Friday, June 10, 2011

Can there be both 'rational and irrational' numbers ?

Everyone knows, from the school, that square root 2 is irrational . I still remember the fashionable way in which my school teacher proved it. (for those who do not remember http://en.wikipedia.org/wiki/Irrational_number ). As all know, an irrational number cannot be written as a/b where a and b are integers. We know following facts which are proven in fashion in real analysis.
1. Any real number is rational or irrational
2. There are infinitely rational and irrational numbers
3. There are infinitely integers
'Can we say a given number is rational or irrational ?'. (If it is given in decimal points, number is rational for sure) . What I really mean is if I marked a random number on number line (see how irrational numbers are marked on number line at http://www.ehow.com/how_4455801_graph-irrational-numbers-number-line.html), who on the earth can say it is rational or irrational ? Since there are infinitely integers it can be a ratio of any two of them. Or since almost all the real numbers are transcendental (http://en.wikipedia.org/wiki/Transcendental_number)  and all transcendental numbers are irrational, there is a higher probability of randomly marked number is being irrational.
Therefore until being confirmed (or observed) we cannot comment about the rationality of the number. But we can give a probability distribution for the rationality of the randomly marked number.
According to quantum mechanics, we can say this number is in superposition of states.  i.e. the number is both rational and irrational. Described probability distribution is the wave function.

1. Well, I had this thought too though I wasn't able to prove it. My concern was, if it can be represented in a line, why cannot we measure it exactly. If we can measure it exactly, we can give it as a rational number.

2. Yes. As you might know measurement problem is one of the key concepts in quantum mechanics. You can never measure a irrational number(it is impossible to make a measuring tool for this) according to Heisenberg uncertainty principal. There is always an uncertainty of at least of h(plank constant). Hence, with any sophisticated technique, you can only measure an approximated value of the number.Let say you have a very accurate ruler. But when you go to the electron level, ruler gets changing randomly. So you cannot measure.