I ran into a math subproblem in the course of a bigger problem. I thought I'd give the Internet a shot before posting the answer. Here it is:
Show that for any finite set of positive real numbers, there must exist some number x such that x times each of the numbers in the set produces an irrational number.
Or rephrased:
If n runners start out together on an infinite race, each at a constant speed, which is different from the others' speeds, show that at some time each one will be an irrational distance from the start line.
Just a reminder: irrational numbers are those that cannot be written as a fraction a/b with a and b both whole numbers.
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3 comments:
Interesting problem!
Could you take sqrt.(x-(sqrt.(x))+pi) ?
Not the most simple or elegant:)
Natalie,
That function is very interesting, because for any rational input x, the result of the function is irrational. However, the function is too sophisticated! This question is asking only about a function of the form f(x) = bx, where "b" is some constant.
Jonathan
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