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Is it possible to geometrically construct any given real number? [closed]

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2

$begingroup$

I know some real numbers, like sqrt(2) can be constructed geometrically. But what about other real numbers, like 'e'? Is it possible to geometrically construct any given real number?

geometry analysis

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asked May 5 at 12:16

user15740user15740

1141

$endgroup$

closed as too broad by YuiTo Cheng, RRL, Yanior Weg, Cesareo, mrtaurho May 7 at 13:43

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  See this question for a discussion of whether $e$ is constructible by means other than what I mentioned.
  $endgroup$
  – 雨が好きな人
  May 5 at 12:23
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You should be precise about the meaning of "geometric construction". With a compass and (unmarked) strightedge even $sqrt[3]2$ is not constructible.
  $endgroup$
  – user
  May 5 at 12:32
  

  
  
  
  

add a comment | 

2

$begingroup$

I know some real numbers, like sqrt(2) can be constructed geometrically. But what about other real numbers, like 'e'? Is it possible to geometrically construct any given real number?

geometry analysis

share|cite|improve this question

asked May 5 at 12:16

user15740user15740

1141

$endgroup$

closed as too broad by YuiTo Cheng, RRL, Yanior Weg, Cesareo, mrtaurho May 7 at 13:43

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  See this question for a discussion of whether $e$ is constructible by means other than what I mentioned.
  $endgroup$
  – 雨が好きな人
  May 5 at 12:23
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You should be precise about the meaning of "geometric construction". With a compass and (unmarked) strightedge even $sqrt[3]2$ is not constructible.
  $endgroup$
  – user
  May 5 at 12:32
  

  
  
  
  

add a comment | 

$begingroup$

I know some real numbers, like sqrt(2) can be constructed geometrically. But what about other real numbers, like 'e'? Is it possible to geometrically construct any given real number?

geometry analysis

share|cite|improve this question

asked May 5 at 12:16

user15740user15740

1141

$endgroup$

share|cite|improve this question

asked May 5 at 12:16

user15740user15740

1141

share|cite|improve this question

asked May 5 at 12:16

user15740user15740

1141

asked May 5 at 12:16

user15740user15740

1141

asked May 5 at 12:16

user15740user15740

1141

2 Answers
2

active

oldest

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8

$begingroup$

No, it is not possible if you are only using a compass and straightedge. See constructible numbers.

$sqrt2$ can be constructed, as you say, but there are other real numbers such as $e$ that cannot. The ancient problem of squaring the circle is impossible precisely because $sqrtpi$ is not a constructible number.

share|cite|improve this answer

edited May 5 at 12:24

answered May 5 at 12:19

雨が好きな人雨が好きな人

2,097317

$endgroup$

add a comment | 

2

$begingroup$

The set of constructable numbers is the same as the closure of Z to ÷ and sqrt(). That is, you can construct ratios of two given lengths, and you can convert any given rectangle to a square.

For the polygons, a polygon n can be constructed if the euler totient ø(n) is a power of 2. This means 2^x . 3. 5. 17. 257. 65537.

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answered May 5 at 12:32

wendy.kriegerwendy.krieger

5,89311427

$endgroup$

add a comment | 

8

$begingroup$

No, it is not possible if you are only using a compass and straightedge. See constructible numbers.

$sqrt2$ can be constructed, as you say, but there are other real numbers such as $e$ that cannot. The ancient problem of squaring the circle is impossible precisely because $sqrtpi$ is not a constructible number.

share|cite|improve this answer

edited May 5 at 12:24

answered May 5 at 12:19

雨が好きな人雨が好きな人

2,097317

$endgroup$

add a comment | 

8

$begingroup$

No, it is not possible if you are only using a compass and straightedge. See constructible numbers.

$sqrt2$ can be constructed, as you say, but there are other real numbers such as $e$ that cannot. The ancient problem of squaring the circle is impossible precisely because $sqrtpi$ is not a constructible number.

share|cite|improve this answer

edited May 5 at 12:24

answered May 5 at 12:19

雨が好きな人雨が好きな人

2,097317

$endgroup$

add a comment | 

$begingroup$

No, it is not possible if you are only using a compass and straightedge. See constructible numbers.

$sqrt2$ can be constructed, as you say, but there are other real numbers such as $e$ that cannot. The ancient problem of squaring the circle is impossible precisely because $sqrtpi$ is not a constructible number.

share|cite|improve this answer

edited May 5 at 12:24

answered May 5 at 12:19

雨が好きな人雨が好きな人

2,097317

$endgroup$

share|cite|improve this answer

edited May 5 at 12:24

answered May 5 at 12:19

雨が好きな人雨が好きな人

2,097317

answered May 5 at 12:19

雨が好きな人雨が好きな人

2,097317

answered May 5 at 12:19

雨が好きな人雨が好きな人

2,097317

2

$begingroup$

The set of constructable numbers is the same as the closure of Z to ÷ and sqrt(). That is, you can construct ratios of two given lengths, and you can convert any given rectangle to a square.

For the polygons, a polygon n can be constructed if the euler totient ø(n) is a power of 2. This means 2^x . 3. 5. 17. 257. 65537.

share|cite|improve this answer

answered May 5 at 12:32

wendy.kriegerwendy.krieger

5,89311427

$endgroup$

add a comment | 

2

$begingroup$

The set of constructable numbers is the same as the closure of Z to ÷ and sqrt(). That is, you can construct ratios of two given lengths, and you can convert any given rectangle to a square.

For the polygons, a polygon n can be constructed if the euler totient ø(n) is a power of 2. This means 2^x . 3. 5. 17. 257. 65537.

share|cite|improve this answer

answered May 5 at 12:32

wendy.kriegerwendy.krieger

5,89311427

$endgroup$

add a comment | 

$begingroup$

The set of constructable numbers is the same as the closure of Z to ÷ and sqrt(). That is, you can construct ratios of two given lengths, and you can convert any given rectangle to a square.

For the polygons, a polygon n can be constructed if the euler totient ø(n) is a power of 2. This means 2^x . 3. 5. 17. 257. 65537.

share|cite|improve this answer

answered May 5 at 12:32

wendy.kriegerwendy.krieger

5,89311427

$endgroup$

share|cite|improve this answer

answered May 5 at 12:32

wendy.kriegerwendy.krieger

5,89311427

answered May 5 at 12:32

wendy.kriegerwendy.krieger

5,89311427

answered May 5 at 12:32

wendy.kriegerwendy.krieger

5,89311427