Irrational, negative and complex numbers make a fascinating story.
This story is told in Kline
[6], but let me give some of the highlights here.
The Greeks defined a
number to be a multitude of units i.e a whole number > 1. The early
Pythagoreans had extended this to include ratios of positive whole numbers,
the positive rationals. They had a mystic belief that these numbers were the
true forms from which nature was constructed. It is said that a mathematician
named Hipassus discovered the irrationality of 
while on board a ship.
When he communicated his discovery, he was thrown overboard. In Euclid,
irrational numbers are handled by basing algebra on line segments rather than
numbers, so that a product of two numbers was an area and a product of three
numbers was a volume and a product of four numbers was impossible.
The Alexandrian mathematician Heron of the first century A.D. gave
as the area of a triangle with sides a, b, c and
perimeter 2s. This was one of the first leaks in the Greek dike protecting
the idea that formulas could not be used without a geometric
conceptualization of
the meaning of their operations. Fortunately there was no little Greek boy
who could plug the hole; what began as a trickle, eventually became a flood.
Algebra became a game of manipulating symbols which were at the time regarded
as meaningless.
The first known use of negative numbers was by the Indian mathematician, Brahmagupta circa A.D. 628. The great Indian mathematician, Bhaskara, said, after giving 50 and -5 as the solutions to an equation, ``The second value is in this case not to be taken, for it is inadequate; people do not approve of negative solutions.'' However, negative numbers continued to be used even by those who considered them meaningless symbols. When they were incorporated into the number system, they could be operated on using the laws of algebra, and these operations seemed to be consistent even if the objects themselves were meaningless symbols.
Antoine Arnauld, a close friend of Pascal, questioned that -1:1 = 1:-1,
because, he said, -1 is less than +1; hence how could the smaller be to
the greater as the greater to the smaller [6, page 115,]. If
I could paraphrase this argument, it seems to me that he is saying that
since numbers satisfy the axiom, 
, there can be be no negative numbers because they do not
satisfy this axiom.
d'Alembert (1717-1783) wrote in his famous dictionary that ``a problem leading to a negative solution means that some part of the hypothesis was false but assumed to be true.'' Leibnitz agreed that there is something fishy about negatives and imaginaries, but argued that one can calculate with them because their form is correct.
The objections to negative numbers were but chaff compared to the
controversy over complex numbers. In 1545, Cardan, in his book
Ars Magna,
considers the problem of finding two numbers whose sum is 10 and whose
product is 40. He solved the equation x(x-10)= 40 and got the solutions,
and then says that these are
``sophistic quantities which though
ingenious are useless.'' Cardan's solution of cubic equations required taking
cube roots of expressions involving square roots. Sometimes these square
roots are imaginary, even when the ultimate solution works out to be a real
number. Again, even though people continued to manipulate complex numbers
formally, there was considerable uneasiness over their use. According to
Leibnitz, ``The divine spirit found a sublime outlet in that wonder of
analysis, that portent of the ideal world, that amphibian between being and
non-being, which we call the imaginary root of negative unity.'' Descartes
rejected complex roots and coined the derogatory term imaginary to describe
.
The controversy was not finally settled until the beginning of the 19th
century when Gauss, Hamilton, and others
settled the matter by getting abstract.
For modern mathematics, the question is not whether or not there are
Platonic ideals corresponding to certain words, but just whether it is
consistent and useful to use these words. Is there an algebraic system
in which -1 has a square root satisfying much of the basic laws of algebra?
Yes, of course there is, consider
all pairs of real numbers with the operations +, -, 
, and 
defined as follows:
There is no need to debate about Platonic ideals, just define the
complex numbers to be this system. Note that if we define 
,
and identify 
with the real number x, then 
.
Setting up the axioms for complex numbers
involves a systematic organization of algebra. A
field is a model
for the axioms of field theory. First we define the language of field
theory to be
constant symbols 0 and 1,
the two-place relation symbol = and the two-place function symbols,
+, , and the one-place function symbols,
. The axioms of field theory are the equality axioms listed
on page 
together with
the following algebraic axioms:
In addition to the field axioms,
the complex numbers satisfy two more principles,
each of which requires an infinite number of axioms to state. Firstly,
no matter how many times you add 1 to itself, you never get 0 and secondly,
a very deep fact first proven by Gauss, every polynomial of positive
degree has a root. To state the first of these, we use the infinite
axiom list (where conforming to common practice, we abbreviate 
by 
),
etc.
To state the second principle, first note that for degree 1, it is a
consequence of the field axioms, so we need only state the principle
for degree 2 and up. Each degree takes a separate axiom. (Again, we
follow customary practice; this time by using 
in place of
. Likewise for higher powers.)
etc.
All these axioms taken together form what is known as the axioms for algebraically closed fields of characteristic 0.
The real numbers also have an axiom set. They are what is called
a real closed field.
Let us go into what this means. In addition
to being a field, they interpret one more relation symbol, <, so as
to satisfy the axioms for an ordered field. In the following axioms
we again follow common practice and abbreviate 
by 
:
A system satisfying these axioms is called an ordered field. To axiomatize the real closed fields, we add in more axioms stating that every positive element has a square root and every polynomial of odd degree has a root. The first of these is just one axiom, but the second is a whole infinite class, just like algebraically closed fields. Here are the axioms:
etc.
These axioms are often called RCF standing for real closed fields.
