- #1

newton1

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we know e (exponential) is a irrational number...

how can we prove it??

how can we prove it??

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- Thread starter newton1
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In summary, we know that e (exponential) is an irrational number and we can prove it through various methods. One way is by calculating e using an infinite series of non-repeating rational numbers, which results in an irrational sum. Another way is by proving that e cannot be written as a fraction through the use of Taylor's series. It can also be shown using a theorem that if there exists a function that is continuous and positive on a certain interval and its iterated anti-derivatives can be taken to be integer valued at the endpoints, then the number is irrational. Finally, we can also use a proof similar to the one used for proving pi is irrational to show that e is irrational. In conclusion, e cannot be written

- #1

newton1

- 152

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we know e (exponential) is a irrational number...

how can we prove it??

how can we prove it??

Last edited by a moderator:

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- #2

emu

One way yields an infinite series of non-repeating rational numbers. The sum is therefore irrational.

Try proving the sqrt(5) is irrational.

- #3

HallsofIvy

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One standard method is to use the Taylor's series (which may be what emu meant): e= 1+ 1/2 + 1/6+ ...+ 1/n!+ ...

If j is any positve integer then e*j!= integer+ 1/(q+1)+ 1/(q+1)(q+2)+ ... which is not an integer so e cannot be written as a fraction with denominator j for any j.

A theorem I saw years ago was this: If c> 0, and there exist a function f(x), continuous on [0,c], positive on (0,c) and such that f(x) and its iterated anti-derivatives can be taken to be integer valued at both 0 and c, the c is irrational!

Taking f(x)= sin(x) in this theorem shows that pi is irrational.

It can also be used to prove: If c is a positive number other than 1 and ln(c) is rational, then c is irrational.

Since e is a positive number, not equal to 1, and ln(e)= 1 is rational, it follows that e is irrational.

- #4

Frac

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- #5

Paradox

- #6

Hurkyl

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e*j!= integer+ 1/(q+1)+ 1/(q+1)(q+2)+ ... which is not an integer

That's not obvious... I don't see why the infinite sequence there cannot add up to an integral value.

Hurkyl

- #7

bogdan

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It's quite simple to prove that 1/(n+1)!<e-An<1/(n!*n);

Let's suppose e is rational, so it's equal to p/q, where p and q are integers.

1/(n+1)!<e-(1+1/2!+...+1/n!)<1/(n!*n); | *n!;

1/(n+1)<n!*p/q-n!*(1+1/2!+...+1/n!)<1/n;

But between 1/(n+1) and 1/n is no integer...

n!*p/q must be an integer because for n big enough n! is a multiple of q;

So e is not rational...

QED

- #8

climbhi

- #9

newton1

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thank you...:)

- #10

Hurkyl

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That bit isn't obvious either... but the ordinary taylor remainder formula gives e/(n+1)! for that term which is sufficient for the proof. Don't tell me how to get that end of the inequality, it'd be a good exercise to figure it out myself!

Hurkyl

An irrational number is a real number that cannot be expressed as a ratio of two integers. This means that it cannot be written as a simple fraction, and its decimal representation is non-terminating and non-repeating.

e, also known as Euler's number, is a mathematical constant that is approximately equal to 2.71828. It is a fundamental constant in calculus and is often used in exponential growth and decay equations.

The proof that e is an irrational number was first given by Swiss mathematician Leonhard Euler in the 18th century. It involves using the properties of limits and infinite series to show that e cannot be expressed as a ratio of two integers.

Knowing that e is irrational is important in mathematics because it allows us to use it as a base for logarithms and exponential functions. It also has applications in fields such as physics, engineering, and finance.

Yes, there are several other irrational numbers that are related to e, such as pi (π) and the golden ratio (φ). These numbers have their own unique properties and are also commonly used in mathematical equations and formulas.

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