Please don't answer that this is because it have non-ending and non-repeating decimal. Please answer this question in science nature why pi is an irrational number.
2006-09-09 15:54:41 · 11 answers · asked by trey 1 in Science & Mathematics ➔ Mathematics
An irrational number is one that cannot be written as the ratio of integers. Pi is one such number. It represents the ratio between the circumerence of a circle and its diameter, which implies that both the circumference and the diameter cannot be integers at the same time. There is no perfect answer to why this ratio has to be an irrational number. That's just the way it is. The "natural" number e is also an irrational number. "Irrational" in this context doesn't mean crazy or incomprehensible, just that it's not the ratio of integers. Nature appears to be okay with irrational numbers.
2006-09-09 16:23:41 · answer #1 · answered by just♪wondering 7 · 0⤊ 1⤋
I know you don't like your answer about non-repeating decimals, but that's really kinda the heart of the issue. If it were rational, then there would exist integers p,q such that
p/q = pi
But that means that p = (pi)q. Since p is an integer, that would mean that q(pi) is an integer. But because pi is a non-ending, non-repeating decimal, you can never actually make this happen (if multiplication by an integer would yield an integer, than that would imply that either the decimal expansion terminates or repeats after a finite number of places). I'm sorry that's not an answer you like, but it's really the best one that doesn't invoke the transcendence of pi (which is a very non-trivial fact; it is MUCH more difficult than proving that e is transcendental...in fact showing e is transcendental is probably the easiest transcendental proof there is).
2006-09-10 02:46:55 · answer #2 · answered by wlfgngpck 4 · 1⤊ 0⤋
You answered your own question. PI is IRRATIONAL because it is a non-repeating non-terminating decimal. Otherwise, it would be a rational number.
What is your question? What do you mean "answer this question in science"?
You have to realize that its not like we "chose" to make PI irrational. Pi just IS irrational. Numbers are not "made" irrational or rational. They just have this property. Pi is the ratio of a circle's circumference to its diameter and it just happens to be irrational.
If you are asking for a proof, you are going to need a bit more of math before you can understand the proof.
2006-09-10 00:38:24 · answer #3 · answered by The Prince 6 · 0⤊ 0⤋
It isn't the easiest thing in the world to do, to prove pi is irrational. e is fairly simple; if it were p/q, multiply by q!, expand the series, and get an integer between 0 and 1.
Pi seems to require a proof in reverse. First show that if x is an algebraic number, then cos(x) is transcendental. Then substitute pi for x. If pi were algebraic, then cos(pi) would be transcendental, but actually, cos(pi) = -1. This not only proves pi to be irrational, it shows that it is transcendental.
2006-09-09 23:56:38 · answer #4 · answered by alnitaka 4 · 0⤊ 0⤋
According to Merriam-Webster's defintion of irrational numbers...
"It is a number that can be expressed as an infinite decimal with no set of consecutive digits repeating itself indefinitely and that cannot be expressed as the quotient of two integers"
and since pi is an infinite decimal with no set of consecutive digits repeating itself and cannot be expressed as the quotient of two integers.. pi is irrational..
remember that math is a branch of science.. so.. a mathematical nature is also science nature..:D
2006-09-09 23:21:39 · answer #5 · answered by noturordinaryguy 1 · 0⤊ 0⤋
Now that you mention it, I don't know why pi is irrational. I know that when Pi was finally proved to be transcendental it was heralded as a great achievement. If you go to wikipedia you can find a proof that e is irrational--maybe a modification of this proof would show that pi is also irrational.
2006-09-09 23:05:48 · answer #6 · answered by bruinfan 7 · 0⤊ 0⤋
The reason it's irrational is because the only ways to calculate it involve taking square roots of numbers that don't have rational roots. In other words, it's irrational for the same reason sqrt(2) is.
2006-09-09 23:47:47 · answer #7 · answered by Will 6 · 0⤊ 0⤋
Pi is ratio of circumfarence of a circle to diameter and it is not 3.14159. It is represented to be that way. It is irrational can be proved and proof is in the following site. In case you have a problem in understanding the proof please e-mail me.
2006-09-09 23:06:10 · answer #8 · answered by Mein Hoon Na 7 · 0⤊ 0⤋
Pi is irrational because it can not be found by multiplying or dividing two rational integers.
2006-09-09 23:19:47 · answer #9 · answered by عبد الله (ドラゴン) 5 · 0⤊ 0⤋
You must study that is named or called on a certain class of semi-simple subalgebras of the construction of finite local -rings ,cyclic abelian p-groups
2006-09-10 00:57:20 · answer #10 · answered by heinrich m 1 · 0⤊ 0⤋