12 Appendix: the decimal system
12.1 The decimal representation of natural numbers
Definition 12.1 (Digits) A digit is any one of the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Definition 12.2 (Positive integer decimals) We equate each positive integer decimal with a positive integer by the formula \[ a_n a_{n-1}\dots a_0 \equiv \sum_{i=0}^n a_i \times \text{ten}^i. \]
Theorem 12.1 (Uniqueness and existence of decimal representtions) Every positive integer \(m\) is equal to exactly one positive integer decimal.
The theorem can be proven with the strong principle of induction (Chapter 1
12.2 The decimal representation of real numbers
Definition 12.3 (Real decimals) The decimal is equated to the real number \[ \pm a_n\dots a_0.a_{-1}a_{-2}\dots \equiv \pm 1\times \sum_{i=-\infty}^n a_i \times 10^i. \]
Theorem 12.2 (Existence of decimal representations) Every real number \(x\) has at least one decimal representation \[ x=\pm a_n\dots a_0.a_{-1}a_{-2}\dots \]
Proposition 12.1 (Failure of uniqueness of decimal representation) The number \(1\) has two different decimal representations: \(1.000\dots\) and \(0.999\dots\)
Exercise 12.1 If \(a_n\dots a_0.a_{-1}a_{-2}\dots\) is a real decimal, show that the series \(\sum_{i=-\infty}^n a_i \times 10^i\) is absolutely convergent.
Exercise 12.2 Show that the only decimal representations \[ 1=\pm a_n\dots a_0.a_{-1}a_{-2}\dots \] of \(1\) are \(1=1.000\dots\) and \(0.999\dots\).
Exercise 12.3 A real number \(x\) is said to be terminating decimal if we have \(x=n/10^{-m}\) for some integers \(n,m\). Show that if \(x\) is a terminating decimal, then \(x\) has exactly two decimal representations, while if \(x\) is not at terminating decimal, then \(x\) has exactly one decimal representation.
Exercise 12.4 Rewrite