Is it easier to prove that pi is transcendental or not constructible?

In summary, we discuss the concept of constructible numbers and their relation to algebraic numbers. It is popularly known that pi is a transcendental number and therefore not constructible. Many people tried to prove this fact for over 2000 years, but it was only with the development of new methods that it was finally proven. To show that pi is transcendental, one can use the Lindemann-Weierstrass theorem. It is also possible to show that pi is not algebraic, which is a simpler proof.
  • #1
eddybob123
178
0
hi. I've been working on a project lately about pi. and its unconstructiveness doesn't make sense. can you think of a way to possibly do this?
 
Mathematics news on Phys.org
  • #2
The fact that something doesn't make sense to you does not mean it isn't true! (It never made sense to me that George W. Bush was elected president of the United States, but...)

Do you understand what "constructible numbers" are? The only constructible numbers, in the sense of geometry (given a line segment of length "1" we can construct a line segment of this length using only straightedge and compasses), are those that are "algebraic of order a power of 2" (algebraic of order 1, 2, 4, 8, ...). [itex]\pi[/itex] is a transcendental number, not algebraic of any order, and so is not constructible.
 
  • #3
eddybob123 said:
hi. I've been working on a project lately about pi. and its unconstructiveness doesn't make sense. can you think of a way to possibly do this?

Well, you're in good company. The unconstructiveness of pi didn't make sense to a lot of people. In fact, for longer than 2000 year, people tried to prove it in one way or another. Starting from the ancient greeks with their question of squaring the circle. In attempts to show that pi was constructive, people invented variants of integral calculus before integrals were around! So the unconstructiveness of pi was really a very popular problem, and people couldn't imagine that it wasn't true.

Therefore, I consider it a real triomph of mathematics that pi was shown to be transcendental. For over 2000 years people have struggled to find a solution for the problem, and it was only with the new developed methods that they could find an answer. And from that moment on it appeared to people that they really could find answers to these problems, using these methods. The solution to this problem (and other related problems: the solvability of the quintic, the parallel postulate,...) is the start of modern mathematics (in my opinion).
 
  • #4
but can you prove that it is transcendental? I have already come up with a small method.
 
  • #5
eddybob123 said:
but can you prove that it is transcendental? I have already come up with a small method.

Yes, you can prove that it's transcendental. A proof is given in the following link: http://myyn.org/m/article/proof-of-lindemann-weierstrass-theorem-and-that-e-and-pi-are-transcendental2/ [Broken]
Beware however, since the proof is quite involved...
 
Last edited by a moderator:
  • #6
The construcable numbers are of a special type and do not include all algebraic numbers. Is it easier to show that pi is not algebraic? Showing that it is transcendental seems to be overkill.
 
  • #7
lavinia said:
The construcable numbers are of a special type and do not include all algebraic numbers. Is it easier to show that pi is not algebraic? Showing that it is transcendental seems to be overkill.

To show that an algebraic number is not constructible one usually shows that the degree of its algebraic extension of Q over Q is not a power of two. My point is that to show non-constructibility usually requires its irreducible polynomial over Q. So I wouldn't be surprised that its anymore difficult to show that a number is not algebraic than not constructible.
 

1. How is pi calculated?

The value of pi is calculated by dividing the circumference of a circle by its diameter. This ratio remains constant for all circles, and is approximately equal to 3.14159.

2. Who discovered pi?

The concept of pi has been studied and used by various ancient civilizations, but the first known calculation of its value was done by the Greek mathematician Archimedes around 250 BCE.

3. What is the significance of pi?

Pi is an irrational number, meaning it cannot be expressed as a fraction of two integers. It is a fundamental constant in mathematics and has numerous applications in geometry, trigonometry, and physics.

4. Can pi be calculated to infinite digits?

Technically, yes. Pi is an irrational number and its decimal representation continues infinitely without repeating. However, for practical purposes, the value of pi is usually rounded to a certain number of digits, such as 3.14 or 3.14159.

5. How has the calculation and understanding of pi evolved over time?

The calculation and understanding of pi has evolved significantly over time, with many mathematicians and scientists contributing to its development. From early approximations by ancient civilizations to modern calculations using computers, our understanding of pi has become more precise and has expanded to include its role in complex mathematical concepts.

Similar threads

Replies
4
Views
284
  • General Math
Replies
7
Views
1K
Replies
9
Views
323
Replies
6
Views
1K
Replies
3
Views
1K
Replies
16
Views
4K
Replies
5
Views
1K
  • Topology and Analysis
Replies
8
Views
395
Replies
13
Views
3K
  • Topology and Analysis
Replies
4
Views
205
Back
Top