Its like this...
pi ~ 3.14159..... e ~ 2.71828..... (eulers number)
It is unknown whether x= pi +e and y= pi - e both are irrational numbers.... but at LEAST ONE of these 2 numbers is irrational .... WHY????? I have spent 4 days and 4 nights thinking... but.... maths.... keeps on killing me,......
2007-03-06 19:41:26 · 8 answers · asked by mathsbiochemcompecons 1 in Science & Mathematics ➔ Mathematics
Let's assume that neither x nor y were irrational. If x is rational and y is rational, then x+y should be rational too (after all, you can always write a rational number as a fraction, and then add two different fractions together to get a bigger fraction).
But x + y = 2pi. Since pi is irrational, so is 2pi. Therefore, our original assumption is wrong and x or y (or both) must be irrational.
2007-03-06 19:52:27 · answer #1 · answered by Anonymous · 2⤊ 1⤋
To show that at least one of pi+e and pi-e are irrational, suppose that they are both rational. If so then their sum must also be rational. But the sum is 2pi, so this means that pi must also be rational, which is a contradiction, so at leat one of pi +e and pi - e must be irrational.
Hmmm, now why am I hungry for actual pie now?
2007-03-06 19:54:36 · answer #2 · answered by Phineas Bogg 6 · 0⤊ 1⤋
What is more fun is that at least one of pi+e and pi*e is irrational. In fact, if both were rational, the both e and pi would be roots of the polynomial with rational coefficients
(x-pi)(x-e)=x^2-(pi+e)x+pi*e.
But both pi and e are transcendental, so this can't happen.
In both your problem and the one I gave, the expectation is that both are irrational (in fact, transcendental), but in neither problem is that known.
2007-03-07 00:12:48 · answer #3 · answered by mathematician 7 · 2⤊ 0⤋
Both pi and e have long been proved to be irrational, and indeed transcendental.
But if x and y were rational, then so would be x + y and x - y, yielding two immediate contradictions, which is twice as many as you need to do the proof.
2007-03-06 21:15:57 · answer #4 · answered by Curt Monash 7 · 0⤊ 2⤋
Suppose neither of them are irrational numbers...
then... we mean both of them are rational number...
so... x+y is also rational number...
x+y=pi+e+pi-e=2*pi
then 2*pi is not rational number,as pi is not...
so we've faced the bad-contradiction against our suppose...
and we've got the answer...
thank you...
2007-03-06 19:54:35 · answer #5 · answered by QuizBox 2 · 0⤊ 1⤋
i could gown up like a splash lady, with clothing, crinolines and white socks with ruffle trim...my hair in pigtails. Kinda like Dorothy...and Alice. it may be plenty exciting!
2016-11-23 12:43:27 · answer #6 · answered by ? 4 · 0⤊ 0⤋
If they were both rationals, then their sum (=2pi) would be rational, and that is not the case.
2007-03-06 19:53:41 · answer #7 · answered by 11:11 3 · 0⤊ 1⤋
then asnser is 5
2007-03-14 06:29:43 · answer #8 · answered by Anonymous · 0⤊ 0⤋