Proof of the Irrationality of e - attributed to Joseph Fourier

In summary, in Chapter 4 of Julian Havil's book, "The Irrationals: The Story of the Numbers You Can't Count On," he gives a proof of the irrationality of e which was attributed to Joseph Fourier. The proof is by contradiction and demonstrates that R is an integer by showing that it is the difference between two integers. There is some confusion about the ellipses in the expression, but it is most likely a typo.
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(NOTE: Maybe this post belongs in the Number Theory Forum? Apologies if it is wrongly located!)

I am reading Julian Havil's book, "The Irrationals: The Story of the Numbers You Can't Count On"

In Chapter 4: Irrationals, Old and New, Havil gives a proof of the irrationality of e which was attributed to Joseph Fourier ... the proof is by contradiction and reads as follows:View attachment 3571In the above proof, Havil writes:

"Then

\(\displaystyle n!e = n! \frac{m}{n} = (n-1)! m\)

\(\displaystyle = ( n! + \frac{n!}{1!} + \frac{n!}{2!} + \frac{n!}{3!} + \frac{n!}{4!} + \ ... \ ... \ \frac{n!}{n!} ) + \ ... \ ... \ + R\)

which makes \(\displaystyle R \ne 0\) the difference between two integers and so an integer itself."Can someone please explain how the expression above demonstrates that R is an integer?

Peter***EDIT***

oh! just saw the answer I think ... I was thrown by the ellipses ( that is the ... ... ) after the term in parentheses (brackets) in the expression:

\(\displaystyle = ( n! + \frac{n!}{1!} + \frac{n!}{2!} + \frac{n!}{3!} + \frac{n!}{4!} + \ ... \ ... \ \frac{n!}{n!} ) + \ ... \ ... \ + R\)

I thought there was some term there ... but did not know what ... but now believe that there is nothing there ...

That is I think maybe it should read:

\(\displaystyle n!e = ( n! + \frac{n!}{1!} + \frac{n!}{2!} + \frac{n!}{3!} + \frac{n!}{4!} + \ ... \ ... \ \frac{n!}{n!} ) + R\)

BUT ... why did Havil include the ... ... in the expression?

... just a typo? ... or is there actually a term there?
 
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  • #2
Hi Peter,

I'm not sure why he wrote the ... between the expansion and R, but you got the idea.
 
  • #3
Fallen Angel said:
Hi Peter,

I'm not sure why he wrote the ... between the expansion and R, but you got the idea.

Thanks Fallen Angel ... glad you can confirm it is mysterious ... or probably a typo ...

Peter
 

Related to Proof of the Irrationality of e - attributed to Joseph Fourier

1. What is the proof of the irrationality of e attributed to Joseph Fourier?

The proof of the irrationality of e attributed to Joseph Fourier is a mathematical proof that shows that the number e, also known as Euler's number, is an irrational number. This means that it cannot be expressed as a ratio of two integers and has an infinite number of non-repeating decimal digits.

2. Who is Joseph Fourier and why is he associated with this proof?

Joseph Fourier was a French mathematician and physicist who lived in the 18th and 19th centuries. He is best known for his work in Fourier analysis and heat transfer. He is associated with the proof of the irrationality of e because he published a paper in 1827 that presented this proof.

3. What is the significance of proving the irrationality of e?

Proving the irrationality of e has several important implications in mathematics. Firstly, it demonstrates the infinite nature of e, which is a fundamental constant in many areas of mathematics and science. Additionally, it helps to solidify the concept of irrational numbers and their role in mathematics. Finally, it has practical applications in fields such as cryptography and signal processing.

4. How is the proof of the irrationality of e attributed to Joseph Fourier carried out?

The proof of the irrationality of e attributed to Joseph Fourier is a proof by contradiction. It begins by assuming that e is a rational number, and then uses mathematical techniques to show that this assumption leads to a contradiction. This contradiction proves that e cannot be a rational number and must therefore be irrational.

5. Is the proof of the irrationality of e attributed to Joseph Fourier widely accepted?

Yes, the proof of the irrationality of e attributed to Joseph Fourier is widely accepted by the mathematical community. It has been studied and verified by numerous mathematicians and has not been disproven. It is considered a fundamental proof in the field of mathematics and is often taught in undergraduate mathematics courses.

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