id: 06439701
dt: j
an: 2015d.00564
au: Deakin, Michael A. B.
ti: History of mathematics: the real numbers ‒ making them respectable.
so: Parabola 50, No. 3, 16-25 (2014).
py: 2014
pu: AMT Publishing, Australian Mathematics Trust, University of Canberra,
Canberra; School of Mathematics \& Statistics, University of New South
Wales, Sydney
la: EN
cc: F50 A30 E40
ut: real numbers; history of mathematics; decimal expansion; decimal numbers;
rational numbers; recurring decimals; irrational numbers; non-recurring
decimals; separator; approximation; Dedekind cut; Cauchy sequences;
limits; equivalent approaches; countable sets; arithmetic laws
ci: ME 2015a.00495; ME 2006d.02290
li:
ab: From the text: In my last column [the author, ibid. 50, No. 2, 3‒10
(2014; ME 2015a.00495)], I described how, at the cost of some apparent
artificiality and seemingly needless complication, the imaginary
numbers eventually became respectable. Here I describe the analogous
process with the real numbers. Paradoxically, the story of how the real
numbers themselves became respectable is more convoluted and the
processes involved took thousands rather than hundreds of years.
Moreover the topic still causes controversy today (although no one is
claiming that the reals aren’t respectable). But let us begin at the
beginning. The real numbers are made up of two sets: the rational
numbers and the irrational ones. In [the author, ibid. 41, No. 2,
13‒20 (2005; ME 2006d.02290)], I described the discovery (by the
ancient Greeks) of the irrational numbers. In what follows, I will
describe several approaches to the real numbers, but will concentrate
on one in particular. This is the one that will almost certainly be the
most familiar to readers of this journal. Certainly it was my own and
it is described by the Wikipedia entry “Construction of the real
numbers" as “[having] the advantage that it is close to the way we
are used to thinking of real numbers". I shall begin by looking at the
decimal expansions of the different numbers. It was first put forward
by the Dutch polymath Simon Stevin (1548‒1620), who was an early
champion of the decimal system.
rv: