Skip to content

Real Number Construction

Decimal expansion

Given \(r\in\mathbb{R}^+\)

  1. find \(q\in\mathbb{N}^+, q\leq r\le q+1\)
  2. So we find next decimal place \(d_1/10 \leq r- q < d_1 / 10 + 1/10\)
  3. repeat \(\frac{d_k}{10^k} \leq r - q - \sum _1^{k-1} \frac{d^m}{10^m} < \frac{d_k}{10^k} + 10^{-k}\)

So that \(r = q.d_1d_2d_3...\)

Def'n. Terminating and repeating

terminating decimal expansion \(q.d_1d_2...d_{m_0}\)
repeating decimal expansion \(q.d_1...d_k\overline{d_{k+1}...d_n}\)(ex. \(1/35 = 0.0287154\overline{287154}\))

Thrm. 1

Claim \(\forall x\in \mathbb{R}^+\) is rational IFF \(x\) has a decimal expansion that is either terminating or repeating

proof.
\(\Leftarrow\)
Assume \(x\) is terminating,

\[x = q.d_1...d_{n_0} = q + \sum_1^n \frac{d_m}{10^m} \in \mathbb{Q}\]

Assume \(x\) is repeating,

\[q.d_1...d_k\overline{d_{k+1}...d_m}\]

Known that \(d_1...d_k\) part is rational, the remaining \(0.0...0\overline{d_{k+1}...d_m}\) equals

\[10^{-k}(\sum_{m=1}^n \sum_{l=0}^\infty \frac{d'_m}{10^{nl + m}})\]

since the number is repeating, we denote \(d'_0,...,d'_n\) be the repeated digits

\[10^{-k}\sum_{m=1}^n d'_m 10^{-m} (\sum_{l=0}^\infty 10^{-nl})\]

Using geometeric series, we have

\[10^{-k}\sum_{m=1}^n d'_m 10^{-m} (1 - 10^{-n})^{-1} = \sum_{m=1}^n\frac{d'_m 10^n}{10^{m+k}(10^n - 1)}\]

\(\Rightarrow\)
Take \(l, m \in\mathbb{Z}^+， x = l/m\).
By Euclidean division, \(l = d_0m + r_0/m\) where \(d_0\in\mathbb{Z}^+\) is the quotient, \(r_0\in\mathbb{Z}^+\) is the remainder, \(r_0<m\).

Lemma 2

\[\forall n\in \mathbb{N}. l/m = \sum_{m=0}^n \frac{d_m}{ 10^{m}} + \frac{r_n} {10^{n}m}\]

proof. (induction by further Euclidean division)

Suppose \(\exists i, r_i = 0\), then is terminated
Suppose \(\forall k\) WTS repeating.

  • since \(r_k\) is a remainder, it can only choose from \(r_k \in \{1, ..., m - 1\}\). Then \(\exists k_1, k_2. r_{k_1} = r_{k_2}\).

Then by uniqueness of Euclidean division, it is repeated.

Def'n. Irrational Numbers

\(x\) is irrational if \(x\in\mathbb{Q}^c\), i.e. \(\not\exists l, m\in\mathbb{Z}^+ s.t. x = l/m\)

Claim 3

\(\forall x, y \in \mathbb{R}, x < y \Rightarrow \exists r\in\mathbb{R}. x < r < y\) and \(r\) is terminating.

proof. consider the decimal expansions of \(x = x_0.x_1..., y = y_0.y_1...\), then exists the smallest \(k\) where \(x_k + 1 \leq y_k\), then find the next \(m>k\) where \(x_m \neq 9\). Then construct \(r = x_0.x_1...[x_m + 1]y_{m+1}y_{m+2}...\)

Construction From Cauchy Sequence

Consider the space of Cauchy sequence \(C_Q = \{(y_n): y_n\in \mathbb{Q}\}\)

\(x_n, y_n\) Cauchy, then

Proposition 1 \(x_n + y_n\) Cauchy.
proof. Let \(\epsilon > 0, N_\epsilon = \max(N^x_{\epsilon/2}, N^y_{\epsilon/2})\)

Proposition 2 \(x_ny_n\) Cauchy
proof.

\[\begin{align*} |x_ny_n - x_my_m| &= |x_ny_m - x_my_n + x_my_n - x_my_m| \\ &\leq |y_n(x_n-x_m)| + |x_m(y_n - y_m)|\\ &\leq B_2|x_n - x_m| B_1|y_n - y_m| &\text{Cauchy implies bounded}\\ \end{align*}\]

Want this to be less than \(\epsilon > 0\). Therefore, take \(|x_n - x_m| \leq \epsilon / 2B_2, |y_n - y_m|\leq \epsilon/2B_1\)

Proposition 3 Additive identity \(r_0=(0,0,0,...)\), multiplicative identity \(r_1 = (1,1,1,...)\)

\[a+r_o = \{a_1 + 0, a_2 + 0,...\}=\{a_n\} = a\]
\[ar_1 = \{a_11, a_21,...\} = \{a_n\} = a\]

Def'n. Equivalence Class

\((x_n)\sim (y_n)\) IFF \(\lim_{n\rightarrow \infty}|x_n -y_n| = 0\)

Example \(x_n = 1/n, y_n = 0\)

Example Let \(\pi=p_0.p_1p_2...\), take \(x_n = p_0.p_1...p_n, y_n = p_0.p_1...p_n1 = x_n + 10^{-(n+1)}\).

Let \(\mathbb{R}:=\)equivalence class on \(C_Q\), so for any \(x\in\mathbb{R}\) you can find a Cauchy sequence \((x_n)\) in \(\mathbb{Q}\) s.t. \(\lim_{n\rightarrow\infty}|x_n - x| = 0\)