BASIC NUMBER THEORY

12 PROPERTIES

DEFINITION (12 properties of numbers)

We should take a close look at the basic properties of numbers and properly define our mathematical language before we can do any analysis on these numbers. Here I will list the 12 basic properties of numbers, and talk about the implications of these properties later. Each letter (\(a\), \(b\), \(c\), etc.) denotes any number:
(P1) \(a + (b + c) = (a + b) + c\)
(P2) \(a + 0 = 0 + a = a\)
(P3) \(a + (-a) = (-a) + a = 0\)
(P4) \(a + b = b + a\)
(P5) \(a \cdot (b \cdot c) = (a \cdot b) \cdot c\)
(P6) \(a \cdot 1 = 1 \cdot a = a, 1 \not= 0\)
(P7) \(a \cdot a^{-1} = a^{-1} \cdot a = 1\), for \(a \not=0\)
(P8) \(a \cdot b = b \cdot a\)
(P9) \(a \cdot (b + c) = a \cdot b + a \cdot c\)

For the following properties, \(P\) is the collection of all positive numbers
(P10) \(\forall a\), one and only one of the following holds:

\[ \qquad\begin{aligned} \text{(i)}\qquad&a = 0\text{,}\\ \qquad \text{(ii)}\qquad&a\text{ is in the collection }P\text{,}\\ \qquad \text{(iii)}\qquad&-a\text{ is in the collection }P\end{aligned} \]

(P11) If \(a\) and \(b\) are both in the collection \(P\), then \(a + b\) is in \(P\)
(P12) If \(a\) and \(b\) are both in the collection \(P\), then \(a \cdot b\) is in \(P\)