A geometry theory without irrational numbers? The 2019 Stack Overflow Developer Survey Results Are InDo mathematicians, in the end, always agree?Real Numbers to Irrational PowersInfinite irrational number sequences?Do irrational numbers have equivalence classes the way rational numbers do?Are there numbers that if proven rational (or irrational) will have important consequences to mathematics?Are irrational numbers irrational by nature?Rational mean of irrational numbers?Is there a “positive” definition for irrational numbers?Geometric proofs outside euclidean geometryHow many Irrational numbers?Continued fractions of rational vs irrational numbers
Why don't Unix/Linux systems traverse through directories until they find the required version of a linked library?
Access elements in std::string where positon of string is greater than its size
Output the Arecibo Message
Is there a name of the flying bionic bird?
JSON.serialize: is it possible to suppress null values of a map?
What is the motivation for a law requiring 2 parties to consent for recording a conversation
Is it possible for the two major parties in the UK to form a coalition with each other instead of a much smaller party?
Limit the amount of RAM Mathematica may access?
On the insanity of kings as an argument against monarchy
What is a mixture ratio of propellant?
What does Linus Torvalds means when he says that git "never ever" tracks a file?
Landlord wants to switch my lease to a "Land contract" to "get back at the city"
How come people say “Would of”?
If the Wish spell is used to duplicate the effect of Simulacrum, are existing duplicates destroyed?
Does light intensity oscillate really fast since it is a wave?
Pristine Bit Checking
Extreme, unacceptable situation and I can't attend work tomorrow morning
What is the meaning of Triage in Cybersec world?
I looked up a future colleague on LinkedIn before I started a job. I told my colleague about it and he seemed surprised. Should I apologize?
What are the motivations for publishing new editions of an existing textbook, beyond new discoveries in a field?
Confusion about non-derivable continuous functions
What could be the right powersource for 15 seconds lifespan disposable giant chainsaw?
The difference between dialogue marks
Realistic Alternatives to Dust: What Else Could Feed a Plankton Bloom?
A geometry theory without irrational numbers?
The 2019 Stack Overflow Developer Survey Results Are InDo mathematicians, in the end, always agree?Real Numbers to Irrational PowersInfinite irrational number sequences?Do irrational numbers have equivalence classes the way rational numbers do?Are there numbers that if proven rational (or irrational) will have important consequences to mathematics?Are irrational numbers irrational by nature?Rational mean of irrational numbers?Is there a “positive” definition for irrational numbers?Geometric proofs outside euclidean geometryHow many Irrational numbers?Continued fractions of rational vs irrational numbers
0
$begingroup$
Is there any theory or theorem of geometry -- whether used in practice or not -- which denies or forbids the use of irrational numbers?
If not, were there any notable attempts at it?
Disclaimer: I am not looking for a proof for the existence of irrational number.
geometry math-history irrational-numbers
asked Apr 4 at 13:54
Eyal RothEyal Roth
1757
New contributor
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
|
show 10 more comments
0
$begingroup$
Is there any theory or theorem of geometry -- whether used in practice or not -- which denies or forbids the use of irrational numbers?
If not, were there any notable attempts at it?
Disclaimer: I am not looking for a proof for the existence of irrational number.
geometry math-history irrational-numbers
asked Apr 4 at 13:54
Eyal RothEyal Roth
1757
New contributor
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
2
$begingroup$
A geometrically interesting subset of the real numbers are the constructible numbers, you can find some information on that on Wikipedia and read into it from there if interested. However, these also include some irrational numbers (but not all).
$endgroup$
– Dirk
Apr 4 at 13:58
2
$begingroup$
Have you heard of finite geometry, as in: en.wikipedia.org/wiki/Finite_geometry ? This is geometry where there are only a fintie numbre of points, hence you can assign them all natural numbers.
$endgroup$
– quarague
Apr 4 at 13:59
1
$begingroup$
@EyalRoth That is surely a matter of opinion :)
$endgroup$
– Hans Engler
Apr 4 at 14:28
1
$begingroup$
I recall in the book Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity, which was about the mathematicians who first promoted the idea of infinity and set theory, and their religious proclivities, that at a large math conference at the time (around 1880?) one of the great mathematicians proclaimed that all of math would be described using "integer alone". Sorry I can't give you a better reference, but it would be worth reading the whole book on its own, if not only to find the reference.
$endgroup$
– user151841
Apr 4 at 18:15
1
$begingroup$
@EyalRoth The author, Loren Graham, seems to have good credentials: history.mit.edu/people/loren-r-graham
$endgroup$
– user151841
yesterday
|
show 10 more comments
0
0
0
$begingroup$
Is there any theory or theorem of geometry -- whether used in practice or not -- which denies or forbids the use of irrational numbers?
If not, were there any notable attempts at it?
Disclaimer: I am not looking for a proof for the existence of irrational number.
geometry math-history irrational-numbers
asked Apr 4 at 13:54
Eyal RothEyal Roth
1757
New contributor
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Is there any theory or theorem of geometry -- whether used in practice or not -- which denies or forbids the use of irrational numbers?
If not, were there any notable attempts at it?
Disclaimer: I am not looking for a proof for the existence of irrational number.
geometry math-history irrational-numbers
geometry math-history irrational-numbers
asked Apr 4 at 13:54
Eyal RothEyal Roth
1757
New contributor
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Apr 4 at 13:54
Eyal RothEyal Roth
1757
New contributor
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Apr 4 at 14:05
Eyal Roth
asked Apr 4 at 13:54
Eyal RothEyal Roth
1757
New contributor
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Apr 4 at 13:54
Eyal RothEyal Roth
1757
asked Apr 4 at 13:54
Eyal RothEyal Roth
1757
1757
New contributor
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
2
$begingroup$
A geometrically interesting subset of the real numbers are the constructible numbers, you can find some information on that on Wikipedia and read into it from there if interested. However, these also include some irrational numbers (but not all).
$endgroup$
– Dirk
Apr 4 at 13:58
2
$begingroup$
Have you heard of finite geometry, as in: en.wikipedia.org/wiki/Finite_geometry ? This is geometry where there are only a fintie numbre of points, hence you can assign them all natural numbers.
$endgroup$
– quarague
Apr 4 at 13:59
1
$begingroup$
@EyalRoth That is surely a matter of opinion :)
$endgroup$
– Hans Engler
Apr 4 at 14:28
1
$begingroup$
I recall in the book Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity, which was about the mathematicians who first promoted the idea of infinity and set theory, and their religious proclivities, that at a large math conference at the time (around 1880?) one of the great mathematicians proclaimed that all of math would be described using "integer alone". Sorry I can't give you a better reference, but it would be worth reading the whole book on its own, if not only to find the reference.
$endgroup$
– user151841
Apr 4 at 18:15
1
$begingroup$
@EyalRoth The author, Loren Graham, seems to have good credentials: history.mit.edu/people/loren-r-graham
$endgroup$
– user151841
yesterday
|
show 10 more comments
2
$begingroup$
A geometrically interesting subset of the real numbers are the constructible numbers, you can find some information on that on Wikipedia and read into it from there if interested. However, these also include some irrational numbers (but not all).
$endgroup$
– Dirk
Apr 4 at 13:58
2
$begingroup$
Have you heard of finite geometry, as in: en.wikipedia.org/wiki/Finite_geometry ? This is geometry where there are only a fintie numbre of points, hence you can assign them all natural numbers.
$endgroup$
– quarague
Apr 4 at 13:59
1
$begingroup$
@EyalRoth That is surely a matter of opinion :)
$endgroup$
– Hans Engler
Apr 4 at 14:28
1
$begingroup$
I recall in the book Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity, which was about the mathematicians who first promoted the idea of infinity and set theory, and their religious proclivities, that at a large math conference at the time (around 1880?) one of the great mathematicians proclaimed that all of math would be described using "integer alone". Sorry I can't give you a better reference, but it would be worth reading the whole book on its own, if not only to find the reference.
$endgroup$
– user151841
Apr 4 at 18:15
1
$begingroup$
@EyalRoth The author, Loren Graham, seems to have good credentials: history.mit.edu/people/loren-r-graham
$endgroup$
– user151841
yesterday
2
2
$begingroup$
A geometrically interesting subset of the real numbers are the constructible numbers, you can find some information on that on Wikipedia and read into it from there if interested. However, these also include some irrational numbers (but not all).
$endgroup$
– Dirk
Apr 4 at 13:58
$begingroup$
A geometrically interesting subset of the real numbers are the constructible numbers, you can find some information on that on Wikipedia and read into it from there if interested. However, these also include some irrational numbers (but not all).
$endgroup$
– Dirk
Apr 4 at 13:58
2
2
$begingroup$
Have you heard of finite geometry, as in: en.wikipedia.org/wiki/Finite_geometry ? This is geometry where there are only a fintie numbre of points, hence you can assign them all natural numbers.
$endgroup$
– quarague
Apr 4 at 13:59
$begingroup$
Have you heard of finite geometry, as in: en.wikipedia.org/wiki/Finite_geometry ? This is geometry where there are only a fintie numbre of points, hence you can assign them all natural numbers.
$endgroup$
– quarague
Apr 4 at 13:59
1
1
$begingroup$
@EyalRoth That is surely a matter of opinion :)
$endgroup$
– Hans Engler
Apr 4 at 14:28
$begingroup$
@EyalRoth That is surely a matter of opinion :)
$endgroup$
– Hans Engler
Apr 4 at 14:28
1
1
$begingroup$
I recall in the book Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity, which was about the mathematicians who first promoted the idea of infinity and set theory, and their religious proclivities, that at a large math conference at the time (around 1880?) one of the great mathematicians proclaimed that all of math would be described using "integer alone". Sorry I can't give you a better reference, but it would be worth reading the whole book on its own, if not only to find the reference.
$endgroup$
– user151841
Apr 4 at 18:15
$begingroup$
I recall in the book Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity, which was about the mathematicians who first promoted the idea of infinity and set theory, and their religious proclivities, that at a large math conference at the time (around 1880?) one of the great mathematicians proclaimed that all of math would be described using "integer alone". Sorry I can't give you a better reference, but it would be worth reading the whole book on its own, if not only to find the reference.
$endgroup$
– user151841
Apr 4 at 18:15
1
1
$begingroup$
@EyalRoth The author, Loren Graham, seems to have good credentials: history.mit.edu/people/loren-r-graham
$endgroup$
– user151841
yesterday
$begingroup$
@EyalRoth The author, Loren Graham, seems to have good credentials: history.mit.edu/people/loren-r-graham
$endgroup$
– user151841
yesterday
|
show 10 more comments
2 Answers
2
active
oldest
votes
4
$begingroup$
I don't know how helpful you will find it, but there are videos on YouTube by njwildberger on rational trigonometry. The main idea is to avoid taking square roots and deal with squares of lengths and ratios between them. He calls it quadrance.
https://www.youtube.com/watch?v=GGj399xIssQ&list=PL3C58498718451C47
http://www.wildegg.com/intro-rational-trig.html
Trouble is, the irrational approach seems to be working fine so there is no reason to completely overhaul the system.
answered Apr 4 at 14:00
Chris MoorheadChris Moorhead
1176
$endgroup$
8
$begingroup$
It should also be mentioned, however, the njwildberger is considered a bit of a contrarian on the fringes and that one should be ready with a grain of salt when consuming his material. If you (eyal roth, the original poster) do not have a lot of mathematical maturity, his message might be more confusing/distracting than informative. I'm far from an expert on his subject area though, and maybe some of it stands up better than the negative parts I have heard about.
$endgroup$
– rschwieb
Apr 4 at 14:52
$begingroup$
@rschwieb Thanks for the warning. I'm quite agnostic in nature, so I tend to employ a lot of critical thinking and try to figure out things on my own before I accept a proposition.
$endgroup$
– Eyal Roth
Apr 4 at 14:58
1
$begingroup$
@EyalRoth That's good, but even so, keep an eye on your watch as you budget time to sink into that material.
$endgroup$
– rschwieb
Apr 4 at 15:00
$begingroup$
I agree, he is somewhat eccentric, but I can see the rationale behind some of his objections. I think the rational trig idea is more that he thinks it would be easier to teach because it is more intuitive and teaches you a geometry closer to the Greek's understanding. But for someone who has learned the existing system, it is like trying to learn to write with your other hand.
$endgroup$
– Chris Moorhead
Apr 4 at 15:05
add a comment |
4
$begingroup$
Have you heard of finite geometry, as in: en.wikipedia.org/wiki/Finite_geometry ? This is geometry where there are only a finite number of points. So you don't even need rationals, natural numbers suffice.
answered Apr 4 at 14:07
quaraguequarague
693312
$endgroup$
$begingroup$
Well, the natural numbers "sort of" suffice. The things that are being used as coordinates in finite geometries aren't really like natural numbers either (there's no order, for example.) . But in terms of there only being finitely many things in the field, yeah, you wouldn't need "as many" things in your system of numbers.
$endgroup$
– rschwieb
Apr 4 at 14:56
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
);
);
Eyal Roth is a new contributor. Be nice, and check out our Code of Conduct.
draft saved
draft discarded
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%2f3174657%2fa-geometry-theory-without-irrational-numbers%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
4
$begingroup$
I don't know how helpful you will find it, but there are videos on YouTube by njwildberger on rational trigonometry. The main idea is to avoid taking square roots and deal with squares of lengths and ratios between them. He calls it quadrance.
https://www.youtube.com/watch?v=GGj399xIssQ&list=PL3C58498718451C47
http://www.wildegg.com/intro-rational-trig.html
Trouble is, the irrational approach seems to be working fine so there is no reason to completely overhaul the system.
answered Apr 4 at 14:00
Chris MoorheadChris Moorhead
1176
$endgroup$
8
$begingroup$
It should also be mentioned, however, the njwildberger is considered a bit of a contrarian on the fringes and that one should be ready with a grain of salt when consuming his material. If you (eyal roth, the original poster) do not have a lot of mathematical maturity, his message might be more confusing/distracting than informative. I'm far from an expert on his subject area though, and maybe some of it stands up better than the negative parts I have heard about.
$endgroup$
– rschwieb
Apr 4 at 14:52
$begingroup$
@rschwieb Thanks for the warning. I'm quite agnostic in nature, so I tend to employ a lot of critical thinking and try to figure out things on my own before I accept a proposition.
$endgroup$
– Eyal Roth
Apr 4 at 14:58
1
$begingroup$
@EyalRoth That's good, but even so, keep an eye on your watch as you budget time to sink into that material.
$endgroup$
– rschwieb
Apr 4 at 15:00
$begingroup$
I agree, he is somewhat eccentric, but I can see the rationale behind some of his objections. I think the rational trig idea is more that he thinks it would be easier to teach because it is more intuitive and teaches you a geometry closer to the Greek's understanding. But for someone who has learned the existing system, it is like trying to learn to write with your other hand.
$endgroup$
– Chris Moorhead
Apr 4 at 15:05
add a comment |
4
$begingroup$
I don't know how helpful you will find it, but there are videos on YouTube by njwildberger on rational trigonometry. The main idea is to avoid taking square roots and deal with squares of lengths and ratios between them. He calls it quadrance.
https://www.youtube.com/watch?v=GGj399xIssQ&list=PL3C58498718451C47
http://www.wildegg.com/intro-rational-trig.html
Trouble is, the irrational approach seems to be working fine so there is no reason to completely overhaul the system.
answered Apr 4 at 14:00
Chris MoorheadChris Moorhead
1176
$endgroup$
8
$begingroup$
It should also be mentioned, however, the njwildberger is considered a bit of a contrarian on the fringes and that one should be ready with a grain of salt when consuming his material. If you (eyal roth, the original poster) do not have a lot of mathematical maturity, his message might be more confusing/distracting than informative. I'm far from an expert on his subject area though, and maybe some of it stands up better than the negative parts I have heard about.
$endgroup$
– rschwieb
Apr 4 at 14:52
$begingroup$
@rschwieb Thanks for the warning. I'm quite agnostic in nature, so I tend to employ a lot of critical thinking and try to figure out things on my own before I accept a proposition.
$endgroup$
– Eyal Roth
Apr 4 at 14:58
1
$begingroup$
@EyalRoth That's good, but even so, keep an eye on your watch as you budget time to sink into that material.
$endgroup$
– rschwieb
Apr 4 at 15:00
$begingroup$
I agree, he is somewhat eccentric, but I can see the rationale behind some of his objections. I think the rational trig idea is more that he thinks it would be easier to teach because it is more intuitive and teaches you a geometry closer to the Greek's understanding. But for someone who has learned the existing system, it is like trying to learn to write with your other hand.
$endgroup$
– Chris Moorhead
Apr 4 at 15:05
add a comment |
4
4
4
$begingroup$
I don't know how helpful you will find it, but there are videos on YouTube by njwildberger on rational trigonometry. The main idea is to avoid taking square roots and deal with squares of lengths and ratios between them. He calls it quadrance.
https://www.youtube.com/watch?v=GGj399xIssQ&list=PL3C58498718451C47
http://www.wildegg.com/intro-rational-trig.html
Trouble is, the irrational approach seems to be working fine so there is no reason to completely overhaul the system.
answered Apr 4 at 14:00
Chris MoorheadChris Moorhead
1176
$endgroup$
I don't know how helpful you will find it, but there are videos on YouTube by njwildberger on rational trigonometry. The main idea is to avoid taking square roots and deal with squares of lengths and ratios between them. He calls it quadrance.
https://www.youtube.com/watch?v=GGj399xIssQ&list=PL3C58498718451C47
http://www.wildegg.com/intro-rational-trig.html
Trouble is, the irrational approach seems to be working fine so there is no reason to completely overhaul the system.
answered Apr 4 at 14:00
Chris MoorheadChris Moorhead
1176
answered Apr 4 at 14:00
Chris MoorheadChris Moorhead
1176
answered Apr 4 at 14:00
Chris MoorheadChris Moorhead
1176
answered Apr 4 at 14:00
Chris MoorheadChris Moorhead
1176
1176
8
$begingroup$
It should also be mentioned, however, the njwildberger is considered a bit of a contrarian on the fringes and that one should be ready with a grain of salt when consuming his material. If you (eyal roth, the original poster) do not have a lot of mathematical maturity, his message might be more confusing/distracting than informative. I'm far from an expert on his subject area though, and maybe some of it stands up better than the negative parts I have heard about.
$endgroup$
– rschwieb
Apr 4 at 14:52
$begingroup$
@rschwieb Thanks for the warning. I'm quite agnostic in nature, so I tend to employ a lot of critical thinking and try to figure out things on my own before I accept a proposition.
$endgroup$
– Eyal Roth
Apr 4 at 14:58
1
$begingroup$
@EyalRoth That's good, but even so, keep an eye on your watch as you budget time to sink into that material.
$endgroup$
– rschwieb
Apr 4 at 15:00
$begingroup$
I agree, he is somewhat eccentric, but I can see the rationale behind some of his objections. I think the rational trig idea is more that he thinks it would be easier to teach because it is more intuitive and teaches you a geometry closer to the Greek's understanding. But for someone who has learned the existing system, it is like trying to learn to write with your other hand.
$endgroup$
– Chris Moorhead
Apr 4 at 15:05
add a comment |
8
$begingroup$
It should also be mentioned, however, the njwildberger is considered a bit of a contrarian on the fringes and that one should be ready with a grain of salt when consuming his material. If you (eyal roth, the original poster) do not have a lot of mathematical maturity, his message might be more confusing/distracting than informative. I'm far from an expert on his subject area though, and maybe some of it stands up better than the negative parts I have heard about.
$endgroup$
– rschwieb
Apr 4 at 14:52
$begingroup$
@rschwieb Thanks for the warning. I'm quite agnostic in nature, so I tend to employ a lot of critical thinking and try to figure out things on my own before I accept a proposition.
$endgroup$
– Eyal Roth
Apr 4 at 14:58
1
$begingroup$
@EyalRoth That's good, but even so, keep an eye on your watch as you budget time to sink into that material.
$endgroup$
– rschwieb
Apr 4 at 15:00
$begingroup$
I agree, he is somewhat eccentric, but I can see the rationale behind some of his objections. I think the rational trig idea is more that he thinks it would be easier to teach because it is more intuitive and teaches you a geometry closer to the Greek's understanding. But for someone who has learned the existing system, it is like trying to learn to write with your other hand.
$endgroup$
– Chris Moorhead
Apr 4 at 15:05
8
8
$begingroup$
It should also be mentioned, however, the njwildberger is considered a bit of a contrarian on the fringes and that one should be ready with a grain of salt when consuming his material. If you (eyal roth, the original poster) do not have a lot of mathematical maturity, his message might be more confusing/distracting than informative. I'm far from an expert on his subject area though, and maybe some of it stands up better than the negative parts I have heard about.
$endgroup$
– rschwieb
Apr 4 at 14:52
$begingroup$
It should also be mentioned, however, the njwildberger is considered a bit of a contrarian on the fringes and that one should be ready with a grain of salt when consuming his material. If you (eyal roth, the original poster) do not have a lot of mathematical maturity, his message might be more confusing/distracting than informative. I'm far from an expert on his subject area though, and maybe some of it stands up better than the negative parts I have heard about.
$endgroup$
– rschwieb
Apr 4 at 14:52
$begingroup$
@rschwieb Thanks for the warning. I'm quite agnostic in nature, so I tend to employ a lot of critical thinking and try to figure out things on my own before I accept a proposition.
$endgroup$
– Eyal Roth
Apr 4 at 14:58
$begingroup$
@rschwieb Thanks for the warning. I'm quite agnostic in nature, so I tend to employ a lot of critical thinking and try to figure out things on my own before I accept a proposition.
$endgroup$
– Eyal Roth
Apr 4 at 14:58
1
1
$begingroup$
@EyalRoth That's good, but even so, keep an eye on your watch as you budget time to sink into that material.
$endgroup$
– rschwieb
Apr 4 at 15:00
$begingroup$
@EyalRoth That's good, but even so, keep an eye on your watch as you budget time to sink into that material.
$endgroup$
– rschwieb
Apr 4 at 15:00
$begingroup$
I agree, he is somewhat eccentric, but I can see the rationale behind some of his objections. I think the rational trig idea is more that he thinks it would be easier to teach because it is more intuitive and teaches you a geometry closer to the Greek's understanding. But for someone who has learned the existing system, it is like trying to learn to write with your other hand.
$endgroup$
– Chris Moorhead
Apr 4 at 15:05
$begingroup$
I agree, he is somewhat eccentric, but I can see the rationale behind some of his objections. I think the rational trig idea is more that he thinks it would be easier to teach because it is more intuitive and teaches you a geometry closer to the Greek's understanding. But for someone who has learned the existing system, it is like trying to learn to write with your other hand.
$endgroup$
– Chris Moorhead
Apr 4 at 15:05
add a comment |
4
$begingroup$
Have you heard of finite geometry, as in: en.wikipedia.org/wiki/Finite_geometry ? This is geometry where there are only a finite number of points. So you don't even need rationals, natural numbers suffice.
answered Apr 4 at 14:07
quaraguequarague
693312
$endgroup$
$begingroup$
Well, the natural numbers "sort of" suffice. The things that are being used as coordinates in finite geometries aren't really like natural numbers either (there's no order, for example.) . But in terms of there only being finitely many things in the field, yeah, you wouldn't need "as many" things in your system of numbers.
$endgroup$
– rschwieb
Apr 4 at 14:56
add a comment |
4
$begingroup$
Have you heard of finite geometry, as in: en.wikipedia.org/wiki/Finite_geometry ? This is geometry where there are only a finite number of points. So you don't even need rationals, natural numbers suffice.
answered Apr 4 at 14:07
quaraguequarague
693312
$endgroup$
$begingroup$
Well, the natural numbers "sort of" suffice. The things that are being used as coordinates in finite geometries aren't really like natural numbers either (there's no order, for example.) . But in terms of there only being finitely many things in the field, yeah, you wouldn't need "as many" things in your system of numbers.
$endgroup$
– rschwieb
Apr 4 at 14:56
add a comment |
4
4
4
$begingroup$
Have you heard of finite geometry, as in: en.wikipedia.org/wiki/Finite_geometry ? This is geometry where there are only a finite number of points. So you don't even need rationals, natural numbers suffice.
answered Apr 4 at 14:07
quaraguequarague
693312
$endgroup$
Have you heard of finite geometry, as in: en.wikipedia.org/wiki/Finite_geometry ? This is geometry where there are only a finite number of points. So you don't even need rationals, natural numbers suffice.
answered Apr 4 at 14:07
quaraguequarague
693312
edited Apr 4 at 14:41
answered Apr 4 at 14:07
quaraguequarague
693312
answered Apr 4 at 14:07
quaraguequarague
693312
answered Apr 4 at 14:07
quaraguequarague
693312
693312
$begingroup$
Well, the natural numbers "sort of" suffice. The things that are being used as coordinates in finite geometries aren't really like natural numbers either (there's no order, for example.) . But in terms of there only being finitely many things in the field, yeah, you wouldn't need "as many" things in your system of numbers.
$endgroup$
– rschwieb
Apr 4 at 14:56
add a comment |
$begingroup$
Well, the natural numbers "sort of" suffice. The things that are being used as coordinates in finite geometries aren't really like natural numbers either (there's no order, for example.) . But in terms of there only being finitely many things in the field, yeah, you wouldn't need "as many" things in your system of numbers.
$endgroup$
– rschwieb
Apr 4 at 14:56
$begingroup$
Well, the natural numbers "sort of" suffice. The things that are being used as coordinates in finite geometries aren't really like natural numbers either (there's no order, for example.) . But in terms of there only being finitely many things in the field, yeah, you wouldn't need "as many" things in your system of numbers.
$endgroup$
– rschwieb
Apr 4 at 14:56
$begingroup$
Well, the natural numbers "sort of" suffice. The things that are being used as coordinates in finite geometries aren't really like natural numbers either (there's no order, for example.) . But in terms of there only being finitely many things in the field, yeah, you wouldn't need "as many" things in your system of numbers.
$endgroup$
– rschwieb
Apr 4 at 14:56
add a comment |
Eyal Roth is a new contributor. Be nice, and check out our Code of Conduct.
draft saved
draft discarded
Eyal Roth is a new contributor. Be nice, and check out our Code of Conduct.
Eyal Roth is a new contributor. Be nice, and check out our Code of Conduct.
Eyal Roth is a new contributor. Be nice, and check out our Code of Conduct.
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.
draft saved
draft discarded
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%2f3174657%2fa-geometry-theory-without-irrational-numbers%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
2
$begingroup$
A geometrically interesting subset of the real numbers are the constructible numbers, you can find some information on that on Wikipedia and read into it from there if interested. However, these also include some irrational numbers (but not all).
$endgroup$
– Dirk
Apr 4 at 13:58
2
$begingroup$
Have you heard of finite geometry, as in: en.wikipedia.org/wiki/Finite_geometry ? This is geometry where there are only a fintie numbre of points, hence you can assign them all natural numbers.
$endgroup$
– quarague
Apr 4 at 13:59
1
$begingroup$
@EyalRoth That is surely a matter of opinion :)
$endgroup$
– Hans Engler
Apr 4 at 14:28
1
$begingroup$
I recall in the book Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity, which was about the mathematicians who first promoted the idea of infinity and set theory, and their religious proclivities, that at a large math conference at the time (around 1880?) one of the great mathematicians proclaimed that all of math would be described using "integer alone". Sorry I can't give you a better reference, but it would be worth reading the whole book on its own, if not only to find the reference.
$endgroup$
– user151841
Apr 4 at 18:15
1
$begingroup$
@EyalRoth The author, Loren Graham, seems to have good credentials: history.mit.edu/people/loren-r-graham
$endgroup$
– user151841
yesterday