Most people's first exposure to algebra is through simple equations like x+3=7. One thinks of x as a `mystery' number, and one solves the mystery by working out what x must be from the information given. When one is thoroughly used to this idea, one can then be taught how to solve more sophisticated kinds of equations, such as quadratic equations or simultaneous linear equations in two or three variables.

The positive integers are so basic that, when we
give x=4 as a solution to the equation x+3=7, we have
obviously achieved something: we have discovered that
x, which we didn't know, turns out to be 4, which is a
number with which we already had some familiarity.
However, when we leave behind the integers, it is
not always as easy to say what our achievement is.
Consider, for example, the equation x^{2}=2.
What is the solution? Well, there are two, but the
more obvious one is the (positive) square root of 2.
And what does that mean? Well, the positive square
root of two is the positive real number that squares
to 2. So what have we achieved? We have `learned' that
the positive solution to the equation x^{2}=2
is the positive real number that squares to 2.

This is a technique of wide applicability. For example, the largest real solution of the quintic equation

x^{5}-13x^{4}+2x^{2}
-7x-1=0

is nothing other than {[-13,0,2,-7,-1]}. And what is the meaning of the expression {[-13,0,2,-7,-1]}? Well, for real numbers a,b,c,d and e I define {[a,b,c,d,e]} to be the largest real solution of the quintic equation

x^{5}+ax^{4}+bx^{3}
+cx^{2}+dx+e=0

By a similar method I can integrate e^{-x2}
(just define a function Phi(x) to be the integral of
e^{-t2} from minus infinity to x),
solve unpleasant partial differential equations and so on.
Or, at a more basic level, I can solve the equations
x+5=0 (by setting x to be -5, which is defined as the
`additive inverse' of 5 - that is, the number which gives
0 when you add 5 to it) and 3x=1 (by setting x=1/3, the
`reciprocal' of 3 - that is, the number which, when
you multiply it by 3, gives 1).

Of course, a game that is too easy to play is an uninteresting game, and yet most of the time when we solve equations we feel that we are doing some work, and learning something that we didn't know. So what is the right way to view what we do? There are two different answers to this question, both important.

It is when we focus on rather simple equations such as
x^{2}=2, 3x=1 and x+5=0 that we are most likely to
sense a certain circularity in what we do. However, in
each case there * is * something non-trivial in
what is going on. It is not a trivial statement that
there is a real number that squares to 2 (as I have
discussed in my pages on the
square root of two and
real numbers as infinite decimals ). So one way
of justifying our usual practice when we `solve'
the equation x^{2}=2 is to say that, interestingly,
we can prove that a unique positive real solution exists
and root two is the name that we give to this solution.

This sort of answer doesn't work as well for x+5=0. It is not clear what it would mean to prove that -5 exists. However, even here there is a non-trivial statement involved, namely that the positive integers can be embedded into a larger set, which we call the integers, in which addition and multiplication can be defined in a natural way. When defined on this larger set they still have familiar properties such as commutativity, but they have the additional very useful property that every number has an additive inverse.

One can say something similar about the equation 3x=1.
What is interesting about the solution is not that, amazingly,
3 has a multiplicative inverse and we have * managed to
find it *, but rather that the ring of integers
can be embedded into a field, Q.

In fact, algebraists (as opposed to analysts) take this
attitude even to the square root of two. What is interesting
about this number is less that it is approximately
1.4142135... and more that if we declare x to have
the property that x^{2}=2 (without worrying
about what x is - though we know that it is not a
rational number) then the set of all numbers of the
form a+bx, with a and b rational, forms a field,
which we call Q(root two). This second, more abstract,
approach to equation solving is of great importance when
we consider the equation x^{2}+1=0. Again it
looks circular to say that the solution is i, where
i is the square root of -1, but again what is really
interesting is the existence of a certain algebraic
structure - this time the field of complex numbers,
which of course has all sorts of interesting properties,
not all of them algebraic.

The examples I have looked at so far suggest the following answer to what it means to solve an equation. The equation draws attention to an inadequacy in a certain number system (it does not contain a solution to the equation) and one is therefore driven to extend the number system by introducing, or `adjoining', a solution. With luck, the extended system has the good properties of the original one, with the added advantage that more equations can be solved.

Sometimes we may take the attitude that a larger
number system has already been defined, and that to
solve an equation means to prove that it has a solution
in the larger number system. In other words, what is
interesting is the * existence * (and after that,
properties) of a solution rather than a neat formula for
it. This switch in attitude is particularly important
with differential equations - where of course now I am
talking not of number systems but of systems of functions.
It is known that the equation f'(x)=e^{-x2}
has no solution in terms of functions such as polynomials
exponentials and trigonometric functions. However, a
solution certainly exists (even if we can't antidifferentiate,
we can Riemann integrate) and it has very interesting
properties, as any probabilist will tell you.

However, neither of the above views really seems to
capture what is going on when we solve a quadratic equation
such as x^{2}=x+1, obtaining the solution
(1 +/- root 5)/2. What do we achieve by completing
the square, or by applying the formula?

The situation is undoubtedly different, because
although (1 +/- root 5)/2 is the solution of the
equation, this is certainly not true * by definition
*. On the other hand, part of the solution, namely
the square root of 5, * is * a by-definition
solution. So it seems that what we have done is to
take for granted that we can solve the equation
x^{2}=5 (and similar ones) and to use that
interesting ability to solve an equation which is not
of such a simple form.

In other words, when we `solve' the quadratic, what
we are really doing is showing that the problem can
be reduced to that of solving a particularly simple
quadratic - that is, one of the form x^{2}=c.
A similar remark can be made about the equation
3x=7. Suppose we know that multiplication and addition
satisfy all the axioms for a field. That allows us
to solve, by definition, an equation like 3x=1, but
to solve the equation 3x=7 we must take the additional
step of multiplying the multiplicative inverse of 3 by
7. Thus, from the existence of multiplicative inverses
we can deduce the solutions of * all * equations
of the form ax=b (with a not equal to zero).

Similarly, one might wish to say that e^{x}
is by definition the solution of the differential equation
f'(x)=f(x) (with f(0)=1). Once that is done it is interesting
to know that no * new * definitions are needed in
order to solve equations such as f''(x)-3f'(x)+2f(x)=0.

This point of view is the one taken when one says that
quintic equations cannot be solved. It can be shown that
solutions exist (as real or complex numbers, or as
abstract objects adjoined to the rationals) but it is
not possible to reduce the solution of the quintic to
the solution of equations of the form x^{m}=c.
To be a little more precise, there is no formula for
the solution of a general quintic, where the formula
expresses a function of the coefficients and the function
is a composition of the usual arithmetical operations
together with the extra operation of taking roots.