what do infinitesimals look like?
$begingroup$
i was looking at infinitesimals https://en.wikipedia.org/wiki/Hyperreal_number, and i have a few questions
(1) is every finite number in the nonstandards reals of the form x+epsilon, where x is a normal real number and epsilon is an infinitesimal?
(2) what does the nonstandard real line look like? it reminds me of the cantor set, because for every infinitesimal epsilon u can have another layer of infinitesimal with respect to it (epsilon^2), in a fractal fashion. is the nonstandard real line totally disconnected?
(3) what's the point of using this anyway? it seems to be totally disconnected and doesnt seem to have anything to do with our intuition of what a continuous line is. how can you even do analysis with it? in my brief contact with intro analysis the conditions of continuity/completeness/compactness are essential, yet this nonstandard real line looks extremely pathological.
(4) is it possible to define an ordered field structure on RxR with the lexicographical order? because in this case, u get each real number is surround by a line of "infinitesimals" and you don't get the fractal behaviour of "infinitesimal of infinitesimal" the other construction has
real-numbers infinitesimals
asked Jan 19 at 3:37
coping roidcelcoping roidcel
233
$endgroup$
add a comment |
$begingroup$
i was looking at infinitesimals https://en.wikipedia.org/wiki/Hyperreal_number, and i have a few questions
(1) is every finite number in the nonstandards reals of the form x+epsilon, where x is a normal real number and epsilon is an infinitesimal?
(2) what does the nonstandard real line look like? it reminds me of the cantor set, because for every infinitesimal epsilon u can have another layer of infinitesimal with respect to it (epsilon^2), in a fractal fashion. is the nonstandard real line totally disconnected?
(3) what's the point of using this anyway? it seems to be totally disconnected and doesnt seem to have anything to do with our intuition of what a continuous line is. how can you even do analysis with it? in my brief contact with intro analysis the conditions of continuity/completeness/compactness are essential, yet this nonstandard real line looks extremely pathological.
(4) is it possible to define an ordered field structure on RxR with the lexicographical order? because in this case, u get each real number is surround by a line of "infinitesimals" and you don't get the fractal behaviour of "infinitesimal of infinitesimal" the other construction has
real-numbers infinitesimals
asked Jan 19 at 3:37
coping roidcelcoping roidcel
233
$endgroup$
$begingroup$
You can actually compute the real derivative as the standard part of $frac{f(x+h) - f(x)} h$ with $h$ infinitesimal.
$endgroup$
– Matt Samuel
Jan 19 at 3:45
add a comment |
$begingroup$
i was looking at infinitesimals https://en.wikipedia.org/wiki/Hyperreal_number, and i have a few questions
(1) is every finite number in the nonstandards reals of the form x+epsilon, where x is a normal real number and epsilon is an infinitesimal?
(2) what does the nonstandard real line look like? it reminds me of the cantor set, because for every infinitesimal epsilon u can have another layer of infinitesimal with respect to it (epsilon^2), in a fractal fashion. is the nonstandard real line totally disconnected?
(3) what's the point of using this anyway? it seems to be totally disconnected and doesnt seem to have anything to do with our intuition of what a continuous line is. how can you even do analysis with it? in my brief contact with intro analysis the conditions of continuity/completeness/compactness are essential, yet this nonstandard real line looks extremely pathological.
(4) is it possible to define an ordered field structure on RxR with the lexicographical order? because in this case, u get each real number is surround by a line of "infinitesimals" and you don't get the fractal behaviour of "infinitesimal of infinitesimal" the other construction has
real-numbers infinitesimals
asked Jan 19 at 3:37
coping roidcelcoping roidcel
233
$endgroup$
i was looking at infinitesimals https://en.wikipedia.org/wiki/Hyperreal_number, and i have a few questions
(1) is every finite number in the nonstandards reals of the form x+epsilon, where x is a normal real number and epsilon is an infinitesimal?
(2) what does the nonstandard real line look like? it reminds me of the cantor set, because for every infinitesimal epsilon u can have another layer of infinitesimal with respect to it (epsilon^2), in a fractal fashion. is the nonstandard real line totally disconnected?
(3) what's the point of using this anyway? it seems to be totally disconnected and doesnt seem to have anything to do with our intuition of what a continuous line is. how can you even do analysis with it? in my brief contact with intro analysis the conditions of continuity/completeness/compactness are essential, yet this nonstandard real line looks extremely pathological.
(4) is it possible to define an ordered field structure on RxR with the lexicographical order? because in this case, u get each real number is surround by a line of "infinitesimals" and you don't get the fractal behaviour of "infinitesimal of infinitesimal" the other construction has
real-numbers infinitesimals
real-numbers infinitesimals
asked Jan 19 at 3:37
coping roidcelcoping roidcel
233
asked Jan 19 at 3:37
coping roidcelcoping roidcel
233
edited Jan 19 at 3:44
coping roidcel
asked Jan 19 at 3:37
coping roidcelcoping roidcel
233
asked Jan 19 at 3:37
coping roidcelcoping roidcel
233
asked Jan 19 at 3:37
coping roidcelcoping roidcel
233
233
$begingroup$
You can actually compute the real derivative as the standard part of $frac{f(x+h) - f(x)} h$ with $h$ infinitesimal.
$endgroup$
– Matt Samuel
Jan 19 at 3:45
add a comment |
$begingroup$
You can actually compute the real derivative as the standard part of $frac{f(x+h) - f(x)} h$ with $h$ infinitesimal.
$endgroup$
– Matt Samuel
Jan 19 at 3:45
$begingroup$
You can actually compute the real derivative as the standard part of $frac{f(x+h) - f(x)} h$ with $h$ infinitesimal.
$endgroup$
– Matt Samuel
Jan 19 at 3:45
$begingroup$
You can actually compute the real derivative as the standard part of $frac{f(x+h) - f(x)} h$ with $h$ infinitesimal.
$endgroup$
– Matt Samuel
Jan 19 at 3:45
add a comment |
0
active
oldest
votes
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%2f3078996%2fwhat-do-infinitesimals-look-like%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3078996%2fwhat-do-infinitesimals-look-like%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
$begingroup$
You can actually compute the real derivative as the standard part of $frac{f(x+h) - f(x)} h$ with $h$ infinitesimal.
$endgroup$
– Matt Samuel
Jan 19 at 3:45