Section 10 Number Systems

10.1 Integers \(\mathbb{Z}\)

The integers \[ \mathbb{Z}= \{ \ldots, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, \ldots\} \] form a cyclic group (and thus abelian group) under addition with identity 0. There are two generators \(\mathbb{Z}= \langle 1 \rangle = \langle -1 \rangle\).

Under multiplication, the only invertible elements (units) are \(1\) and \(-1\), and \(\mathbb{Z}^\ast = \{-1,1\}\) is a group of order two under addition.

10.2 Rational Numbers \(\mathbb{Q}\)

The rational numbers \[ \mathbb{Q}= \left\{ \frac{p}{q} \mid p,q \in \mathbb{Z}, q \not=0 \right\} \] also constitute an abelian group under addition with idenity 0. Every rational number except 0 has a multiplicative inverse, and thus \[ \mathbb{Q}^\ast = \mathbb{Q}\setminus \{0\} \] is a group under multiplication with multiplicative identity 1. The positive rational numbers are also closed under multiplication and taking multiplicative inverses, so if \(\mathbb{Q}_{>0} = \{ a \in \mathbb{Q}\mid a > 0\}\), then \[ \mathbb{Q}_{>0} \le \mathbb{Q}^\ast. \]

10.3 Real Numbers \(\mathbb{R}\)

We’ll leave it to Math 377 to define the real numbers \(\mathbb{R}\). The real numbers form a group under addition, and the only real number that does not have an inverse is 0, so \[ \mathbb{R}^\ast = \mathbb{R}\setminus \{0\} \] is an abelian group under multiplication with identity 1. As with the rational numbers, the positive reals also form a multiplicative subgroup \[ \mathbb{R}_{>0} \le \mathbb{R}^\ast. \]

10.4 Complex numbers \(\mathbb{C}\)

The complex numbers consists of pairs of real numbers \[ \mathbb{C}= \left\{ a + b i \mid a, b \in\mathbb{R}\right\} \] with \(i^2 = -1\). It is a group under addition, \[ (a + b i) + (c + d i) = (a+c) + (b+d) i. \] with identity \(0 = 0 + 0 i\).

Multiplication in \(\mathbb{C}\) is best understood using polar coordinates. We write \[ a + b i = r (\cos(\theta) + i \sin(\theta)) = r e^{i \theta} \] where \[ \begin{array}{lcl} r = \sqrt{a^2 + b^2} & & a = r \cos(theta)\\ \theta = \arctan(b/a) && b = r \sin(\theta). \end{array} \] In polar notation \[ \text{ if } \quad z_1 = r_1 e^{i \theta_1} \quad \text{ and }\quad z_2 = r_2 e^{i \theta_2} \] then \[ z_1z_2 = (r_1 e^{i \theta_1})(r_2 e^{i \theta_2}) = r_1 r_2 e^{i (\theta_1 + \theta_2)} \] so that their angles add and their lengths multiply.

Every complex number other than \(0 = 0 + 0i\) has a multiplicative inverse, and \[ \mathbb{C}^\ast = \mathbb{C}\setminus \{0\} \] is a multiplicative group with identity \(1 = 1 + 0i\).

The circle group is the set of complex numbers on the unit circle, \[ C = \{ e^{i \theta} \mid 0 \le \theta < 2 \pi\}. \] Since lengths multiply and angles add when we multiply complex nubers, the circle group is a multiplicative subgroup \[ C \le \mathbb{C}^\ast \]

As additive groups, we have the following containment of subgroups, \[ \mathbb{Z}< \mathbb{Q}< \mathbb{R}< \mathbb{C}. \] And as multiplicative groups, the units in each of these number systems give us containments of subgroups \[ \{1,-1\} < \mathbb{Q}^\ast < \mathbb{R}^\ast < \mathbb{C}^\ast. \]