Example 1.1.1: Properties of number systems

  1. a natural number without an additive inverse
  2. an integer without a multiplicative inverse
  3. a sequence of rational numbers that converges but the limit is not rational
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1. In N no number except zero has an additive inverse. For example, there is no natural number you can add to 2 to get zero.

2. In Z no number except for 1 has a
multiplicative inverse. For example, there is no integer you can multiply with 2 to get 1.

3. In Q every number has additive and multiplicative inverses, but let's consider the sequence {xn} where

x1 = 2
xn = 1/2 (xn + 2/xn) for n > 1

Each of the xn is clearly a rational number, but it can be shown that

xn =

But the square root of 2 is not rational and hence is not in Q.

Interactive Complex Analysis, ver. 1.0.0
(c) 2006-2007, Bert G. Wachsmuth
Page last modified: May 29, 2007