2007-06-18 05:54:25 · 9 answers · asked by soran s 1 in Science & Mathematics ➔ Mathematics
π - 1 = 2.14159...
2007-06-18 05:58:21 · answer #1 · answered by MamaMia © 7 · 2⤊ 0⤋
The answer depends on whether you want trivia,
or proof.
π - 1 is of course irrational, but I doubt that many
people around here can prove that this number
is irrational, or even that it falls between 2 and 3.
On the other hand √5 is sure kill.
First, since 2² < √5² < 3², we have 2 < √5 < 3.
Second, lets assume that n/m = √5, and gcd(n,m) = 1.
Then n²/m² = 5, then n² = 5m², therefore gcd(n,m) = 5.
Contradiciton.
2007-06-18 06:08:47 · answer #2 · answered by Alexander 6 · 0⤊ 0⤋
A good example is e = 2.7182818.... which can be defined as The sum (n=0 -> infinity) of 1/ n factorial (1 +1 + 1/2 + 1/2*3 + 1/2*3*4 + ....
The proof that e is irrational goes like this.
If e = a/b then generate e * b! which must be a whole number.
BUT when expanding e you get .....
.... + b!/b! + b!/(b+1)! + b!(b+2!)
which comes out as 1/(b+1) + 1/(b+1).(b+2) + 1/(b+1)(b+2)(b+3)
This fraction part is less than 1/b + 1/b^2 + 1/b^3 + ... which I am sure you recognise as a geometrical series which can be summed to less than one (1/b)* (1/(1-1/b)) = 1/(b-1)
Hence if e is rational then e.b! is a whole number but it is not a whole number - proof by contradiction.
2007-06-19 03:59:47 · answer #3 · answered by welcome news 6 · 1⤊ 0⤋
an irrational # cannot be written as a ratio of two integers (a/b) a transendtal # is not the solution to any polynomial with rational coefficients so there is no problem in ax^2 + bx + c form the pi is a solution to, therefore pi is transendental (as well as irrational) hope that helped a little
2016-04-01 03:41:00 · answer #4 · answered by Anonymous · 0⤊ 0⤋
There are infinite irrational numbers between 2 and 3.
e is one, pi - 1 is one, so are sqrt 5, sqrt 6,sqrt 7, sqrt 8
You can take any other combination like (sqrt 5 + sqrt 6)/2 etc etc
2007-06-18 06:06:03 · answer #5 · answered by astrokid 4 · 1⤊ 0⤋
The square-root of 5 and the square-root of 6 are two such numbers
2007-06-18 06:00:37 · answer #6 · answered by Devarat 7 · 1⤊ 0⤋
the definition of an irrational number is that square rooting it will not give an intiger, therefore, though they have a solution, they are unlikely to have rational ones.
2007-06-18 06:06:14 · answer #7 · answered by Kit Fang 7 · 0⤊ 2⤋
e, which is the natural, is the easiest to write down
2007-06-18 06:02:45 · answer #8 · answered by fastspawn 2 · 1⤊ 0⤋
its 2.32435435 or anything u can write like 2.99999999999
2007-06-18 06:02:50 · answer #9 · answered by CHINTU 2 · 0⤊ 3⤋