§4. Differentiable functions
251
which allows us to define correctly, i.e. without recourse to the symbol Q, a
dual number as a pair (a, b) of complex numbers, the rules for manipulating
these pairs being as in the formulae above. It all works out perfectly except
that, in contrast to the complex numbers, the dual numbers do not form
a field since Q2 = O.
If one has a polynomial f(x) = L;anx n with complex coefficients, one
can then define its "value" for any dual number x+hQ by routinely applying
the formula which defines f. Now
(x + hQ)n = xn + nxn-1hQ + ... = xn + nxn-1hQ
since the unwritten terms contain Q2 as a factor. It follows that
f(x + hQ) = E an(X n + nxn-1hQ) = f(x) + f'(x)hQ
where!, is the usual derivative of f.
This exactly is what Newton constantly did in calculating with his
x + 3;0 - do not confuse his 0 with ours, and do not confuse ours with
the expression o(h) - according to the strict rules of algebra and then
suppressing all terms containing a factor of at least 0 2 • The difference is
that, for us, the symbol Q represents not an "infinitely small increment" of
time like the 0 of Newton, but an element of a ring in which one calculates
according to rules prescribed in advance.
This artifice, which allows one to algebraise the concept of derivative,
was developed in the 1950s by Andre Weil in an incomparably more general
and difficult framework - the "infinitesimal extensions of higher order of
differential varieties" - in a work, unfortunately not published, where he
writes c (why not?) for what we write Q. I didn't think to ask him if he had
been inspired by reading Newton; but knowing his pronounced taste for
the history of mathematics, the reply would hardly have been in doubt.
The reader may amuse himself by introducing, in the same spirit, another symbol Q which, for example, satisfies only the relation Q4 = 0, and
"numbers" x + hlQ+ h2Q2 + h 3Q3 with coefficients in C and, given a polynomial, calculate its "value" at such a number as a function of its derivatives
at the point x as above for the case of dual numbers. The reader who knows
that this is the quotient of a ring by an ideal will notice that systems of
this kind are just quotients of q X], the ring of polynomials in one variable
with complex coefficients, by the ideal of multiples of X2 in the first case,
of X4 in the second, and of XP in the general case. The second person to
give a perfectly correct definition of the complex numbers (the first was
Hamilton), namely Cauchy in 1847, constructed them by considerin f the
quotient of R[X] by the ideal of polynomials that are multiples of X + 1,
a method preferable by reason of the vast generalisations, unknown to
Cauchy, to which it is susceptible, mainly to the algebraic extensions of
commutative fields. See Serge Lang, Algebm, among many other possible
references.
As we have already said (Chap. II, nO 11), there are close connections
between the concepts of derivative and of integral: if one has a continuous
function f on an interval I and if, for an a E I, one denotes by F(x) the integral of f taken over the interval with end points a and x, then F'(x) = f(x);
this is the "fundamental theorem" of the integral calculus (Chap. V). Consider, for example, for x> 0, the function f(x) = l/x. We proved in Chap. II,
nO 11, that, for 0 < a < b,
251
which allows us to define correctly, i.e. without recourse to the symbol Q, a
dual number as a pair (a, b) of complex numbers, the rules for manipulating
these pairs being as in the formulae above. It all works out perfectly except
that, in contrast to the complex numbers, the dual numbers do not form
a field since Q2 = O.
If one has a polynomial f(x) = L;anx n with complex coefficients, one
can then define its "value" for any dual number x+hQ by routinely applying
the formula which defines f. Now
(x + hQ)n = xn + nxn-1hQ + ... = xn + nxn-1hQ
since the unwritten terms contain Q2 as a factor. It follows that
f(x + hQ) = E an(X n + nxn-1hQ) = f(x) + f'(x)hQ
where!, is the usual derivative of f.
This exactly is what Newton constantly did in calculating with his
x + 3;0 - do not confuse his 0 with ours, and do not confuse ours with
the expression o(h) - according to the strict rules of algebra and then
suppressing all terms containing a factor of at least 0 2 • The difference is
that, for us, the symbol Q represents not an "infinitely small increment" of
time like the 0 of Newton, but an element of a ring in which one calculates
according to rules prescribed in advance.
This artifice, which allows one to algebraise the concept of derivative,
was developed in the 1950s by Andre Weil in an incomparably more general
and difficult framework - the "infinitesimal extensions of higher order of
differential varieties" - in a work, unfortunately not published, where he
writes c (why not?) for what we write Q. I didn't think to ask him if he had
been inspired by reading Newton; but knowing his pronounced taste for
the history of mathematics, the reply would hardly have been in doubt.
The reader may amuse himself by introducing, in the same spirit, another symbol Q which, for example, satisfies only the relation Q4 = 0, and
"numbers" x + hlQ+ h2Q2 + h 3Q3 with coefficients in C and, given a polynomial, calculate its "value" at such a number as a function of its derivatives
at the point x as above for the case of dual numbers. The reader who knows
that this is the quotient of a ring by an ideal will notice that systems of
this kind are just quotients of q X], the ring of polynomials in one variable
with complex coefficients, by the ideal of multiples of X2 in the first case,
of X4 in the second, and of XP in the general case. The second person to
give a perfectly correct definition of the complex numbers (the first was
Hamilton), namely Cauchy in 1847, constructed them by considerin f the
quotient of R[X] by the ideal of polynomials that are multiples of X + 1,
a method preferable by reason of the vast generalisations, unknown to
Cauchy, to which it is susceptible, mainly to the algebraic extensions of
commutative fields. See Serge Lang, Algebm, among many other possible
references.
As we have already said (Chap. II, nO 11), there are close connections
between the concepts of derivative and of integral: if one has a continuous
function f on an interval I and if, for an a E I, one denotes by F(x) the integral of f taken over the interval with end points a and x, then F'(x) = f(x);
this is the "fundamental theorem" of the integral calculus (Chap. V). Consider, for example, for x> 0, the function f(x) = l/x. We proved in Chap. II,
nO 11, that, for 0 < a < b,