# 2 Complex numbers

## 2.7 Arithmetical properties of complex numbers

The set of complex numbers satisfies all the properties previously given for arithmetic in . We state (but do not prove) these properties here.

In particular, 0 = 0 + 0*i* plays the same role in as the real number 0 does in , and 1 = 1 + 0*i* plays the same role as 1. These numbers are called *identities* for addition and multiplication respectively.

We also have that the *additive inverse* (or negative) of *z* = *x* + *yi* is −*z* = −*x* − *yi*, and the *multiplicative inverse* (or reciprocal) of *z* = *x* + *yi* is

There is one important difference between the set of real numbers and the set of complex numbers, however; namely that, unlike the real numbers, the complex numbers are not ordered.

For any two real numbers *a* and *b*, exactly one of the three properties

is true. But this is not the case for the complex numbers; we cannot say, for example, that

Inequalities involving complex numbers make sense only if they are inequalities between real quantities, such as the moduli (moduli is the plural of modulus) of the complex numbers. For example, inequalities such as

are valid.