INTERNATIONAL JOURNAL OF GEOMETRY Vol. 1 (2012), No. 2, 65 - 78 A GEOMETRIC MODEL OF THE FIELD OF COMPLEX NUMBERS CONSTANTIN LUPU Abstract. In this paper we will construct a geometric model of the eld of complex numbers by using the elementary plane Euclidean geometry notions. 1. Introduction Considering the set of complex numbers, introduced by the axiomatic method, it can be build a geometric model of this set of numbers by using elementary notions of the plane Euclidean geometry (segment, segments measure, angle, angles measure) and alsothe properties of some elementary geometric transformations (homothety, rotation, symmetry). Denition 1.1 ([2], [3]). The triplet (K; ; ) ; where K 6= ?, is called the eld of complex numbers if the following conditions (axioms) are true: I: (K; ; ) is a commutative field; II: The field (K; ; ) is an extension of the eld of real numbers (R; +; ); III. There exists an element i in K, with these properties: 1. i i = i 2 = 1 2 R; 2. for every element z in K there exist the real numbers x and y so that: z = x i y. Keywords and phrases: complex number, set, model. (2010)Mathematics Subject Classication: 51M04, 51M25. Received: 21.06.2012. In revised form: 4.07.2012. Accepted: 13.08.2012.
Transcript
INTERNATIONAL JOURNAL OF GEOMETRYVol. 1 (2012), No. 2, 65 - 78
A GEOMETRIC MODEL OF THE FIELD
OF COMPLEX NUMBERS
CONSTANTIN LUPU
Abstract. In this paper we will construct a geometric model of the�eld of complex numbers by using the elementary plane Euclidean geometrynotions.
1. Introduction
Considering the set of complex numbers, introduced by the axiomaticmethod, it can be build a geometric model of this set of numbers by usingelementary notions of the plane Euclidean geometry (segment, segment�smeasure, angle, angle�s measure) and also the properties of some elementarygeometric transformations (homothety, rotation, symmetry).
De�nition 1.1 ([2], [3]). The triplet (K;�;�) ; where K 6= ?, is called the�eld of complex numbers if the following conditions (axioms) are true:
I: (K;�;�) is a commutative field;II: The field (K;�;�) is an extension of the �eld of real numbers (R;+; �);III. There exists an element i in K, with these properties:1. i� i = i2 = �1 2 R;2. for every element z in K there exist the real numbers x and y so that:
Remark 1.1. The �eld�s (K;�;�) property of being real numbers �eldextension has the following meaning: the �eld (K;�;�) contains an sub-�eld (I;�;�) which is isomorph with the �eld (R;+; �) through a function' : I ! R, so that '(u) = x, for every u 2 I; the function�s properties allowthe �elds (I;�;�) and (R;+; �) to identify: if u, u1 and u2 are elements inI so that '(u) = x, '(u1) = x2 and '(u2) = x2, then u = x,
u1 � u2 = x1 + x2and
u1 � u2 = u1 � u2;furthermore, each element u 2 I can be replaced by the real number x = '(u)and in the calculations we consider
x� y (in I) = x+ y (in R)and
x� y (in I) = x � y (in R):
Remark 1.2. In the equalities from the �rst de�nition we have
i� i = i2 = �1 2 Rand z = x� i�y; where the real numbers �1; x and y represent, in fact, theuniquely determined elements � = i2, � and from the sub�eld (I;�;�) sothat '(�) = �1 ('(�) = '(i2) = i2 = �1); '(�) = � = x and '( ) = = y.
Constructing a model of complex numbers set means to specify a construc-tion process of a nonempty K set and to endow it with two composition laws(noted, for example, with the symbols � and �) so that the triplet (K;�;�)must verify the conditions I, II and III from De�nition 1.1. In the case of realnumbers set, there are known few processes from which we get the modelsof complex numbers set:1. matrix process (using subsets of the second order quadratic matrix,
with real numbers elements; see [1] or [4]);2. the quadratic expansion process or the process with ordered pairs of
real numbers (based on the quadratic expansion of the �eld of real numbers(R;+; �); see [2] or [4]);3. the factorisation process (based on the ring of polynomial factorisation
with the R[X] with the real coe¢ cients through its prime ideal; see [5]);
We will build, in the following lines, a geometric model of complex num-bers set by using the Euclidean plan�s elementary geometry notions.In the Euclidean plane "2, which is endowed with a Cartesian system of
coordinates (XOY ), we consider the set K = fz j z = ��!OM; M 2 "2g. In
this set we are consider:- the element (the null vector)
��!OO from K is noted with 0K ;
- for each element z =��!OM in K n f0Kg is associated with the unique
real number t 2 0; 2�), where t = arg z = ��\OX;OM
�, where the angle�
\OX;OM�is positively oriented (see Figure 1);
A geometric model of the �eld of complex numbers 67
Figure 1- the number t = arg z is called the reduced argument and the set Argz =
farg z + 2k�; k 2 Zg is called the extended argument for the element z;- the element 0K =
��!OO has the reduced argument undetermined, which
means arg 0K = t, for all t 2 [0; 2�);
The notion of argument allows the next simple remarks for the elementsz1 =
��!OM and z2 =
��!ON in K n f0Kg:
a) z1 and z2 have the same direction and the same way if and only ifarg z1 = arg z2 (see Figure 2);
Figure 2 Figure 3
b) z1 and z2 have opposite directions if and only if arg z2 � arg z1 =� (mod 2�) (see Figure 3);
c) z1 = z2 if and only if jz1j = jz2j and arg z1 = arg z2;
Also, if z =���!OM is an element from Kn f0Kg ; then:
d) M 2 OX+ , arg z = 0 and M 2 OX� , arg z = �;
e) M 2 OY+ , arg z = �2 and M 2 OY� , arg z = 3�
2 .
We will remind the de�nitions of some elementary geometric transforma-tions that will be used in the construction of the proposed model:
68 Constantin Lupu
Let be the Euclidean plane "2, which is endowed with a Cartesian systemof coordinates (XOY ) :
De�nition 1.2. A homothety of center O and ratio (power) k 2 R�+ is thegeometric transformation � : "2 ! "2 which satis�es the following condi-tions:
1) �(O) = O; were O is the origin of the Cartesian system of coordinates(XOY ) ;2) if M 6= O is a point from "2 and M 0 = � (M), then:a) the point M 0 is situated on the ray (OM ;b) jOM 0j = k � jOM j : (see Figure 4).
Figure 4
De�nition 1.3. The rotation of center O and angle positively oriented �is the geometric transformation < : "2 ! "2 which satis�es the followingconditions:
1) < (O) = O, where O is the origin of the cartesian system of coordinates(XOY ) ;2) if M 6= O is a point from "2 and M 0 = < (M), then:a) �
�\MOM 0
�= �;
b) jOM j = jOM 0j : (see Figure 5).
Figure 5
A geometric model of the �eld of complex numbers 69
De�nition 1.4. The symmetry with respect to a line d is the geometrictransformation Sd : "2 ! "2 which satis�es the following conditions:
1) Sd (A) = A; 8A 2 d;2) if M =2 d and M 0 = Sd (M), then:a) MM 0 ? db) jMP j = jPM 0j ; where MM 0 \ d = fPg. (see Figure 6).
Figure 6
2. Main Result
Theorem 2.1. In the Euclidean plane "2 endowed with a Cartesian systemof coordinates (XOY ) ; we consider the set of vectors K =
nzjz = ��!OM; M 2 "2
oand the operations � : K �K ! K, � : K �K ! K where its are de�nedas follows:
1. the operation � represents the usual addition of �xed vectors
(1) from plane (XOY ); with the application point at the origin O;
2. a) if z1 =��!OM and z2 =
��!ON are elements from Kn f0Kg, then
z1 � z2 = z =��!OP; where
jzj =�����!OP ��� = jz1j � jz2j
and
(2) arg z = (arg z1 + arg z2) (mod 2�)
b) if z1 = 0K and z2 6= 0K ; or z1 6= 0K and z2 = 0K ; orz1 = z2 = 0K , then
(3) z1 � z2 = 0K
70 Constantin Lupu
Theorem 2.2. The triplet (K;�;�) represents a model of the set of thecomplex numbers.
Proof. We will show that the triplet (K;�;�) verify the conditions I,II andIII from De�nition 1.1. First ,we indicate the geometrical procedures for theobtaining of the sum and the product of two elements from Kn f0Kg. Letz1 =
��!OM and z2 =
��!ON are elements of the set Kn f0Kg. Then the element
z = z1 � z2 is obtained by parallelogram rule (see Figure 7),
Figure 7
and the element w = z1 � z2 it is obtained in two steps, as follows:- we consider the homothety � of center O and ratio k = jz1j ; and we
obtain the point N 0 = � (N) so that N 0 2 (ON and jON 0j = jz1j � jON j =jz1j � jz2j;- we consider the rotation < of center O and angle of measure arg z1,
and we obtain the point Q = <(N 0) so that jOQj = jON 0j = jz1j � jz2j and��\N 0OQ
�= arg z = �: The vector w =
��!OQ represents the element z1 � z2
because jwj = jz1j � jz2j and argw = (arg z1+arg z2)(mod 2�) (see Figure 8).
Figure 8
So, if we represent in the plane "2 the elements z1 and z2 from Kn f0Kg,then the element z1 � z2 is obtained by the rule of the parallelogram and
A geometric model of the �eld of complex numbers 71
the lement z1� z2 is obtained by composing the rotation < by the center Oand angle � = arg z1 with the homotety by center O and ratio k = jz1j.I. The triplet (K;�;�) is a commutative �eld.(K1) : The composition laws � and � are always de�ned on the set K:
if z1 and z2 are elements from K, then, according to rules (1), (2) and (3),we deduce that z1 � z2 and z1 � z2 are elements from K;(K2) : The composition laws � and � are commutative: if z1; z2 2 K;
then
z1 � z2 = z2 � z1
(property of usual vector addition); considering u = z1� z2 and v = z2� z1,we deduce that
so u = v.(K3) : The laws � and � are associative: if z1; z2; z3 2 K; then
(z1 � z2)� z3 = z1 � (z2 � z3)
(the associativity of usual vector addition); considering u = (z1 � z2) � z3and v = z1 � (z2 � z3) and applying the laws (1), (2) and (3) we easilyconclude that u= v;(K4) : 1) The element 0K =
��!OO from K is neutral element with respect
to the law �: If z = ��!OM 2 K; then z � 0K = 0K � z = z;2) The element 1K =
��!OU fromK; with U 2 OX+ and
�����!OU ��� = 1 is neutralelement with respect to the law �: If z 2 K and z0 = z � 1K ; z00 = 1K � z;then ��z0�� = jzj � j1K j = jzj ; ��z00�� = j1K j � jzj = jzj ;
arg z0 = (arg z + arg 1K)mod 2� = (arg z + 0)mod 2� = arg z
It follows that z0 = z00 = z; hence z � 1K = 1K � z = z:(K5) : 1) For every z =
��!OM 2 K there exists an opposite element z =
���!OM 0 2 K, with
������!OM 0��� = �����!OM ��� and arg���!OM 0 = (arg z + �)mod 2� (see
Figure 9).
72 Constantin Lupu
Figure 9 Figure 10
If z 2 K and z =��!OM; z =
���!OM 0 with jzj = jzj and
argz = (arg z + �)mod 2�;
then the vectors��!OM and
���!OM 0 are collinear, they have tha same length but
opposite directions, so
z � (z) = (z)� z = 0K :
2) For every z =��!OM 2 Knf0Kg there exists an inverse element z�1 =���!
OM0 2 K, with ��z�1�� = 1
jzj
and
arg z�1 = (2� � arg z)mod 2�
(see Figure 10).If w = z � z�1; then
jwj = jzj ���z�1�� = jzj � 1jzj = 1
and
argw =�arg z + arg z�1
�mod2� = (2�)mod 2� = 0;
we get that w = 1K : Similarly we �nd that z�1 � z = 1K :(K6) The law � is distributive over the law �: Let the vectors z1 =
���!OM1,
z2 =���!OM2 and z =
��!OM be from Knf0Kg: We show that
z � (z1 � z2) = (z � z1)� (z � z2):
Representing in the Euclidean plane "2 the vectors z =��!OM; z1 =
���!OM1,
z2 =���!OM2 and z1 � z2 =
��!OP , it is obvious that the quadrilateral OM1PM2
is a parallelogram (see Figure 11).
A geometric model of the �eld of complex numbers 73
Figure 11
a) Considering the homothety � of centre O and power jzj, ifM 01 = �(M1),
M02 = �(M2) and we take the parallelogram OM
01QM
02 (see Figure12) with���OM 0
1
��� = jzj � jz1j,���M 0
1Q��� = jzj � jz2j we deduce that the triangles OM1P
(Figure 11) and OM01Q (Figure 12) are similar.
Figure 12We �nd that
jOP jjOQj =
jz1jjzj � jz1j
i.e.jz1 � z2jjOQj =
1
jzj ;
so jOQj = jzj � jz1 � z2j; in these conditions we obtain that Q = �(P );
74 Constantin Lupu
b) We consider the rotation < with the centre O and angle � = arg z. IfM"1 = <(M
01), Q
0= <(Q) and M"
2 = <(M02) it results that the quadrilateral
OM"1Q
0M"2 is a parallelogram (see Figure 13), with
���!OM"
1 = z � z1,��!OQ
0=
z � (z1 � z2) and���!OM"
2 = z � z2.
Figure 13
Applying the addition rule for the vectors���!OM"
1 and���!OM"
2 , it follows that��!OQ0 = (z � z1)� (z � z2), so
��!OQ0 = z � (z1 � z2) = (z � z1)� (z � z2):
II. The �eld (K;�;�) is an extension of the �eld of real numbers (R;+; �).a) We consider the set I � K, with I = fz 2 K j z = ��!OM , M 2 OXg.
We see that z is an element from I if and only if arg z = 0 or arg z = �(excepting the element 0K which has undertermined argument) We simplyshow that the triplet (I;�;�) is an sub�eld of the �eld K because the nextconditions are satis�ed:1) If z and v are elements from I, then z v is an element from I (the
property results by applying the usual rule of vector addition);2) If z =
��!OM and v =
��!ON 6= 0K are elements from I, then the element
z � v�1 is from I; indeed, since v 2 I; v�1 is an element from I because
A geometric model of the �eld of complex numbers 75
b) Let ' : (I;�;�) �! (R;+; �) be a function de�ned as follows: ifz =
��!OM is an element from I, then
'(z) = x =
8>><>>:jzj =
�����!OM ��� ;0;
� jzj = ������!OM ��� ;
if M 2 OX+if M = Oif M 2 OX�
The function ' is an isomorphism between the �elds (I;�;�) and (R;+; �)because the next conditions are ful�lled:1) ' is a bijective function (acount of the axiom of construction of a
segment with a precised length and of ');2) '(z1 � z2) = '(z1) + '(z2);3) '(z1 � z2) = '(z1) � '(z2) for any z1 and z2 from I.Let�s assume, without restrict the generality, that z1 =
��!OM and z2 =
��!ON ,
are elements from I with the property that M;N 2 OX+. If jz1j = x1 > 0and jz2j = x2 > 0, then
'(z1 � z2) = '(��!OM ���!ON) = '(��!OP );
with P 2 OX+ and�����!OP ��� = �����!OM ��� + �����!ON ��� = x1 + x2 (the rule of addition
of the vectors with the extremities on the axis OX). Therefore,
'(z1 � z2) = '(z1) + '(z2):
The property 2) can be proved in the same manner and in the cases M 2Ox+, N 2 Ox�, or M 2 Ox� and N 2 Ox�; also, in these conditions, if
Therefore that any element z =��!OM from I can be identi�ed through the
real number x = '(z) (it�s written simpli�ed z = x) and sums, respectiveproducts of the types z1 � z2 and z1 � z2 from I its can identify with sums,respective products of the types x1 + x2 , respective x1 � x2 from R (wewritten z1 � z2 = x1 + x2 and z1 � z2 = x1 � x2). In conclusion , the �eld(K;�;�) it is an extesion of the �eld of the real numbers (R;+; �), the setR thus considered that a subset of K.III. There is an element i in K with the properties:1: i� i = i2 = �1 2 R;2: For every element z from K, it can be indicated the real numbers x
and y, so that z = x� i� y.We consider the element i from K with i =
��!OM 2 Knf0Kg. We will demostrate that the real numbers
x and y are existing, so that z = x� i� y;a) We will assume that the extremity M of the vector z =
��!OM is posi-
tioned in the �rst or the second quadrant (see Figure 14) and let M1 andM2 are the projections of M on the axis OX, respectively OY .
Figure 14
According to the rule of addition of the vectors, it results that:
(4)��!OM =
���!OM1 �
���!OM2
Since, the point M1 is situated on the axis OX, the vector���!OM1 it is iden-
ti�ed through the real number x = �������!OM1
���, as M1 2 OX+ or M1 2 OX�,so
(5)���!OM1 = x:
Now we consider the simmetry Sd, where d represents the �rst bisetrix and
let be M02 = Sd(M2) 2 OX (see Figure14), so
�������!OM02
���� = ������!OM2
���. In theeseconditions,
���!OM2 = i�
���!OM
02 because
������!OM2
��� = jij � ������!OM 02
��� = ������!OM 02
���, and(arg i+ arg
���!OM 0
2)mod 2� =�
2+ 0 =
�
2= arg
���!OM2:
A geometric model of the �eld of complex numbers 77
Since, the vector���!OM
02, is indeti�ed through the real number y =
�������!OM02
���� =������!OM2
���, it results that(6)
���!OM2 = i� y:
Replacing (5) and (6) in (4), we deduce that:
z =��!OM =
���!OM1 �
���!OM2 = x� i� y:
b)We assume now that the extremityM of z =��!OM is positioned in the third
or the fourth quadrant (see Figure15). Let M1and M2 be the projections ofthe point M on the axis OX and OY respectively.
Figure 15
Considering M02 = Sd(M2), we deduce that M
02 2 OX� and
���!OM
02 vector
is identi�ed through the real number y < 0, with y = ��������!OM
02
���� = � ������!OM2
���.But
���!OM2 = i�
���!OM
02 because
������!OM2
��� = jij � ���OM 0
2
��� andarg(i�
���!OM
02) = (arg i+ arg
���!OM
02)mod 2� = (
�
2+ �)mod 2� =
3�
2:
Since���!OM
02 = y (through ' isomorphism), it results that
���!OM2 = i � y.
Therefore, replacing in the equality��!OM =
���!OM1 +
���!OM2,
���!OM1 with x and���!
OM2 with i� y, we obtain z =��!OM =
���!OM1 �
���!OM2 = x� i� y: �
Being met the �rst, second, and third conditions from the de�nition ofthe complex numbers set, we deduce that the ensemble (K;�;�) is a modelof the complex numbers set and they contain the follow conditions:- K set is noted with the symbol C;- the operation � and � are noted with the classic symbols "+" and "�";- the triplet (C,+,�) is named the complex numbers set;- the elements of C set are named complex numbers;- the �eld (C , +, �) is named the complex numbers �eld
78 Constantin Lupu
Acknowledgement. The author is grateful to the referees for theiruseful suggestions.
