Title: Theory / Unimportant / Math / Number Theory / Hardy - Wright{ / 09 decimals
Tags: hasSeenConjDef

Name: The decimal representation of \(x\)
Definition: \(x=\sum_{i=s}^{-\infty}A_ir^i\)
Page: 138, 144
Chapter: 9
Section: 1
Variables: \(x\in\mathbb R\)
\(r\in\mathbb N^{>1}\)
Kind: Theo
Definition2: an infinite number of non-\(r-1\) digit
Conjdef: A sequence \(A_s\dots A_0.A_{-1}\dots \) with \(A_i\in[0,r-1]\)

Name: The number represented by \(A_s\dots A_0.A_{-1}\dots \)
Definition: \(\sum_{i=s}^{-\infty}A_i10^i\)
Page: 138
Chapter: 9
Section: 1
Variables: \(A_s\dots A_0.A_{-1}\dots \) with \(A_i\in[0,9]\)
Kind: Theo

Name: A number whose decimate terminates
Definition: A number whose decimals are ultimately 0
Page: 140
Chapter: 9
Section: 1

Name: A number whose decimate recur in radix \(r\)
Definition: whose decimals are ultimately periodic in radix \(r\)
Page: 141, 143
Chapter: 9
Section: 2
Variables: \(\mathcal P=\{p_i\}\) a finite family of prime
\(r=\prod_{p_i\in\mathcal P} p_i^{\alpha_i}\) a radix
Index: 135
Kind: Theo
Definition2: \(\frac{n}{\prod_{p_i\in\mathcal P}p_i^{\beta_i}}\) with \(n,\beta_i\in\mathbb N\)
Conjdef: A number, equivalently
Denoted by: A dot on the first and last decimal of the recurring sequence

Name: A number whose decimate are pure recur
Definition: A number whose decimals are periodic
Page: 141
Chapter: 9
Section: 2

Name: A number whose decimate are mixed recurring
Definition: A number whose decimals are recurring but not pure recurring
Page: 141
Chapter: 9
Section: 2

Question: The decimate in the scale of \(r\) recur after
Answer: \(\max(\lceil\frac{\beta_i}{\alpha_i}\rceil)\) digits
Page: 143, 144
Chapter: 9
Section: 2
Variables: \(\frac{p}{\prod p_i^{\beta_i}}\) a rational
\(r=\prod p_i^{\alpha_i}\)
Index: 135
Kind: Theo