Irrational NumbersDate: 01/07/98 at 03:59:41 From: she jianwei Subject: Irrational numbers Are pi and the square root of 2 irrational numbers? Why? What about 1.33333... and 1.181818... ? Date: 01/07/98 at 06:08:17 From: Doctor Mitteldorf Subject: Re: Irrational numbers Dear Xi, A rational number is one that can be expressed as A/B where A and B are integers. 1.33333...= 4/3 and 1.1818... = 13/11. Sqrt(2) cannot be expressed as A/B. Here's the way the proof is usually written: Start with two numbers A and B so A/B = sqrt(2). Then reduce the fraction to lowest terms in the usual way, so there is no number that can be divided into both A and B. Now A/B = sqrt(2), so A^2/B^2 = 2. This means that A^2 must be even. So A must be even. So A^2 must really be divisible by 4. So B^2 must be even as well. So B must be even. But then A/B must not be a fraction in lowest terms, and I asked you before to put the fraction in lowest terms. That's the proof - it finds a hidden contradiction in the assumption that sqrt(2) can be written as a fraction A/B in lowest terms. It's more difficult to prove that pi is irrational, but it is. In fact, pi isn't even the square root or the cube root of anything. There's no equation that you can write that has only rational numbers in it where pi is the solution to the equation. This kind of number is called "transcendental." -Doctor Mitteldorf, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 01/10/98 at 06:17:01 From: she jianwei Subject: Irrational numbers Are 1.33333333 and 1.1818181818 (repeated decimals) rational? Why? Date: 01/14/98 at 04:55:45 From: Doctor Mitteldorf Subject: Re: Irrational numbers Dear Xi, When you divide one integer by another, and compute the result as a decimal, there are two things that can happen. First, it could come out even after a while: for example, 1/8 is exactly 0.125. If you tried to compute more decimal places, they would all be zeros. The second possibility is a repeating pattern. If you divide 4 by 3, you get 1.33333... The 3's go on forever. The definition of a rational number is any number that can be written as a fraction. Hence 4/3 is rational, and 1.33333... is 4/3. -Doctor Mitteldorf, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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