Constructing a Line Segment Equal to a Circle's Circumference?

In summary, the conversation discusses the possibility of constructing a line segment with a length equal to the circumference of a circle using only a ruler and compass. The answer is "no" due to the transcendency of pi, which cannot be constructed with straightedge and compass. The concept of constructible lengths being included in certain Galois extensions of the rational numbers is also mentioned.
  • #1
Trysse
46
10
Is there any way to construct a line segment, that has the lenght of the circumference of a circle using only a ruler and a compass?

My intuition says "no"

Or phrasing the question in another way: given two line segments, can I prove, that the longer line segment has the length of the circumference of a circle to which the shorter line segment is the radius/diameter?

Or In another way:, can I construct a circle, that has a circumference equal to a given line segment?

My intuition for "no" is based in the irrationality of Pi.
 
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  • #2
Trysse said:
My intuition for "no" is based in the irrationality of Pi.
Not the irrationality. Square root of two is irrational, but you can construct it. It is the diagonal of a square with unit length side.
 
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  • #3
"Squareing the circle" is known to be impossible with ruler and compass. I suspect this is the problem here, since having squared the circle (ie. constructed a line segment of length [itex]\sqrt{\pi}r[/itex] you can with ruler and compass construct a line segment of length [itex]\pi r^2[/itex], then from that and a line segment of length [itex]r[/itex] construct a segment of length [itex]\pi r[/itex], and finally double that to get a line segment of length [itex]2\pi r[/itex].
 
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  • #4
Trysse said:
Is there any way to construct a line segment, that has the lenght of the circumference of a circle using only a ruler and a compass?

My intuition says "no"

Or phrasing the question in another way: given two line segments, can I prove, that the longer line segment has the length of the circumference of a circle to which the shorter line segment is the radius/diameter?

Or In another way:, can I construct a circle, that has a circumference equal to a given line segment?

My intuition for "no" is based in the irrationality of Pi.
The answer is "no" due to the transcendency of ##\pi##. This means that ##\pi## cannot be written as
$$
0=a_n\pi^n +a_{n-1}\pi^{n-1}+\ldots+a_2\pi^2+a_1\pi+a_0
$$
with integers ##a_0,\ldots,a_n.##

The technical reason is: ##\pi## is not included in any Galois extension of the rational numbers.

Constructible lengths are included in certain Galois extensions of the rational numbers, namely those of degrees being a power of two, IIRC.
 
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  • #5
Thanks, that was helpful.
martinbn said:
Not the irrationality. Square root of two is irrational, but you can construct it.
Good point.
pasmith said:
"Squareing the circle" is known to be impossible with ruler and compass.
I knew that "squaring the circle" was impossible, but I did not make the connection.
fresh_42 said:
The answer is "no" due to the transcendency of π.
I knew the term transcendency but was unaware that this had a geometric consequence.
 
  • #6
fresh_42 said:
Constructible lengths are included in certain Galois extensions of the rational numbers, namely those of degrees being a power of two, IIRC.
It's even stricter than this. Operations with a straightedge and compass can only introduce square roots, so a complex number number is constructible if and only if it is in a field extension of ##\mathbb{Q}## generated by iterated square roots. Not every extension of degree ##2^n## is a tower of quadratic extensions (and I also don't see why the extensions being Galois should be relevant).
 
  • #7
Infrared said:
It's even stricter than this. Operations with a straightedge and compass can only introduce square roots, so a complex number number is constructible if and only if it is in a field extension of ##\mathbb{Q}## generated by iterated square roots. Not every extension of degree ##2^n## is a tower of quadratic extensions (and I also don't see why the extensions being Galois should be relevant).
IIRC was short for: "I am too lazy to look it up." van der Waerden had it around Galois-theory, so I took what was left in my memory. The statement itself wasn't false.
 
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1. How do you construct a line segment equal to a circle's circumference?

To construct a line segment equal to a circle's circumference, you will need a compass and a straightedge. First, draw a circle with the desired circumference. Then, place the compass at any point on the circle and draw an arc that intersects the circle at two points. Without changing the compass width, place the compass at the other intersection point and draw another arc that intersects the first arc. Finally, use the straightedge to connect the two intersection points. This line segment will be equal to the circle's circumference.

2. Why is it important to construct a line segment equal to a circle's circumference?

Constructing a line segment equal to a circle's circumference is important for various mathematical and engineering applications. It allows for accurate measurements and calculations involving circles, such as finding the area and circumference of a circle, or creating geometric shapes with precise proportions.

3. Can a line segment be constructed to have a circumference greater than a circle's circumference?

No, a line segment cannot have a circumference greater than a circle's circumference. This is because a circle is a closed curve with a constant radius, while a line segment is a straight line with no curvature. Therefore, the length of a line segment will always be shorter than the circumference of a circle with the same radius.

4. Is it possible to construct a line segment equal to a circle's circumference without using a compass?

Yes, it is possible to construct a line segment equal to a circle's circumference without using a compass. One method is to use a string or a strip of paper to measure the circumference of the circle, and then use the same length to mark the line segment on a straightedge. Another method is to use a ruler to measure the diameter of the circle, and then multiply it by pi (π) to get the circumference, which can then be marked on the straightedge.

5. Can a line segment be constructed to have a circumference less than a circle's circumference?

Yes, a line segment can be constructed to have a circumference less than a circle's circumference. This can be achieved by drawing a circle with a larger radius and following the same steps as in question 1 to construct a line segment with a length equal to the original circle's circumference. This line segment will have a smaller circumference compared to the original circle.

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