Decimal representation of the set [0,1)
Multi tool use
$begingroup$
I have encountered the next statement in statistics lecture (translated from german):
"From the analysis
you know that all but a countable number of $$w ∈ [0, 1)$$ represent a unique decimal representation
$$ω = 0.x_1x_2x_3...$$
for the countably many exceptions we choose that representation without period 9".
That sounds very counterintuitive and unprovable to me, is this true?
examples-counterexamples decimal-expansion
edited Dec 4 '18 at 14:26
José Carlos Santos
164k22131234
asked Dec 4 '18 at 14:15
AromaTheLoopAromaTheLoop
444
$endgroup$
add a comment |
$begingroup$
I have encountered the next statement in statistics lecture (translated from german):
"From the analysis
you know that all but a countable number of $$w ∈ [0, 1)$$ represent a unique decimal representation
$$ω = 0.x_1x_2x_3...$$
for the countably many exceptions we choose that representation without period 9".
That sounds very counterintuitive and unprovable to me, is this true?
examples-counterexamples decimal-expansion
edited Dec 4 '18 at 14:26
José Carlos Santos
164k22131234
asked Dec 4 '18 at 14:15
AromaTheLoopAromaTheLoop
444
$endgroup$
$begingroup$
"countable" means having the cardinality of $mathbb{N}$. in german: "abzählbar".
$endgroup$
– trancelocation
Dec 4 '18 at 14:19
$begingroup$
What exactly sounds counterintuitive and unprovable? The countability maybe?
$endgroup$
– drhab
Dec 4 '18 at 14:21
$begingroup$
@drhab José Carlos Santos answered my question below.
$endgroup$
– AromaTheLoop
Dec 4 '18 at 17:56
add a comment |
$begingroup$
I have encountered the next statement in statistics lecture (translated from german):
"From the analysis
you know that all but a countable number of $$w ∈ [0, 1)$$ represent a unique decimal representation
$$ω = 0.x_1x_2x_3...$$
for the countably many exceptions we choose that representation without period 9".
That sounds very counterintuitive and unprovable to me, is this true?
examples-counterexamples decimal-expansion
edited Dec 4 '18 at 14:26
José Carlos Santos
164k22131234
asked Dec 4 '18 at 14:15
AromaTheLoopAromaTheLoop
444
$endgroup$
I have encountered the next statement in statistics lecture (translated from german):
"From the analysis
you know that all but a countable number of $$w ∈ [0, 1)$$ represent a unique decimal representation
$$ω = 0.x_1x_2x_3...$$
for the countably many exceptions we choose that representation without period 9".
That sounds very counterintuitive and unprovable to me, is this true?
examples-counterexamples decimal-expansion
examples-counterexamples decimal-expansion
edited Dec 4 '18 at 14:26
José Carlos Santos
164k22131234
asked Dec 4 '18 at 14:15
AromaTheLoopAromaTheLoop
444
edited Dec 4 '18 at 14:26
José Carlos Santos
164k22131234
asked Dec 4 '18 at 14:15
AromaTheLoopAromaTheLoop
444
edited Dec 4 '18 at 14:26
José Carlos Santos
164k22131234
edited Dec 4 '18 at 14:26
José Carlos Santos
164k22131234
edited Dec 4 '18 at 14:26
José Carlos Santos
164k22131234
164k22131234
asked Dec 4 '18 at 14:15
AromaTheLoopAromaTheLoop
444
asked Dec 4 '18 at 14:15
AromaTheLoopAromaTheLoop
444
asked Dec 4 '18 at 14:15
AromaTheLoopAromaTheLoop
444
444
$begingroup$
"countable" means having the cardinality of $mathbb{N}$. in german: "abzählbar".
$endgroup$
– trancelocation
Dec 4 '18 at 14:19
$begingroup$
What exactly sounds counterintuitive and unprovable? The countability maybe?
$endgroup$
– drhab
Dec 4 '18 at 14:21
$begingroup$
@drhab José Carlos Santos answered my question below.
$endgroup$
– AromaTheLoop
Dec 4 '18 at 17:56
add a comment |
$begingroup$
"countable" means having the cardinality of $mathbb{N}$. in german: "abzählbar".
$endgroup$
– trancelocation
Dec 4 '18 at 14:19
$begingroup$
What exactly sounds counterintuitive and unprovable? The countability maybe?
$endgroup$
– drhab
Dec 4 '18 at 14:21
$begingroup$
@drhab José Carlos Santos answered my question below.
$endgroup$
– AromaTheLoop
Dec 4 '18 at 17:56
$begingroup$
"countable" means having the cardinality of $mathbb{N}$. in german: "abzählbar".
$endgroup$
– trancelocation
Dec 4 '18 at 14:19
$begingroup$
"countable" means having the cardinality of $mathbb{N}$. in german: "abzählbar".
$endgroup$
– trancelocation
Dec 4 '18 at 14:19
$begingroup$
What exactly sounds counterintuitive and unprovable? The countability maybe?
$endgroup$
– drhab
Dec 4 '18 at 14:21
$begingroup$
What exactly sounds counterintuitive and unprovable? The countability maybe?
$endgroup$
– drhab
Dec 4 '18 at 14:21
$begingroup$
@drhab José Carlos Santos answered my question below.
$endgroup$
– AromaTheLoop
Dec 4 '18 at 17:56
$begingroup$
@drhab José Carlos Santos answered my question below.
$endgroup$
– AromaTheLoop
Dec 4 '18 at 17:56
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This is true if “represent a” becomes “is represented by”. The exceptions are the numbers of $(0,1)$ which can be written with finitely many digits: $x=0.d_1d_2ldots d_{n-1}d_n$ (with $d_nin{1,2,3,4,5,6,7,8,9}$), because each such number can be also written as$$0.d_1d_2ldots d_{n-1}(d_n-1)999ldots$$For instance, $0.23=0.22999ldots$ All other numbers have a single decimal expansion.
answered Dec 4 '18 at 14:25
José Carlos SantosJosé Carlos Santos
164k22131234
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3025635%2fdecimal-representation-of-the-set-0-1%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This is true if “represent a” becomes “is represented by”. The exceptions are the numbers of $(0,1)$ which can be written with finitely many digits: $x=0.d_1d_2ldots d_{n-1}d_n$ (with $d_nin{1,2,3,4,5,6,7,8,9}$), because each such number can be also written as$$0.d_1d_2ldots d_{n-1}(d_n-1)999ldots$$For instance, $0.23=0.22999ldots$ All other numbers have a single decimal expansion.
answered Dec 4 '18 at 14:25
José Carlos SantosJosé Carlos Santos
164k22131234
$endgroup$
add a comment |
$begingroup$
This is true if “represent a” becomes “is represented by”. The exceptions are the numbers of $(0,1)$ which can be written with finitely many digits: $x=0.d_1d_2ldots d_{n-1}d_n$ (with $d_nin{1,2,3,4,5,6,7,8,9}$), because each such number can be also written as$$0.d_1d_2ldots d_{n-1}(d_n-1)999ldots$$For instance, $0.23=0.22999ldots$ All other numbers have a single decimal expansion.
answered Dec 4 '18 at 14:25
José Carlos SantosJosé Carlos Santos
164k22131234
$endgroup$
add a comment |
$begingroup$
This is true if “represent a” becomes “is represented by”. The exceptions are the numbers of $(0,1)$ which can be written with finitely many digits: $x=0.d_1d_2ldots d_{n-1}d_n$ (with $d_nin{1,2,3,4,5,6,7,8,9}$), because each such number can be also written as$$0.d_1d_2ldots d_{n-1}(d_n-1)999ldots$$For instance, $0.23=0.22999ldots$ All other numbers have a single decimal expansion.
answered Dec 4 '18 at 14:25
José Carlos SantosJosé Carlos Santos
164k22131234
$endgroup$
This is true if “represent a” becomes “is represented by”. The exceptions are the numbers of $(0,1)$ which can be written with finitely many digits: $x=0.d_1d_2ldots d_{n-1}d_n$ (with $d_nin{1,2,3,4,5,6,7,8,9}$), because each such number can be also written as$$0.d_1d_2ldots d_{n-1}(d_n-1)999ldots$$For instance, $0.23=0.22999ldots$ All other numbers have a single decimal expansion.
answered Dec 4 '18 at 14:25
José Carlos SantosJosé Carlos Santos
164k22131234
answered Dec 4 '18 at 14:25
José Carlos SantosJosé Carlos Santos
164k22131234
answered Dec 4 '18 at 14:25
José Carlos SantosJosé Carlos Santos
164k22131234
answered Dec 4 '18 at 14:25
José Carlos SantosJosé Carlos Santos
164k22131234
164k22131234
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3025635%2fdecimal-representation-of-the-set-0-1%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
17 v eQ3k1K9,PSdQ9L2qS
$begingroup$
"countable" means having the cardinality of $mathbb{N}$. in german: "abzählbar".
$endgroup$
– trancelocation
Dec 4 '18 at 14:19
$begingroup$
What exactly sounds counterintuitive and unprovable? The countability maybe?
$endgroup$
– drhab
Dec 4 '18 at 14:21
$begingroup$
@drhab José Carlos Santos answered my question below.
$endgroup$
– AromaTheLoop
Dec 4 '18 at 17:56