Irrational PowersDate: 8/30/96 at 22:20:49 From: Anonymous Subject: Irrational numbers to irrational powers Does an irrational number to the irrational power yield a rational number? If so, what are the numbers? The two irrational numbers don't need to be the same. I approached the problem through trial and error, but it is too time consuming. Date: 9/5/96 at 10:13:58 From: Doctor Jerry Subject: Re: Irrational numbers to irrational powers In general, an irrational number to an irrational power yields an irrational number. The mathematics needed to prove this is quite complicated. I'll try to explain a little. A mathematician named Ferdinand Lindemann proved that if x1, x2, ..., xn are distinct algebraic numbers (see below) and p1, p2, ...,pn are algebraic numbers, not all zero, then the sum p1*e^(x1) + p2*e^(x2) + ... + pn*e^(xn) cannot be zero. A number is an algebraic number if it is a root of an equation a0 + a1*x^1 + a2*x^2 + ... +an*x^n = 0, where a0, a2, a2, ... , an are rational numbers. All of the rational numbers and many of the irrational numbers are algebraic. For example, the number 2^(1/2) is algebraic since it is a root of x^2 - 2 = 0. If we take n = 2, p1 = 1, and x2 = 0, then e^(x1)+ p2 cannot be zero if x1 is a nonzero algebraic number and p1 is any algebraic number. From this we conclude that e^(x1) cannot be rational, and so must be irrational. The number e is irrational and we may take x1 = 2^(1/2). Hence, an irrational number to an irrational power need not be rational. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 9/5/96 at 10:17:12 From: Doctor Ken Subject: Re: algebra(rational and irrational numbers) Hello - If you're looking for some specific examples of irrational numbers x and y such that x^y is rational, you can probably construct some yourself. Here's how you might go about doing it: take your favorite irrational number and call it x. Now ask yourself what number y would have to be in order to satisfy x^y = 5 (Think logs!). Good luck! -Doctor Ken, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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