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Chapter VII
On imaginary expressions and their moduli.
_______
§ I – General considerations on imaginary expressions
In analysis, we call a symbolic expression or symbol any combination of algebraic
signs that do not mean anything by themselves or to which we attribute a value different
from that which they naturally have. Likewise, we call symbolic equations all those that,
taking the letters and the interpretations according to the generally established
conventions, are inexact or do not make sense, but from which we can deduce exact
results in modifying and altering them according to fixed rules either the equations
themselves or the symbols which comprise them. The use of symbolic expressions or
equations is often a means of simplifying calculations and of writing in a short form
results that appear quite complicated. We have already seen this in the second paragraph
of the third chapter where formula (9) gives a very simple symbolic value to the unknown
x satisfying the equations (4). Among the symbolic expressions or equations the
consideration of which is of some importance in analysis, we should especially
distinguish those which we call imaginary. We are going to show how we can put them
to good use.
We know that the sine and the cosine of the arc a + b are given as functions of the
sines and cosines of the arcs a and b by the formulas
(1)

cos a  b   cos a cosb  sin asinb


sin a  b   sin a cosb  sinb cos a
/154
But, without taking the pain to remember these formulas, we have a very simple means of
recovering them at will. It suffices, in fact, to consider the following remark.
Suppose that we multiply together the two symbolic expressions
cos a  1sin a,
cosb  1sinb,
Cours d’analyse – Chap. VII: Imaginary expressions
1
by applying the known rules of algebraic multiplication and knowing that -1 is a real
[?? Actual?] quantity the square of which is equal to -1. The resulting product is
composed of two parts. The one real and the other has a factor -1. The real part gives
the value of cos(a + b) and the coefficient of -1 gives the value of sin(a + b). To
observe this remark, we write the formula
(2)


cos a  b   1sin a  b  cosa  1sina cosb  1sinb

The three expressions that make up the preceding equation, namely
cos a + -1 sin a,
cos b + -1 sin b,
cos(a + b) + -1 sin(a + b),
are three symbolic expressions that cannot be interpreted using the generally established
conventions, and they do not represent anything real. For this reason, they are called
imaginary expressions. The equation (2) itself, taken literally, inexact and it does not
make sense. To get exact results, first we must expand its second part by algebraic
multiplication, and this reduces the expression to
(3)
cos a  b  1sin a  b  cosacosb  sinasinb  1 sinacosb  sinbcosa
Secondly, we must equate the real part of the first member (left hand side) of equation (3)
with the real part of the right hand side, then the coefficient of -1 on the left hand side
with the coefficient of -1 on the right. Thus we recover the equations (1) both of which
we ought to consider as implicitly contained in the formula (2)
In general, we call an imaginary expression any symbolic expression of the form
 +  -1,
where  and  denote real quantities. We say that two expressions
 +  -1 and  + -1
are equal to each other when there is equality between corresponding parts, that is
between the real parts  and  , and between the coefficients of -1, namely  and . We
indicate the equality between two imaginary expression in the same way that we it
between two real quantities, by the symbol =, and this results in what we call an
imaginary equation. This said, any imaginary expression is just the symbolic
representation of two equations between real quantities. For example, the symbolic
equation
 +  -1 =  + -1
Cours d’analyse – Chap. VII: Imaginary expressions
2
is just equivalent to the two real equations
 =  and  = .
When, in the imaginary expression  +  -1, the coefficient  of -1 vanishes,
the term  -1 supposed to reduce to zero, and the expression itself reduces to the real
quantity  . By virtue of this convention, imaginary expressions include, as special
cases, real quantities.
Imaginary expressions and those involving real quantities can be supposed to
have different algebraic operations. In particular, if we do addition, subtraction, or
multiplication /156 of two imaginary expressions, they operate according to the
established rules for real quantities, and we will obtain as a result a new imaginary
expression that we call the sum, the difference or the product of the given expressions,
and the ordinary notations will serve to indicate that sum, difference or product. For
example, if we are given two imaginary expressions,
 +  -1 and  + -1
we will find
(4)
(5)
(6)
   1   
   1   
   1   

1         1 ,
1        
1          1 ,
1 .
It is good to remark that the product of two or more imaginary expressions, like that of
two or more real binomials, remains the same regardless of the order in which we
multiply the different factors.
To divide a first imaginary expression by a second means to find an imaginary
expression which, when multiplied by the second, reproduces the first. The result of this
operation is the quotient of the two given expressions. To indicate this, we use the
ordinary symbol for division. So, for example,
   1
   1
represents the quotient of the two imaginary expressions
 +  -1 and  + -1
Cours d’analyse – Chap. VII: Imaginary expressions
3
To raise an imaginary expression to the power m (where m denotes a integer) means to
form the product of m factors equal to that expression. We write the mth power of
 +  -1 with the notation
  
1
.
m
/158
In the particular case when  is reduced to zero, one or two of these powers can become
real. [What does this mean?]
To raise the imaginary expression  +  -1 to a negative power, – m, or –1/n or
–m/n is to divide 1 by the power m or 1/n or m/n of the same expression. The problem
will have a unique solution in the first case, and several solutions in the two others. We
denote the power of degree – m with the simple notation
  
1

m
while the two notations
   1 ,
   1

1
n

m
n
represent, in the first case, any of the powers of degree =1/n, and in the second case, any
of the powers of degree – m/n.
We say that two imaginary expressions are conjugate to each other when the two
expressions differ only in the signs of the coefficient of -1. The sum of two such
expressions is always real, as is their product. Indeed, the two conjugate imaginary
expressions
 +  -1,  -  -1
have as their sum 2 and as their product  2   2 . The final part of this observation
contained in a theorem about numbers, expressed as
Theorem I: If we multiply together two whole numbers that are each the sum of two
squares, then the product will always be a sum of two squares.
Proof: Let the numbers be 2 + 2 and ’2 + ’2 /159 where 2, 2, ’2, ’2 denote the
perfect squares. We obviously have the two equations
  


1  '  ' 1   '  '  '  '   1
Cours d’analyse – Chap. VII: Imaginary expressions
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   1 '  '

1   '  '  '  '   1
and, by multiplying these, term by term, we will obtain
(7)

2


  2 a'2   '2   '  '   '  '   .
2
2
If we interchange the letters ’ and ’ in this last expression, we get
(8)

2


  2 a'2   '2   '  '     '  ' .
2
2
Thus, in general we have two ways to decompose into two squares the product of
two whole numbers each of which is the sum of two squares. Thus, for example, one
draws from equations (7) and (8)
2
2


 1 32  22  4 2  72  12  8 2 .
We see from these examples that the use of imaginary expressions can be of great use,
not only in ordinary algebra but also in number theory.
Sometimes we represent an imaginary expression by a single letter. It is an
artifice which helps us use the resources of analysis and we will make use of it in what
follows.
159
§ II – On the moduli of imaginary expressions and on reduced expressions
A remarkable property of any imaginary expression
 +  -1
is that it can be put into the form


 cos  1sin ,
/160
 denoting a positive quantity and  a real arc. Indeed, if we write the symbolic equation

   1   cos  1sin

which amounts to the same as the two real equations
   cos 

   sin 
Cours d’analyse – Chap. VII: Imaginary expressions
5
and we find that
 2   2   2 cos2   sin2    2
  2  2
(3)
and, having thus determined the value of the number , all that remains to verify
completely the equations (2) is to find an arc  such that its cosine and sine are
respectively


cos 
2
  2



sin  

2  2

(4)
This last problem is always solvable because each of the quantities

  2
2

  2
2
and
have numerical values less than one and the sum of their squares is equal to 1.
Also, it has infinitely many different solutions because, having calculated one convenient
value of the arc , we can, without changing the value of the sine or the cosine, increase
or decrease this arc by any number of circumferences.
When the imaginary expression  +  -1 is put into the form


 cos  1sin ,
the positive quantity  is called the modulus of this imaginary expression, and what
remains after the suppression of the module, that is /161 to say the factor
cos  1sin
we will call the reduced expression. Since we take the quantities  and  to be known
quantities, we get only one unique value for the modulus  as determined by equation (3),
and as a result the modulus remains the same for any two imaginary expressions that are
equal. Thus we can state the following theorem:
Theorem I: The equality of two imaginary expressions always implies the equality of
their moduli, and as a consequence, the equality of their reduced expressions.
Cours d’analyse – Chap. VII: Imaginary expressions
6
If we compare two conjugate imaginary expressions, we find that their moduli are
equal. The square of their common modulus is just their product.
When in the imaginary expression  +  -1, the second term  vanishes, the
expression reduces to a real quantity . In this case, we get from equations (3) and (4),
first, when  is positive,
  2
cos  = 1, sin  = 0
and so
  2k ,
where k denotes any whole number; or second, when  is negative,
  2
cos  = – 1, sin  = 0
and so
   2k  1 .
Thus the modulus of a real quantity  is just its numeric value  2 , and the reduced
expression that corresponds to such a quantity is always +1 or – 1, since
1  cos 2k   1sin 2k 
/162
which always gives a positive quantity, and
1  cos  2k  1  1sin  2k  1 ,1
which gives a negative quantity.
Any imaginary expression that has modulus zero itself reduces to zero, since its
two terms both must vanish. Conversely, since the cosine and the sine of an arc are never
zero at the same time, it follows that an imaginary expression cannot be reduced to zero
unless its modulus vanishes.
Any imaginary expression which has one as its modulus is necessarily a reduced
expression. Thus, for example,
1
Here, rather than nesting parentheses, Cauchy uses the old notation



1  cos 2k  1  1sin 2k  1

Cours d’analyse – Chap. VII: Imaginary expressions
7
cosa  1sina ,
cosa  1sina ,
cosa  1sina ,
cosa  1sina
are four reduced expressions, two conjugate pairs. Indeed, to derive these four
expressions from the formula
cos  1sin
it is enough to take successively
  2k  a ,
   2k  1  a
  2k  a ,
   2k  1  a ,
where k denotes any whole number.
Calculations involving imaginary expressions can be simplified by using reduced
expressions. It is important to take note of their properties. These properties are
contained in the theorems that I am about to state.
Theorem II: To multiply two reduced expressions
cos  1sin and cos ' 1sin ' ,
it suffices to add the arcs  and ’ to which they correspond.
/163
Proof: Indeed, we have
(5)
cos 


1sin cos ' 1sin '  cos    '  1sin    '.
Corollary: If in the previous theorem we take ’ = – , we find, as expected,
(6)
cos 


1sin cos  1sin  1 .
Theorem III: To multiply together several reduced expressions,
cos  1sin , cos ' 1sin ' , cos '' 1sin '' , …,
it suffices to add the arcs , ’, ’’, … to which they correspond.
Proof: Indeed, we get successively,
[block of formulas]
Cours d’analyse – Chap. VII: Imaginary expressions
8
and, continuing in the same way, we find generally that whatever the number of arcs, ,
’, ’’, … may be,
cos 
(7)



1sin cos ' 1sin ' cos '' 1sin '' L
 cos    '  '' L   1sin    '  '' L .
Corollary: If we expand by ordinary multiplication the left hand side of equation (7), the
expansion will consist of two parts, one real and the other having a factor -1. This said,
the real part will take on the value
cos ( + ’ + ’’ + …),
/164
and the coefficient of -1 in the second part will have the value
sin ( + ’ + ’’ + …).
Let us suppose, for example, that we are considering only three arcs, , ’ and ’’. Then
equation (7) becomes
cos 


1sin cos ' 1sin ' cos '' 1sin ''

 cos    '  ''  1sin    '  '',
and, after expanding the left hand side of this last equation by algebraic multiplication,
we conclude that
cos    '  ''  cos cos 'cos '' cos sin  'sin  ''
  sin  cos 'sin  '' sin  sin  'cos ''
sin    '  ''  sin  cos 'cos '' cos sin  'cos ''
  cos cos 'sin  '' sin  sin  'sin  ''
Theorem IV: To divide the reduced expression
cos  1sin
by another one
cos ' 1sin ' ,
it suffices to subtract the arc ’ corresponding to the second expression from the arc 
that corresponds to the first.
Proof: Let x be the quotient we are seeking, so that
Cours d’analyse – Chap. VII: Imaginary expressions
9
x
cos  1sin
.
cos ' 1sin '
This quotient ought to be a new imaginary expression chosen so that when it is multiplied
by cos ' 1sin ' it makes cos  1sin . In other words, x ought to satisfy the
equation
cos '

1sin ' x  cos  1sin .
To solve this equation for x, it works to multiply both sides by
cos ' 1sin ' .
/165
In this way we reduce the coefficient of x to unity (see Theorem II, Corollary I), and we
find that

 cos 

1sin   cos  ' 
x  cos  1sin  cos ' 1sin  '

1sin  '
 cos    '  1sin    '.
Finally we will get
(8)
cos  1sin
 cos    '  1sin    '.
cos ' 1sin '
Corollary: If in equation (8) we take  = 0, it gives
(9)
1
 cos ' 1sin '.
cos ' 1sin '
Theorem V: To raise the imaginary expression
cos  1sin
to the power m (where m denotes any whole number), it suffices to multiply the arc  in
this expression by m.
Proof: Indeed, the arcs , ’, ’’, … can be taken as in formula (7), and if we suppose
that they are all equal to , and that there are m of them, we find
Cours d’analyse – Chap. VII: Imaginary expressions
10
cos 
(10)
1sin 
  cos m 
m
1sin m .
Corollary: If in equation (10) we take successively  = z and then  = – z, we get the
following two equations:


 cos z  1sin z


 cos z  1sin z

(11)
  cos mz 
  cos mz 
m
1sin mz
m
1sin mz.
The left hand sides of each of these is always the product of m equal factors, can be
expanded by multiplication [what is “immediate multiplication” vs “algebraically”?] of
these factors, or, what amounts to the same thing, by the /166 formula of Newton.2 If,
after expanding the equation, if we equate corresponding parts of each equation, first the
real parts and second the coefficients of -1, we will conclude
(12)
m  m  1 m  2

m
cos z sin 2 z
cos mz  coz z 
1 2

m  m  1m  2 m  3 m  4



cos z sin 4 z  L

1 2  3  4

m
 sin mz  cos z m 1 sin z

1

m  m  1m  2  m  3


cos z sin 3 z  L
1 2  3

Taking m = 2, for example, we will find
cos 2z  cos 2 z  sin 2 z
sin 2z  2 sin z cos z
and supposing m = 23,
cos 3z  cos 3 z  3cos z sin 2 z
sin 3z  3cos 2 z sin z  sin 3 z,
etc.
Theorem VI: To raise the imaginary expression
cos  1sin
2
That is, Newton’s binomial formula.
Cours d’analyse – Chap. VII: Imaginary expressions
11
to the power – m, (where m denotes any whole number), it suffices to multiply in this
expression the arc  by the degree – m.
Proof: Indeed, from the definition we have given of negative powers (see § I), we
get
cos 
1sin 

m

cos 
1
1sin 

m

1
cos m  1sin m
Consequently, using formula (9), we get
(13)
cos 
1sin 

 cos m  1sin m

 cos m   1sin m  .
m
/167
or, what amounts to the same thing,
(14)
cos 
1sin 
m
After establishing, as we have just done, the principle properties of reduced
expressions, it becomes easy to multiply or divide one by the other of two or more
imaginary expressions, if we know their moduli, as well as to raise any imaginary
expression to a power m or – m, (where m denotes a whole number). Indeed, we can
easily perform these different operations with the aid of the following theorems:
Theorem VII: To obtain the product of two or more imaginary expressions, it suffices to
multiply the product of the reduced expressions to which they correspond by the product
of the moduli.
Proof: The stated theorem follows immediately from the principle that the
product of several factors, real or imaginary, remains the same regardless of the order in
which one multiplies them. Indeed, let
(cos  + -1 sin ), (cos  ‘+ -1 sin ’), (cos ’’ + -1 sin ’’), …
be several imaginary expressions, where , ’, ’’, … denote their moduli. When we
want to multiply these expressions together, where each expression is the product of a
modulus and a reduced expression, we can, by virtue of the principle just mentioned,
form as one part the product of the moduli, and as another part that of all the reduced
expressions, then multiply together these two products. We find in this way there results
the final expression
(15)
 '  ''L  cos    '  ''L   1sin    '  ''L  .
Cours d’analyse – Chap. VII: Imaginary expressions
12
Corollary I: The product of several imaginary expressions is a new imaginary expression
which has as its modulus the product of the moduli of all the others.
Corollary II: Since only the modulus of an imaginary expression can ever vanish, /168 in
order to make the product of several moduli vanish, it is necessary that one of them
reduces to zero, and it is clear that one may draw from theorem VII the following
conclusion:
Theorem VIII: To obtain the quotient of two imaginary expressions, it suffices to
multiply the quotient of their corresponding reduced expressions by the quotient of their
moduli.
Proof: Suppose that it is a question of dividing the imaginary expression
(cos  + -1 sin )
where the modulus is,  by the following
’(cos ’ + -1 sin ’)
where the modulus is ’. If we denote by x the desired quotient, then x must be a new
imaginary expression satisfying the equation
’(cos ’ + -1 sin ’) x = (cos  + -1 sin )
To solve this equation for the value of x, we will multiply both sides by the product of the
two factors
1
and cos ' 1sin '
'
and in this way we will find, writing
x
1

in place of  , that
'
'

 cos    '  1sin    ' .

' 
In the final analysis we will have
(16)

 ' cos '
   cos    ' 
1sin  '  ' 
 cos  1sin 
1sin    '
/169
and since, by virtue of theorem IV,
Cours d’analyse – Chap. VII: Imaginary expressions
13
cos    '  1sin    '
is exactly the quotient of the two reduced expressions
cos  + -1 sin  and cos ’ + -1 sin ’,
it is clear that, having established formula (16), we ought to consider proved theorem
VIII.
Corollary: If in equation (16) we take  = 0, it gives
(17)

1
 ' cos ' 1sin '
1


'
cos '

1sin ' .
Theorem IX: To obtain the mth power of an imaginary expression (where m denots any
whole number) it suffices to multiply the mth power of the corresponding reduced
expression by the mth power of the modulus.
Proof: Indeed, if in theorem VII we take the imaginary expressions
(cos  + -1 sin ),
(cos  ‘+ -1 sin ’),
(cos ’’ + -1 sin ’’), …
all to be equal to each other and to be m in number, their product will be equivalent to the
mth power of the first one, that is to say, equal to


m
  cos  1sin   ,


and, with this hypothesis, the expression becomes

 m cos m  1sin m
(18)


m



  cos  1sin    m cos m  1sin m .


/170
The reduced expression
cos m + -1 sin m
is equal to (by virtue of theorem V)
Cours d’analyse – Chap. VII: Imaginary expressions
14
cos 
1sin 
,
m
and, having established formula (10), as a result we ought to consider theorem IX to be
proved.
Theorem X: To raise an imaginary expression to the power – m (where m denotes a
whole number), it suffices to form the same powers of the modulus and of the reduced
expression, then multiplying the two parts together.
Proof: Suppose that it is a question of raising the following imaginary expression
to the power – m
(cos  + -1 sin )
where the modulus is,  by virtue of the definition of negative powers


  cos  1sin  


m



1

  cos  1sin  


1

m
 cos m  1sin m

.
Consequently, making use of formula (17),we find


  cos  1sin 


m

1
m
cos m 
1sin m

or, what amounts to the same thing,
(19)


  cos  1sin 


m

   m cos m  1sin m

This last formula, together with equation (13), furnish the complete proof of theorem X.
171
§ III – On real and imaginary roots of the two quantities + 1 and –1 and on their
fractional powers
Suppose that m and n denote two whole numbers relatively prime to each other.
If we use the notations adopted in § I, the nth roots of unity, or what amounts to the same
1
thing, the powers of degree
will have the different values of the expression
n
Cours d’analyse – Chap. VII: Imaginary expressions
15
1
n
1  1 ,
n
or, in the same way, the fractional positive or negative powers of unity of degree

m
or
n
m
are the various values of
n
m
n
1
or
1

m
n
.
We will conclude that, to determine these roots and powers, it suffices to solve, one after
another, the three following problems.
Problem I: To find the various real and imaginary values of the expression
1
n
1 .
Solution: Let x be one of these values, and in order to present it in a form that
includes the real quantities and the imaginary quantities at the same time, let us suppose
that

x  r cost  1sint

where r denotes a positive quantity and t denotes a real arc. We have, because of the
1
definition of the expression
(1)
1n , that
xn = 1
or, in other words


r n cosnt  1sinnt  1 .
/172
We can draw from this last equation (with the aid of theorem I § II)
rn = 1
cos nt + -1 sin nt = 1
and so,
cos nt = 1,
r=1
sin nt = 0,
2k
t
,
n
nt =  2kp
Cours d’analyse – Chap. VII: Imaginary expressions
16
where k represents any whole number. The quantities r and t are also determined, their
various values that satisfy equation (1) are obviously contained in theformula
(2)
x  cos
2 k
2 k
 1sin
.
n
n
In other words, the various values of
(3)
1
n
1
 cos
1
n
1
are given by the equation
2k
2k
.
 1sin
n
n
Now let h be the whole number closest to the ratio
k
. The difference between
n
k
will be at most equal to ½, and so we will have
n
k
k'
h ,
n
n
the numbers h and
k'
denotes a fraction less than or equal to ½, and, as a consequence, k’ will be a
n
n
whole number less than, or at most equal to . From this we will concluce
2
where
2k
2k ' 
2k
2k
2k ' 
2k ' 
 2h 
, cos
 1sin
 cos
 1sin
.
n
n
n
n
n
n
Consequently, all the values of
cos
1
n
1
will be contained in the formula
2k ' 
2k ' 
 1 sin
,
n
n
/173
n
, or, what amounts to the
2
same thing, in formula (3) if we suppose that k is contained between the same limits.
if we suppose that k’ is contained between the limits of 0 and
Corollary I: When n is even, the various values that the number k can assume
n
without leaving the limits 0 and
are respectively
2
0, 1, 2, …,
n2 n
, .
2
2
Cours d’analyse – Chap. VII: Imaginary expressions
17
For each of these values of k, formula (3) gives in general two conjugate imaginary
1
1n , that is to say, two conjugate imaginary roots of unity of
n
degree n. However, we find for k = 0 but a single real root, +1, and for k  another
2
real root, – 1. In summary, when n is even, the expression
values of the expression
1
1n
admits two real values, namely
+1 and – 1,
along with n – 2 imaginary values, conjugate two by two, namely
(4)
[block of equations.]
The total number of these values, real and imaginary, is equal to n.
Suppose, for example, that n = 2. We find that there exist two values of the
expression
1
n
1 ,
/174
where, what amounts to the same thing, the two values of x are those that satisfy the
equaton
x2 = 1,
and these values, both real, are respectively
+1 and – 1.
Now suppose that n = 4. We find that there are four values of the expression
1
4
1 ,
where, what amounts to the same thing, the four values of x are those that satisfy the
equation
x4 = 1.
Among these four values, two of them are real, namely
Cours d’analyse – Chap. VII: Imaginary expressions
18
+1 and – 1.
The two others are imaginary and are respectively equal, first to
cos

2
 1sin

2
  1 ,
and second to
cos

2
 1sin

2
  1 .
Corollary II: When n is odd, the various values that the number k can assume
n
without leaving the limits 0 and
are respectively
2
0, 1, 2, …,
n 1
.
2
For each of these values of k, formula (3) gives in general two conjugate imaginary
1
values of the expression 1n , that is to say, two conjugate imaginary roots of unity of
degree n. However, we find for k = 0 but a single real root, +1. In summary, when n is
odd, the expression
1
n
1
/175
admits a single real value, namely
+1,
along with n – 1 imaginary values, conjugate two by two, namely
(5)
[block of equations.]
The total number of these values, real and imaginary, is equal to n.
Suppose, for example, that n = 3. We find that there exist three values of the
expression
1
3
1 ,
where, what amounts to the same thing, the three values of x are those that satisfy the
equaton
x3 = 1,
Cours d’analyse – Chap. VII: Imaginary expressions
19
and these values, of which one is real, are respectively
2
2
cos
 1 sin
,
3
3
+1,
cos
2
2
 1 sin
.
3
3
Also, the side of the hexagon is, as we know, equal to its radius, and the supplement of
2
the arc subtended by this side has for its measure
, so we can easily obtain the
3
equations
cos
2
1
 ,
3
2
1
2
sin
2
3
 ,
3
2
1
and, by virtue of these equations, the imaginary values of the expression
1
2
1 3
 
2 2
1 ,
1
2
1 3
 
2 2
13
reduce to
1 .
/176
[4092]
Cours d’analyse – Chap. VII: Imaginary expressions
20