Prove the impossibility of construction with ruler and compass the numbers:

Thread starter aporter1
Start date Jun 13, 2012
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Compass Construction Numbers

In summary, it has been proven that not all numbers can be constructed using only a ruler and compass. The proof for this is known as the "Gauss-Wantzel Theorem" which states that a number can only be constructed if it is a product of powers of 2 and distinct Fermat primes. There are alternative tools that can be used for more complex constructions, such as a straightedge, compasses with adjustable radius, and a marked ruler. While certain constructions may be impossible with ruler and compass, approximations can be made. Understanding the limitations of ruler and compass constructions has practical applications in fields such as geometry, architecture, and computer science.

Jun 13, 2012

aporter1

Homework Statement

∛4 ∛5 ∛3 and ∛a

Homework Equations

The Attempt at a Solution

I have started with the problem but i feel like I'm off track. I'll upload a pic of the work so far.

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Jun 13, 2012

aporter1

here is that I've started on

1. Can all numbers be constructed using only a ruler and compass?

No, not all numbers can be constructed using only a ruler and compass. In fact, it has been proven that only certain numbers can be constructed in this way.

2. What is the proof that certain numbers cannot be constructed with ruler and compass?

The proof is known as the "Gauss-Wantzel Theorem", which states that a number can be constructed with ruler and compass if and only if it is a product of powers of 2 and distinct Fermat primes (numbers of the form 2^(2^n) + 1).

3. Are there any other tools besides a ruler and compass that can be used for constructions?

Yes, there are other tools such as a straightedge, compasses with adjustable radius, and a marked ruler that can be used for more complex constructions.

4. Can the construction of certain numbers be approximated with a ruler and compass?

Yes, with constructions such as angle trisection and squaring the circle, which have been proven to be impossible with ruler and compass, approximations can be made using a ruler and compass. However, these approximations will not be exact.

5. Are there any practical applications for understanding the impossibility of certain constructions with ruler and compass?

While the impossibility of certain constructions with ruler and compass may seem like a purely theoretical concept, it has practical applications in fields such as geometry, architecture, and computer science. Understanding the limitations of ruler and compass constructions can help in solving complex problems and designing efficient algorithms.