## Example 1.1.1: Properties of number systems |

Find |

**1.** In * N* no number except zero has an additive inverse. For
example, there is no natural number you can add to

**2.** In * Z* no number except for

multiplicative inverse. For example, there is no integer you can multiply with

**3.** In * Q* every number has additive and multiplicative inverses,
but let's consider the sequence

x_{1}= 2

x_{n}=^{1}/_{2}(x_{n}+^{2}/_{xn}) for n > 1

Each of the *x _{n}* is clearly a rational number, but
it can be shown that

x_{n}=

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