The discussion revolves around the properties of the sine function, particularly its behavior at integer values, and the relationship between integers and rational numbers. Participants explore whether the absolute value of sin x is greater than zero for all positive integer values of x and the implications of irrational numbers in the context of integers.
Participants express differing views on the behavior of the sine function at integer values and the implications for rationality and irrationality in integers. There is no clear consensus on whether the absolute value of sin x is greater than zero for all positive integers.
Some statements rely on specific interpretations of sine values and the definitions of integers and rational numbers, which may not be universally agreed upon. The discussion includes assumptions about the nature of sine and its values at integer inputs.
Can you convince yourself that this conjecture is equivalent toGunnaSix said:I don't know if you can say that there are no irrational numbers in an infinite list of integers.
GunnaSix said:Is it safe to assume that the absolute value of sin x is greater than zero for all positive integer values of x? I have no real experience in number theory, and I don't know if you can say that there are no irrational numbers in an infinite list of integers.
Eynstone said:If you mean to ask whether the intervals [2npi,(2n+1)pi) (where sinx >0) contain integers, the answer is yes ( as pi>1).
Yes, since any integer, n, can be written as the fraction n/1, all integers are rational numbers. There are no irrational integers. It is true that sin(n) is never 0 for any integer n- but that does NOT mean "greater than 0". sin(4) is negative.GunnaSix said:Is it safe to assume that the absolute value of sin x is greater than zero for all positive integer values of x? I have no real experience in number theory, and I don't know if you can say that there are no irrational numbers in an infinite list of integers.
While I used the language "Sine x" is "greater than or less than 0", the poser asked whether "the absolute value of Sine x is greater than 0 for all integer values of x.". I think you had my language in mind when you overlook the "absolute value" part of the poser's question.HallsofIvy said:Yes, since any integer, n, can be written as the fraction n/1, all integers are rational numbers. There are no irrational integers. It is true that sin(n) is never 0 for any integer n- but that does NOT mean "greater than 0". sin(4) is negative.