§l. Convergent sequences and series
53
c 2 + 1 = 0;
it is thus because this equation has no solution in IR that every non-zero
complex number has an inverse in C.
To conclude the construction, we observe that the mapping x f-+ (x,O) of
R into C is injective and transforms addition and multiplication in IR into the
corresponding operations in C. We may therefore agree to identify each real
number x with the couple (x,O). Since we have
(O,l)(y,O) = (O,y)
by rule (0.3), rule (0.2) then proves that
(x, y) = (x,O) + (0, l)(y, 0) = x + iy
where we define
i = (0,1).
Again by rule (0.3), we then have
(0,1)(0,1) = (-1,0),
i.e. i 2 = -1. No mysteries anymore.
Having done these constructions and verifications, you may forget them,
and start computing mechanically as Euler was doing in 1750.
The geometric representation of complex numbers x + iy by points (x, y)
(or by vectors of origin 0) of a plane allows one to introduce the modulus or
the absolute value
Ix+iyl = JX2+y2
of a complex number. On introducing the conjugate z = x - iy of z = x + iy,
one finds immediately that
zz = Iz12,
whence Iz'z"l = Iz'llz"l and liz = z/lzl2. These summary indications -
plus, of course, the habit of computing complex numbers, which is acquired
by practice - will suffice for nearly all our needs.
To conclude, let us state the wonderful property of complex numbers that
explains their importance everywhere in mathematics: any algebraic equation,
of any degree, with complex coefficients, has complex roots. This cannot be
proved by purely algebraic methods; analysis is needed, as will be seen in
Chap. VII, nO 18.
1 - Algebraic operations and the order relation: axioms of IR
For those who have faith, one can consider the more fundamental properties
of the real numbers as axioms to be accepted without worrying about the
53
c 2 + 1 = 0;
it is thus because this equation has no solution in IR that every non-zero
complex number has an inverse in C.
To conclude the construction, we observe that the mapping x f-+ (x,O) of
R into C is injective and transforms addition and multiplication in IR into the
corresponding operations in C. We may therefore agree to identify each real
number x with the couple (x,O). Since we have
(O,l)(y,O) = (O,y)
by rule (0.3), rule (0.2) then proves that
(x, y) = (x,O) + (0, l)(y, 0) = x + iy
where we define
i = (0,1).
Again by rule (0.3), we then have
(0,1)(0,1) = (-1,0),
i.e. i 2 = -1. No mysteries anymore.
Having done these constructions and verifications, you may forget them,
and start computing mechanically as Euler was doing in 1750.
The geometric representation of complex numbers x + iy by points (x, y)
(or by vectors of origin 0) of a plane allows one to introduce the modulus or
the absolute value
Ix+iyl = JX2+y2
of a complex number. On introducing the conjugate z = x - iy of z = x + iy,
one finds immediately that
zz = Iz12,
whence Iz'z"l = Iz'llz"l and liz = z/lzl2. These summary indications -
plus, of course, the habit of computing complex numbers, which is acquired
by practice - will suffice for nearly all our needs.
To conclude, let us state the wonderful property of complex numbers that
explains their importance everywhere in mathematics: any algebraic equation,
of any degree, with complex coefficients, has complex roots. This cannot be
proved by purely algebraic methods; analysis is needed, as will be seen in
Chap. VII, nO 18.
1 - Algebraic operations and the order relation: axioms of IR
For those who have faith, one can consider the more fundamental properties
of the real numbers as axioms to be accepted without worrying about the