5 Complex Numbers 39Of courseE(t)=etis theexponential function. We will now adopt the usual
notation, but we remark that the definition ofetas a solution of a differential equa-
tion provides a meaning for irrationalt, as well as a simple proof of both the addition
theorem and the exponential series.
The power series foretshows that
et> 1 +t>1foreveryt> 0.Sincee−t =(et)−^1 , it follows that 0< et <1foreveryt <0. Thuset > 0
for allt∈R. Hence, by (3),etis a strictly increasing function. Butet→+∞as
t→+∞andet→0ast→−∞. Consequently, since it is certainly continuous,
the exponential function maps the real lineRbijectively onto the positive half-line
R+={x∈R:x> 0 }.Foranyx>0, the uniquet∈Rsuch thatet=xis denoted
by lnx(thenatural logarithmofx)orsimplylogx.
5 ComplexNumbers
By extending the rational numbers to the real numbers, we ensured that every posi-
tive number had a square root. By further extending the real numbers to the complex
numbers, we will now ensure that all numbers have square roots.
The first use of complex numbers, by Cardano (1545), may have had its origin in
the solution of cubic, rather than quadratic, equations. The cubic polynomial
f(x)=x^3 − 3 px− 2 qhas three real roots ifd:=q^2 −p^3 <0 since then, for largeX>0,
f(−X)< 0 , f(−p^1 /^2 )> 0 , f(p^1 /^2 )< 0 , f(X)> 0.Cardano’s formula for the three roots,
f(x)=
3√
(q+√
d)+
3√
(q−√
d),gives real values, even thoughdis negative, because the two summands are conjugate
complex numbers. This was explicitly stated by Bombelli (1572). It is a curious fact,
first proved by H ̈older (1891), that if a cubic equation has three distinct real roots, then
it is impossible to represent these roots solely by real radicals.
Intuitively, complex numbers are expressions of the forma+ib,whereaandbare
real numbers andi^2 =−1. But what isi? Hamilton (1835) defined complex numbers
as ordered pairs of real numbers, with appropriate rules for addition and multiplication.
Although this approach is similar to that already used in this chapter, and actually was
its first appearance, we now choose a different method.
We d e fi n e acomplex numberto be a 2×2 matrix of the form
A=
(
ab
−ba