
types of cubi gives no met?
Itore so of higher degrees.However, we encounter
and all the rl
dgebraic equations in different branches of engineering, 9 physics. The theory of algebraic equations of an ee n, where n is a positive integer, has required levelop and now constitutes one of the main parts bra taught at universities and pedagogical institutes.
higher degree t mechanics
ani
arbitrary degr centuries
to c
of higher alger
1. Complex Numbers ' of algebraic equations is essentially based on the mplex numbers taught at high school. However, a doubt the justification for introducing these their actual existence. When complex numbers ced, even mathematicians doubted their actual >ce the term "imaginary numbers" which still vever, modern science sees nothing mysterious cc'
x2+
e r......
students ofte; numbers and were introdu existence, he> survives. Hoi
1 =0
t of such equations. Is there a way to expand the tbers so that these equations also possess roots? !dy of mathematics at school the student sees numbers at his disposal constantly extended. He .grai positive numbers in elementary arithmetic. Very .........1.
The theory theory of co:
s.ne. gati.ve..~umber'...a.i.
J
is the simples realm In
of nun his st<
the system of starts with inn c l........
previously had no roots. Thus, the equation 2x 1
d, the equation
=0
acquires a root only after fractions are introduce x+1=0
mbers, and the
has a root equation
after the introduction
of negative nu
x~ 2=0
m the way to ed to a general ted on a given unit
of scale is
can be put in
has a root only after irrational numbers are adde All this completely justifies one more step enlarge the store of numbers. We shall now proce outline of this last step. It is known that if a positive direction is fi, straight line, if the origin 0 is marked, and if a
chosen Fig. 1!, then each point A on this line 01 iIG
l number which nce between A or the distance
correspondence with its coordinate, i. e. with a rea expresses in the chosen units of scale the dista and 0 if A lies to the right of the point 0,
'
~I i ibers in such a st as natural a not constructed obers. which this new
But is it possible to expand the store of nun way that new numbers can be represented in ju manner by the points of a plane? So far we have a system of numbers wider than that of real nur We shall start by indicating the "material" with
hat objects will
system of numbers is to be "constructed", i. e. w act as new numbers. We must also define hov
to
carry
out
implest way is to consider the points of
points of the plane, the s the plane themselves as these points as nuinbers,
the new numbers. To seriously conyider we must merely define how to carry out i them, i. e. which point is to be the of the plane, which one is to be their
algebraic operations witt
sum of two given points product, etc. Just as the position ol defined by a single rea
I' a point on a straight line is completely t number, its coordinate, the position a plane can be defined by a pair of real us take two perpendicular straight lines, at the point 0 and on each of them
of an arbitrary point on numbers.
To do this let
intersecting on the plane
and set olf a unit of scale Fig. 2!. Let
fix the positive direction
FIG. 2
II
II
I]
two positive real numbers, e. g. t e the selected units
is completely defined by number a which gives in from this point to the c
of scale the distance
Irdinate axis the abscissaof point A!, gives in the selected units of scale its
and the number b which distance from the abscis,
sa axis the ordinate of the point A!. a, b! of positive real numbers we can ly defined point
Conversely,for each pair indicate a single
in the first quadrant U
U U UI I U U
precise'
to eral S lrl ants axis the d b
plane. and the pairs of coordinates a, b!, i. e. in order avoid the same pair of coordinates a, b! correspondingto sev distinct points on the plane, we assume the abscissas of point quadrants 11 and 111 and the ordinates of points in quadr III and IV to be negative. Note that points on the abscissa
are given by coordinates of the type a, 0!, and those on ordinate axis by coordinates of the type , b!, where a an are certain
by if a oint iints
nely to !ef s. lntil duct :issa
real numbers,
We are now able to define all the points on the plane pairs of real numbers. This enables us to talk further not <
point A, given by the coordinates a, b!, but simply of a p a, b!.
Let us now define addition and multiplication of the pc on the plane. At first these definitions may seem extrer artificial. However, only such definitions will make it possibh realize our goal of taking square roots of negative real nuinl
Let the points a, b! and c, d! be given on the plane. l. now we did not know how to define the sum and the proi of these points. Let us call their sum the point with the aha a+
c and
the ordinate
b+
d,
e.
a, b! + c, d! = a + c, b + d! >ints e.
t of and law :s it the and
Dn the other hand, let us call the product of the given pi the point with the abscissa ac
bd and the ordinate ad + bc,
be interchanged!,associative i. e. the sum and the produc three points are independentof the position of the brackets! distributive i. e. the brackets can be removed!. Note that the of association for addition and multiplication of points makt possible to introduce in an unambiguous way the sum and product of any finite number of points on the plane. Now we can also perform the operations of subtraction nn
by the divisor yields the dividend!. Thus, a, 6! c, d! = a c, b
a, 6! c,d!
ac+ bd c+d'
d!
bc ad! c+d J
, defined above!
The reader will easily see that the product as
equality by the is even simpler
of the point on the right-hand side of the last
appoint c,d! is indeed equal to thepointa,6!.It
ht-hand
side of
to verify that the sum of the point on the rig
d equal
to the
the first equality and the point c, d! is indee point a, 6!. By applying our definitions to the points on t. e. to the points of the type a, 0!, we obtain.
fie abscissa axis,
a, 0! + ,
0! = a + 6, 0!
a, 0! b, 0! = ab, 0! duce to addition d for subtraction
i. e. addition and multiplication of these points rei and multiplication of their abscissas.The same is vali and division:
a, 0!
b, 0! = a ,
i axis
re resents
If we assume that each
oint
0! a0
6, 0!
6' of the abscissr
in IIIII~ 'll I II IIIIIIIIIlIIIIIIIJIIII the point ,
1!
i. Let us denote
with real numbers. For example, let us consider
which lies at a distance1 upwardof the point C this point by the letter i: i= ,
plication of the
1!
and let us find its square in the senseof multi points on the plane:
axis, not on the 1, i. e.
However, the point
1, 0! lies on the abscissa
ordinate axis, and thus re'presents a real nuinber i~ =
1
>is root is given
Hence, we have found in our new system of m whose square is equal to a real number 1, i. find the square root of 1. Another value of tl
nt ,
by the point
imbers a number e. we can now
1!, which
i the plane, and .maginary unity" ing on the plane. tructed
is more
:d the system of ith the operations s not
difficult
to
by real numbers For example, let of addition, the
i= ,
1!.
Note that the poi
we denoted as i, is a precisely defined point or. the fact that it is usually referred to as an "I does not in the least prevent it from actually exist The system of numbers we have just cons extensive
than that
of real numbers
and is call
complex ppmbers. Thepoints ontheplane togethe w
we have defined are called complex numbers. It i prove that any complex number can be expressed and the nuinber i by means of these operations:
us take point a, b!. By virtue of the definition following is valid:
a, b! = a, 0!+ , therefore a real. ation, the second
The addend a, 0! lies on the abscissaaxis and is number a. By virtue of the definition of rnultiplic; addend can be written
in the form
,
1 <], numbers we can
>ns with complex
b!
'I
b! = b, 0!,
i 1 1] I ~
ll
I!
~ ~
By means of this andard notation of complex immediately rewrite t e above formulas for operatii '.numbers:
a + bi! + c + di! = a + c! + b + 4 c!i
a+ bi! c+ di! = ac bd! + ad + $ a
bi
c+
i'I =
c!+
n of multiplication nent with the law :cond of the above rule of binomial
It should
be noted
that the above definitio
of the points of the plane is in perfect agreer of distribution:
if on the left-hand
side of the @
w of distribution!,
equations we calculate the product by the multiplication which itself stems from the la
luce similar terms,
and then apply the equality i~ =
de of
we shall arrive precisely at the right-hand equation.
the second
I and red
sI
2. Evolution.
Quadratic Equations can extract square
any negative real ' a is a negative
Having complex numbers at our disposal, we roots not only of the number l, but of number, always obtaming two distinct values. II real number, i, e. a > 0, then
V>are root
of
the
:quation with real 0, this equation complex numbers,
b is taken in both
any a and b both :oefficient in i will possesses two values
=+v
wherePa is the positivevalueof the sq
positive number a. Returning to the solution of the quadratic i
coefficients,we can now say that where b2
4ac
also has two distinct roots, this time complex. Now we are able to take square roots of any
rim '~immiIia
s nlI
wherethe positivevalueof the radical Pa +
terms. Of course, the reader will see that for; the first term on the right-hand side and the c
is added
be real numbers.Each of these two radicals pc which are combined with each other accordin rule: if b > 0, then the positive value of one
:gative one to the
to the positive value of the' other, and the ne
~ to the following radical
ar equation are the numbers : 3+i,
xg
1
2i
d that
each
of
these
Therefore, the roots of oi xg =
numbers
indeed
the problem of extracting roots of an index n from complex numbers. It can :omplex number m there exist exactly n
> such that raised to the power n i. e. n factors equal to this number!, each other words, the following I
extremely
It can easily be checke satisfies the equation. I.et
to
theorem
holds.
A root of order n distinct
y applicable to real numbers, which are tplex numbers: the nth root of a real distinct values which in a general case that among these values 'there will be ers, depending on the sign of the number
turn
arbitrary positive integral be proved that for any < distinct complex number< if we take a product of yields the number a. In itn ortant
>f any complex number has exactly n
< n. f one
us now
cont lex values.
This t eorem is equaH
a particular case of con number a has precisely n are complex. We know two, one or no real numb
a and parity of the inde> has three values:
l and 22
i
Thus, the cube root
y~
1,
2
the formula for taking the square root + bi, This
formula
reduces
calculation
quare roots of two positive real numbers. no formula exists which would express
1 2
c
+
In section 1 we gave
of a complex number a of the root to extr'actinge @Unfortunately, for n o 2
bi in tertns of real values
the nth root of a compie
iliary real numbers; it was proved that be:derived. Roots of order n of cotnplex
of radicals of certain aux no such formula can ever
x number
a+
Cubic Equations
g3.
ig quadratic equations is also valid for iird-degree equations, usually called cubic erive a formula, which, although more h radicals the roots of these equations 'his formula is also valid for equations
for solviti
complex coefficients. For th equations, we can also d complicated, expresses wit in terms
of coefficients.
'I
with arbitrary complex co Let an equation
efficients.
- ax
The formula
+ bx + c = 0
x
iis equation, setting
be given. We transform tl
a
x=y
3
a. Substituting this expression of x into cubic equation with respect to y, which
cient of y~ will be zero. The coefficient l the absolute term will be, respectively,
where y is a new unknowi our equation, we obtain a is simpler, since the coeffi of the lirst power of y ani the numbers
tb> written '+
il=
2a 27
ab 3
+c i. e. the equation can be
as + =0
+ p + ' 27+ the three cube radicals
has three values.
iot be combined in an arbitrary manner.
We know that each of
However, these values cani
se two values of the radicals
the number
must be added
root of the equation. Thus we obtain the uation. Therefore, each cubic equation with has three roots, which in a general case are mmeof these roots may coincide, i, e, constitute
P 3 The
together to obtain a three roots of our eq numerical
coefficients
complex; obviously sc a multiple root, The practical sigr small. Indeed, let th<
iificance of the above formula is extremely : coefficients p and q be real numbers. It the equation y +py+
.
can be shown that if
a=0
I roots, then the expression
has three distinct
rea
+ 4 27
:e the expression is under the square root extracting this root will yield a complex !f the two cube root signs. We mentioned of cube roots of complex numbers requires i, but this can be done only approxiinately,
will be negative. Sini sign in the formula,
number under each < above that extraction
trigonometric notatioi by means of tables. Example. The equ
ation x
itive.
19x+
30 = 0
The first
of the cube radicals
i. e. the result is neg: formula yields
in the
3
3
P 27 I' IINSIS
this formula.
But a direct
verification
are the integers 2, 3 and
demonstrates
that these roots
S.
In practice the above formula for solving cubic equationsyields the roots of equations only when the expression +
4 27
is
po-
sitive or equal to zero. In the first instance the equation has one real and two complex roots; in the second instance aH the roots are real but one of thetn is multiple. Kxatnple. We want to solve the cubic equation x'
9x'
+ 36x
80
=0
Setting x=y+3 we obtain the "reduced" equation
ys+ 9y 26 =0 Applying the formula, we obtain += 4 27
P
196 = 14'
and therefore 3
IIIIIIIIIIIIIEIIIITlfffmff ItffffffftffI value of the second radical will be the number the roots of the reduced equation is
yi =3+
1,
and one of
1! =2
Knowing one of the roots of the cubic equation, we can obtain the other two in many different ways. For instance, we
valuesof the secondradical and add up the mutually correspond.ing values of the radicals. Or we may divide the left-hand side of the reduced equation by y 2, after which we only need to solve a quadratic equation. Either of these methods will demonstrate that the other two roots of our reduced equation are the numbers
1+ <+12and 1 i+12 Therefore, the roots of the original cubic equation are the numbers
5,2+ i+12and2 i~12 Of course, calculation of radicals is not always as easy as in the carefully selected example discussed above; much more often they have to be calculated approximately, yielding only approximate values of the roots of an equation.
4. Solution of Equations in Terms
of Radicals
and the Existence
of Roots
of Equations Quartic equations also allow a formula to be worked out which expresses the roots of these equations in terms of their coefficients. Involving still more "multi-storied" radicals, this formula
I] ] l
]
]i ]
']
I] ] '1i] ] '
I i']t
ll
higherdegrees. Note that a general form of an equation of degree n, wheren is a positive integer, is aox" + a,x" '+ a,x"
+
+ a,x+
a=O
The searchcontinued unsuccessfully until the beginningof the 19th century, when the followin s ectacular result was
roved:
n addition, for any n, greater than or equal to five, an ition can be written of degree n with integral coefficients, se roots are not expressible in radicals, however complicated, e radicands involve only integral or fractional numbers. Such ie equation x'
4x
2
I equa who if th is tF
=0
t can be proved that this equation has five roots, three real two coinplex, but none of these roots can be expressed in :als, i. e. this equation is "unsolvable in terms of radicals-.
Ii
'he theory of algebraic numbersis an important branch of algebra;
and radi Ther form are are i mucl term 'I
ian mathematiciansE. I. Zolotarev 847-1878!, G.,F. Voronoi
Russ
!-1908!, N. G. Chebotarev 894-1947!
86l
efore, the store of numbers, both real and complex, which the roots of equationswith integral coefficients such numbers called algebraic as opposed to transcendent numbers which
not the roots of any equations with integral coefficients!,is i greater than that of the numbers which can be written in s of radicals.
inade valuable contribu-
in this field.
tions
,bel 802-1829! proved that deriving general formulas for ng equations of degrees n o 5 in terms of radicals was
solvi
issible. Galois 811-1832! demonstrated the existence of
impc
tions with integral coefficients, unsolvable in terms of radicals. ilso found the conditions under which the equation can be
equa He
shall only mention that presently Soviet mathematicians are :ering in the development of group theory.
but,
pion~
.s far as practical determination of roots of equations is erned,the absenceof formulas for solving nth degreeequations e n!
5 causes
no serious difficulties.
Numerous
methods
of
oximatesolution of equationssuffice,and evenfor cubic equations , methods are much auicker
than th
li
'o
ornn%q
~ ~~ i
conc wher appr these
real tore s 1s any !lds: oots> cide, .bra.
855! cted ,sent rem, gree
If there were equations with numerical coefficients, either or complex, which possessed no real or complex roots, the s of real numbers would have to be extended. However, thi unnecessary since cotnplex numbers are sufhcient to solve equation with numerical coefficients. The following theorem h< Any equation of degreen with any numerical coefficients has n r complex or, in certain cases, real; some of these roots may coin i. e. form multiple roots.
This theorem is called the basic theorem of higher alg~
It was proved by D'Alembert 717-1783! and Gauss 777-1 as early as the 18th century, although these proofs were perfe to complete rigorousness only in the 19th century; at pre there exist several dozen dilferent proofs of this theorem. The concept of a multiple root, mentioned in the basic theo means the following. It can be proved that if an nth de equation
aox" + a1x" '+ ition
has n roots txnz,
+a1x+
a=0
..., |x, then the left-hand side of the equ;
can be factored in the following manner:
.Ja,x" + a,x" '+ side oots
+ a= ao x m,! x mq! ... x 1
Conversely, if such a factorization of our e uation
/5. ions for ems :nts.
+ a,x
the numbers
is given for the left-hand
t«c«
...
will
be the
r
The Number of Real Roots
The basic theorem of higher algebra has important applicat
in theoretical research, but it provides no practical method solving the roots of equations. However, many technical prob] require information about the roots of equations with real coeffich
degree equation be given
uox" + a>x" ' +
+a
Let an nth
ix + a= 0
efficients, We already know that it has n roots. m real roots? If so, how many and approximately located? We can answer these questions as follows. he polynoinial on the left-hand side of our equation
having real cc Are any of the where are they Let us denote t
by f x!, i. e. x! = aox" + a,x"
++
a,x+
a
iiliar with the concept of function will understand he left-hand side of the equation as a function of Taking for x an arbitrary numerical value m and
The reader fan that we treat t the variable x.
into the expressionfor f x!, after performing all the
substituting it i operations, we of the polynon
arrive at a certain
iial f x!
number
which is called the
value
and is denoted as f cx!. Thus, if
f x! = x' Sx'
+ 2x + 1 and m=2,
f!
=2
5
2 + 2. 2+ 1 = 7
: a graph of the polynomial f x!, To do this we ordinateaxes on the plane see above! and, having i value
the
Ix and calculated
s I I I I I]I I
a corres
II I I I I I I I » ]IqlI I I I I i
ondin
value
m
Let us plol choose the coc selected for x
I ]II I I I I I
question. :ly, since there are an infinite number of the values
>t hope to lind the points n, f u!! for all of them satisfied with a finite number of points. For the ity we can first select several positive and negative of »x in succession, mark on the plane the points to them and then draw through them as smooth
Unfortunat» of »x one cann» and must be:
sake of simplic integral values corresponding
and [a~ =
a for a < 0! and A is the greatest of the absolute
values of all the other coefficients a,, a2, B=
A
I ~0I
acr,
then
+1
However,it is often apparent that these bounds are too wide. Example. Plot a graph of the polynomial
f x! = x
5x
m 2x + 1
Here ao! = 1, A = 5, and thus B = 6. Actually, for this particular example we can restrict ourselves to only those values of m, which fall between 1 and 5. Let us compile a table of values of the
polynomial f x! and plot a graph Fig. 3!.
The graph demonstrates that all the three roots a,, e, and a3
i!
li 1I
i!
1 I!
1
I
neighbouringvaluesof a for which the numbersf m! have opposite signs, and thus it was sufficient just to look
ht the table of
values of f a!. If in our example we found less than three points of intersection
of the graph with the abscissaaxis, we might think that owing to the imperfection of our graph we traced the curve knowing only seven of its points!, we could overlook several additional
cated between any given
equation and even the number of roots lo
thods will
numbers here.
not be stated
a and b, where a
b, These me
Sometimes the following theorems are useful since they give some information
on the existence
of real
and even positive roots. Any equation of an odd degree with real coefftcients has at least one real root.
If the leading. coefftcient ao and the absolute term a in an equation with real coefftcients have opposite signs, the equation has at least one positive root. In addition, if our equation is of an even degree,it also has at least one negative root.
Thus, the equation x'
8x
+x
2=0
has at least one positive root, while .the equation -x +2x'
se neighbouring integers ,tion
x
1=0
In the previous section we found tho, between which the real roots of the equa
I0
e roots of this equation :xample, let us take the
+7x
x are located.
The
5x
same method
+2x+ allows
I= th
to be found with greater accuracy. For < If1 4
sive values >scissa axis, f one-tenth. root u2 to
0.9, we can find between which two of these succes
of x the graph of the polynomial f x! intersectsthe af
methods of :ions much :thods and of the Ful to find
i. e. we can now calculate the root u2 to the accuracy o Proceeding further, we can find the value of the the accuracy of one-hundredth, one-thousandth or, tl to any accuracy we want. However, this approac cumbersome calculations which soon becoine practica ageable. This has led to the development of various . calculating approximate values of real roots of equai quicker. Below we present the simplest of these mi immediately apply it to the calculation of the root cubic equation considered above. But first it is use
>ady know,
bounds
oot to the lues of the
0 < m, < 1. For this purpose we shall calculate our r accuracy of one-tenth. If the reader calculates the val polynomial
ieoretically, h involves
lly unman-
for this root
narrower
f x! = x for x = 0.1; 0.2;
'0.9, he will
f.7! re difFerent,
than
the ones we aln
5x + 2x + 1 obtain
= 0.293, f.8!
=
0.088
and therefore,since the signs of these values of f x! a: 0.7 < >x, < 0.8
n is given,
The method is as follows. An equation of degree
The bound c is calculated by means of the formul,
bf a! af ! f a! - f b! i! and f b!
In this case a =0.7, b = 0.8, and the values of f i are given above. Therefore
iction of a new
The formula for the bound d requires the introdv
:re; in essence
concept which will play only an auxiliary role h<
lied differential
it belongs to a dillerent
branch of mathematics cai
calculus.
Let a polynomial of degree n be given zn-sx+
f x! = aox"+a,x" '+ azx" + +
an
azx
Let us call the polynomial of degree n f'4!
as f'{x!.
This
ule: each term
the exponent nity; moreover,
that a= axo. ynomial f ' x!. h is
called
the
voted as f" x!. +2x+
1 we
= nnvx"
' + in 1! a,x" + 2a-zx+
a derivative of this polynomial,
+i
1!
+ [n 21 azx' a -z and denote it
polynomial is derived from j x! by the following r ax" ' of the polynomial f x! is multiplied by
n k of x, while the exponent itself is reduced by u: the absolute term a, disappears, since we can consider We can again take the derivative of the pol
This will be a polynomial of degree n 2!, whic secondderivative of the polynomial f x! and is de> Thus, for the above polynomial f x! = x' 5x obtain
x!
3xz
10x + 2
f" x! = 6x 10
close to each
lIIIIJlllRllllll,llJill lllTIIB I IIllI chosen. If the bounds a, b are chosen sufftciently
I f b! will be 1 f a! are the
other, the secondderivativef" x! will usually have for x = a and x = b, while the signs of f a! am different, as we know, If the signs of f" a! am
i, i. e. the one
same, d must
signs of f" b!
in which the bound a is
bound
andf b! coincide,the secondformula, involving the
the same sign
b, must
be calculated
with
the first
formula
used, however if the
second formula
for
the bound
d must
and f b! negative, the ~ used. Sincef'.8! = <
be
l,08, we obtain
0.088= 0.80.0215...= 0.7784 ... x2 we have found the following : knew
7769
d =0.8
bounds,
Thus narrower
before:
for the root than those wi
< m~ < 0.7784...
ounds somewhat,
or, if we widen
these b
i.7769 < x~ < 0.7785 it if we take for ei~ the arithmetic mean, e calculated u=
It follows, therefore, th; i. e. half the sum, of th
bounds,
0.7777
ed 0.0008, equal to half the difference of
the error will these bounds.
iracy is insuAicient, we could once again >d to the new bounds of the root ~x2. quire much more complicated calculations. >proximate solution of equations are more
If the resulting acci apply the above meth<
III ' III
IIII
III
its of algebraic equations, which we have ove, can be considered in more general ust introduce one of the most important
However,
not exce
this would
re
Other methods of al
IIII
The problem of roc already encountered ab terms. To do so we m
concepts of algebra. ' the following three. systems of numbers: iumbers, the set of all real numbers, and numbers. Without leaving their respective I
Let us first consider the set of all rational t
the set of all complex h
the number
2 cannot
is not always possible for example, ' divided by 5 without a remainder!, as
be
well as from the systetn :tion is not always possible. arming algebraic operations dtiplication of polynomials,
of all positive real numbers, where subtra The reader is already familiar with perfi not on nuinbers such as addition and mt and also addition of forces encountere
d in physics. Incidentally, d to consider addition
in defining complex numbers we also ha multiplication of points of the plane. In general terms, let a set P be l numbers, of geometrical objects, or of so
and
driven, consisting either of me arbitrary objects which I'he operations of addition for each pair of elements ent c from P is indicated,
we shall call the eiements of the set P. '
and multiplication are defined in P if a, b from P one precisely defined clem and called their
sum: c=a+b
and a precisely defined element d froi
n P. called their product:
d =ah
The set P with the operations of additio within it is called a field, if these opera five properties:
n and multiplication defined
:ions possessthe following e, i. e. for any a and
I. Both operations are commutativ
b
=ha
a+b=b+a,
II. Both operations are associative,
i. e. for any a, b and c
Ik
IIII
i. e. for any a and
b
ab
III
I IV. Subtraction
can
Ill
be carried
out
a unique root of the equation a+x=b
I'or any a and b, provided
can be found in P. V Division can be carried
if the equation
a does not equal zero, a unique root c
out, i. e. I
ts existence can be derived
from
Condition
V mentions
zero. I
an arbitrary element of P, then
Conditions I-IV. Indeed, if a is
nite
because of Condition
element
exists
in P which
IV
a defi
satisfies the equation a+»
s element may depend upon the enate it by 0 i. e.
a itself is taken for b!. Since thi
choice of the element a, we desi~ a+0,
hen again there exists one such ,=b
If b is any other element of P, t unique element Ob for which b+0
!
iy a and b, then the existence h plays the role of zero for all ie, will be immediately proved.
If we prove that 0, = Ob for ar in the set P of an element, whic the elements a at the saine tin
Let c be the root of the equi
ition
a+x x IV;
which
hence,
exists because of Conditioi a+c
TIt
illllllTIIII!IIIIll Tl!
I =b
b+0,
d remembering that according to < of the equation b + x = b, we
«II I
II
I 15
««
Comparing this to equation !
an
IV there exists only one solutior linally reach the equality Sl,",hWÃ~'~WIPbWSPl~~ N'i... " "" "" "" "ii" "" ""ii'i" "" "" ":i':i':i':i':i':i'«" "«":i' "«" «":" i'"."i'i«" iwiii i'iNi«" i'«" i'ij!i «
ow proves that in any field P there is a zero element, an element 0 that for all a in P the equality
This n i. e. such
a+0=a
therefore Condition V becomes completely meaningful. :ady have three examples of fields the field of rational
holds and We air
that of real numbers, and that of complex numbers,
numbers,
sets of all integers and of positive real numbers do not fields. Besides these three, an infinite number of other . For instance, many different fields are contained within af real numbers and of complex numbers; these are the iumerical tlelds. In addition some fields are larger than nplex numbers. The elements of these fields are no longer hers,but the fields I'ormed by them are used in mathematical Sere is one example of such a field. consider all possible polynomials
while the, constitute fields exist the fields i so-called i that of cor called num research. I Let us
f x! = aox" + a,x" ' +
+ a,x
+a
rary coinplex coefticients and of arbitrary degrees; for :ero-degree polynonuals will be represented by complex
themselves. Even if we add, subtract and multiply ls with complex coefficients by the rules we already know, ll not obtain a field, since division of a polynomial by olynomial with no remainder is not always possible. us consider ratios of polynomials
with arbit instance, z numbers
polynomia we still wi
another pi Now
let
f x!
if an
f ! 0 ! =e ! q ! Then,
f x! il x!
+
u x! u x!
f x!u x! + g x!u{x! 0 x!v x!
on
:tions whose numerator is equal
The role of zero is played by fra< to ~ero, i. e. fractions of the type
0
x!
a
ype are equal to one another.
Obviously, all fractions of this <
s not equal zero, i. e. u x! g 0,
Finally, if a fraction
u x! Ux
doe
then
f x! v x! g x! « x!
f x! g x!
ie above operations with rational
u x! u x!
' rational functions with complex numbers is totally contained in
It can easily be checked that tt functions satisfy all the requiremi so that we can speak of a /iield o~ coefficients. The field of complex
hose numerator
this field, since a rational
:nts of the definition
of a field;
and denominator
iply a complex number, and any 9 in this form. i< field is either contained in the ains it within itself: some of the inite number of elements.
have to consider equations with inevitably the existence of roots i,m. Thus, in some problems of
function w
are zero-degree polynomials is sin complex number can be presente One should
not think
that an~
field of complex numbers or conl de'erent fields consist only of a l Whenever fields are used, we coefficients from these fields, and
of such equations poses a probe
is with c ellicien
...,,,I...<...,,i,,',,j,,l... + a,x+
a=
aox" + aix"
0
+
ither greater field. At the same d to a field Q in which our
be an equation of degree n witl turns out that this equation ca either in the field P or in any < time the field P can be enlarge
>f which may be multiple!. Even
equation will have n roots some <
i coefficients from this nnot have more than
field. It n roots
w'
regi,hol<js
This field P is called algebraically ciosed. The basic theorem of higher algebra shows that the field of complex numbers belongs to the set of algebraically closed fields.
S. Conclusion
Throughout this booklet we always discussed equations of a
certain degreewith one variable. The study of first-degreeequations is followed by that of quadratic equations in elementary algebra. In addition elementary algebra proceeds from a study of one
first-degreeequation with one variable to a systemof two first-degree equations with two variables and a system of three equations with three variables. A university course in higher algebra continues these trends and teaches the methods for solving any system of n first-degree equations with n variables, and also the methods of solving such systems of first-degree equations in which the nuinber of equations is not equal to the number of variables. The theory of systems of lirst-degree equations and some related theories including the theory of matrices, constitute one special branch of algebra, viz. !ines algebra, which is widely used in geometry and other areas of mathematics, as well as in physics and theoretical mechanics.
It must be remembered, that at present both the theory of
is a set with operations of addition and multiplication, in which Conditions I-IV from the definition of a field are valid; the set of all integers may be cited as an example. We already mentioned another very significant branch of algebra, the group theory. A group is a set with one algebraic operation, multiplication, which must be associative;
division
must be carried
out without
restrictions.
We often encounter noncommutarii:ealgebraic operations, i. e. ones
ematic presentation of the fundamentals of the theory of .c equations and of linear algebra can be found in textbooks ter algebra. The following textbooks are most frequently iended:
i. Kurosh, Higher Algebra,Mir Publishers,1975 in English!. t'a. Okunev, Higher Algebra, "Prosveshchenie", 1966 in !.
elementary presentation of the simplest properties of rings ds, mostly numerical, can be found in; Proskuryakov, Numbers and Polynotniais, "Prosveshchenie", i Russian!. acquaintance with group theory may begin with: , Aleksandrov, Introduction to Group Theory, "Uchpedgiz". i Russian!.
Syst algebrai on higl recomrr A,C
1
Russian An and fiel I. V.
1965 ir
An, P S
1951 ir.
1U 1Ht: Kt ADt:K
Mir Publishers would be grateful for your comments on the content, tnslation and design of this book. We would also be pleased to :eive any other suggestions you may wish to make. Our addres's is: gSR, 129820, Moscow 1-110, GSP rvy Rizhsky Pereulok, 2 ir Publishers
trt re
U,
Pe M
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Contents. Proofs of Identities; Problems of an Arithmetical
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