Transcendental numbers: interesting perspectives

In summary, the professor is presenting a proof that e is transcendental, but wants to spice up the presentation by exploring some interesting perspectives. He has already selected the Lindemann-Weierstrass theorem as a possible candidate, but is looking for other avenues. Some connections that were mentioned include complex analysis and the irrationality measure of e.
  • #1
loom91
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0
Hi,

I'm presenting a proof that e is transcendental, but I want to spice up the presentation a bit by exploring some interesting perspectives, such as connections to topology, abstract algebra etc. I've already selected the Lindemann–Weierstrass theorem as a possible candidate. Can you suggest some other avenues? This is a first-year UG class, so the concepts used cannot be so technical as to be unexplainable in a timeframe of an hour. Thanks!

Molu
 
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  • #2
Something that's stuck with me for a long time is this equation:
[itex]e^{i\pi} + 1 = 0[/itex]
In it appear five of the most important symbols in mathematics. It might strike some of them as very odd that e and [itex]\pi[/itex] cannot be represented with a finite number of decimal digits, and that i is as unreal as a number can get, yet all three combine to make -1.

Maybe some of the less jaded among these students will appreciate this.
 
  • #3
Ok, we could talk about that, but that's rather too basic for a 2nd sem undergrad class. Most of the students have already encountered that material in high-school, they would appreciate something slightly more challenging.
 
  • #4
Anything else?
 
  • #5
Liouville numbers (the first historic transcendentals), irrationality measures? Why does e have irrational measure 2 like algebraic numbers? Continued fraction expansions? Stuff like that?
 
  • #6
Ah, that's what I was looking for. Though irrationality measure seems a very difficult thing to wrap your mind around, so I don't know if we can effectively explain that in a short time period, but let's see. Incidentally, is there any particularly intuitive reason for e having the irrationality measure 2? Thanks.
 
  • #7
It must come from looking at the continued fraction expansion. I've never looked at it, but it might not be too hard. There's a reference in here http://planetmath.org/encyclopedia/IrrationalityMeasure.html
 
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  • #8
Any possible connections with complex analysis?
 

Related to Transcendental numbers: interesting perspectives

1. What are transcendental numbers?

Transcendental numbers are real numbers that cannot be expressed as the root of any polynomial equation with rational coefficients. They are numbers that cannot be written as fractions or decimals with a repeating pattern. Examples of transcendental numbers include pi (π) and e.

2. How are transcendental numbers different from algebraic numbers?

Algebraic numbers are numbers that can be expressed as the root of a polynomial equation with rational coefficients. This means that they can be written as fractions or decimals with a repeating pattern. Transcendental numbers, on the other hand, cannot be expressed in this way.

3. Why are transcendental numbers considered interesting?

Transcendental numbers are considered interesting because they have unique mathematical properties and play an important role in various mathematical and scientific fields, such as geometry, number theory, and physics. They also have practical applications in computer science and engineering.

4. How can transcendental numbers be approximated?

Transcendental numbers can be approximated using various methods, such as continued fractions, infinite series, and iterative algorithms. However, since transcendental numbers are infinite and non-repeating, they can never be represented exactly with a finite number of digits.

5. Are there any famous unsolved problems related to transcendental numbers?

Yes, one famous unsolved problem related to transcendental numbers is the Lindemann–Weierstrass theorem, which states that e raised to any non-zero algebraic number is transcendental. This theorem has many important consequences in mathematics, but a complete proof of it has yet to be found.

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