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A geometry theory without irrational numbers?
Do 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
$begingroup$
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 9 hours ago
Eyal RothEyal Roth
1113
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 8 more comments
$begingroup$
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 9 hours ago
Eyal RothEyal Roth
1113
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
8 hours ago
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
8 hours ago
$begingroup$
@quarague Why not add it as an answer? :)
$endgroup$
– Eyal Roth
8 hours ago
$begingroup$
Irrational numbers were discovered during the early development of geometry (finding lengths of hypotenuses of right triangles). This gives an idea how limiting such a restriction would be.
$endgroup$
– Hans Engler
8 hours ago
1
$begingroup$
@EyalRoth That is surely a matter of opinion :)
$endgroup$
– Hans Engler
8 hours ago
|
show 8 more comments
$begingroup$
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 9 hours ago
Eyal RothEyal Roth
1113
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 9 hours ago
Eyal RothEyal Roth
1113
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 9 hours ago
Eyal RothEyal Roth
1113
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 8 hours ago
Eyal Roth
asked 9 hours ago
Eyal RothEyal Roth
1113
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 9 hours ago
Eyal RothEyal Roth
1113
asked 9 hours ago
Eyal RothEyal Roth
1113
1113
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
8 hours ago
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
8 hours ago
$begingroup$
@quarague Why not add it as an answer? :)
$endgroup$
– Eyal Roth
8 hours ago
$begingroup$
Irrational numbers were discovered during the early development of geometry (finding lengths of hypotenuses of right triangles). This gives an idea how limiting such a restriction would be.
$endgroup$
– Hans Engler
8 hours ago
1
$begingroup$
@EyalRoth That is surely a matter of opinion :)
$endgroup$
– Hans Engler
8 hours ago
|
show 8 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
8 hours ago
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
8 hours ago
$begingroup$
@quarague Why not add it as an answer? :)
$endgroup$
– Eyal Roth
8 hours ago
$begingroup$
Irrational numbers were discovered during the early development of geometry (finding lengths of hypotenuses of right triangles). This gives an idea how limiting such a restriction would be.
$endgroup$
– Hans Engler
8 hours ago
1
$begingroup$
@EyalRoth That is surely a matter of opinion :)
$endgroup$
– Hans Engler
8 hours ago
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
8 hours ago
$begingroup$
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
8 hours ago
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
8 hours ago
$begingroup$
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
8 hours ago
$begingroup$
@quarague Why not add it as an answer? :)
$endgroup$
– Eyal Roth
8 hours ago
$begingroup$
@quarague Why not add it as an answer? :)
$endgroup$
– Eyal Roth
8 hours ago
$begingroup$
Irrational numbers were discovered during the early development of geometry (finding lengths of hypotenuses of right triangles). This gives an idea how limiting such a restriction would be.
$endgroup$
– Hans Engler
8 hours ago
$begingroup$
Irrational numbers were discovered during the early development of geometry (finding lengths of hypotenuses of right triangles). This gives an idea how limiting such a restriction would be.
$endgroup$
– Hans Engler
8 hours ago
1
1
$begingroup$
@EyalRoth That is surely a matter of opinion :)
$endgroup$
– Hans Engler
8 hours ago
$begingroup$
@EyalRoth That is surely a matter of opinion :)
$endgroup$
– Hans Engler
8 hours ago
|
show 8 more comments
2 Answers
2
active
oldest
votes
$begingroup$
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 8 hours ago
Chris MoorheadChris Moorhead
1095
$endgroup$
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
8 hours ago
$begingroup$
@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
7 hours ago
$begingroup$
@EyalRoth That's good, but even so, keep an eye on your watch as you budget time to sink into that material.
$endgroup$
– rschwieb
7 hours ago
$begingroup$
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
7 hours ago
add a comment |
$begingroup$
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 8 hours ago
quaraguequarague
621312
$endgroup$
$begingroup$
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
7 hours ago
add a comment |
Your Answer
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2 Answers
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active
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2 Answers
2
active
oldest
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active
oldest
votes
active
oldest
votes
$begingroup$
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 8 hours ago
Chris MoorheadChris Moorhead
1095
$endgroup$
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
8 hours ago
$begingroup$
@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
7 hours ago
$begingroup$
@EyalRoth That's good, but even so, keep an eye on your watch as you budget time to sink into that material.
$endgroup$
– rschwieb
7 hours ago
$begingroup$
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
7 hours ago
add a comment |
$begingroup$
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 8 hours ago
Chris MoorheadChris Moorhead
1095
$endgroup$
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
8 hours ago
$begingroup$
@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
7 hours ago
$begingroup$
@EyalRoth That's good, but even so, keep an eye on your watch as you budget time to sink into that material.
$endgroup$
– rschwieb
7 hours ago
$begingroup$
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
7 hours ago
add a comment |
$begingroup$
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 8 hours ago
Chris MoorheadChris Moorhead
1095
$endgroup$
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 8 hours ago
Chris MoorheadChris Moorhead
1095
answered 8 hours ago
Chris MoorheadChris Moorhead
1095
answered 8 hours ago
Chris MoorheadChris Moorhead
1095
answered 8 hours ago
Chris MoorheadChris Moorhead
1095
1095
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
8 hours ago
$begingroup$
@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
7 hours ago
$begingroup$
@EyalRoth That's good, but even so, keep an eye on your watch as you budget time to sink into that material.
$endgroup$
– rschwieb
7 hours ago
$begingroup$
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
7 hours ago
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
8 hours ago
$begingroup$
@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
7 hours ago
$begingroup$
@EyalRoth That's good, but even so, keep an eye on your watch as you budget time to sink into that material.
$endgroup$
– rschwieb
7 hours ago
$begingroup$
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
7 hours ago
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
8 hours ago
$begingroup$
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
8 hours ago
$begingroup$
@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
7 hours ago
$begingroup$
@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
7 hours ago
$begingroup$
@EyalRoth That's good, but even so, keep an eye on your watch as you budget time to sink into that material.
$endgroup$
– rschwieb
7 hours ago
$begingroup$
@EyalRoth That's good, but even so, keep an eye on your watch as you budget time to sink into that material.
$endgroup$
– rschwieb
7 hours ago
$begingroup$
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
7 hours ago
$begingroup$
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
7 hours ago
add a comment |
$begingroup$
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 8 hours ago
quaraguequarague
621312
$endgroup$
$begingroup$
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
7 hours ago
add a comment |
$begingroup$
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 8 hours ago
quaraguequarague
621312
$endgroup$
$begingroup$
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
7 hours ago
add a comment |
$begingroup$
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 8 hours ago
quaraguequarague
621312
$endgroup$
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 8 hours ago
quaraguequarague
621312
edited 8 hours ago
answered 8 hours ago
quaraguequarague
621312
answered 8 hours ago
quaraguequarague
621312
answered 8 hours ago
quaraguequarague
621312
621312
$begingroup$
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
7 hours ago
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
7 hours ago
$begingroup$
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.
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– rschwieb
7 hours ago
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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.
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– rschwieb
7 hours ago
add a comment |
Eyal Roth is a new contributor. Be nice, and check out our Code of Conduct.
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2
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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).
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– Dirk
8 hours ago
2
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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.
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– quarague
8 hours ago
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@quarague Why not add it as an answer? :)
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– Eyal Roth
8 hours ago
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Irrational numbers were discovered during the early development of geometry (finding lengths of hypotenuses of right triangles). This gives an idea how limiting such a restriction would be.
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– Hans Engler
8 hours ago
1
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@EyalRoth That is surely a matter of opinion :)
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– Hans Engler
8 hours ago