Irrational numbers in infinite list of integers

  • Thread starter GunnaSix
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  • #1
GunnaSix
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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.
 

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  • #2
Hurkyl
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I don't know if you can say that there are no irrational numbers in an infinite list of integers.
Can you convince yourself that this conjecture is equivalent to
Every integer is rational​
 
  • #3
Tinyboss
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If sin(x)=0 for positive x, then x=k*pi for some positive integer k. If x is an integer, then x/k=pi should raise a red flag.
 
  • #4
ramsey2879
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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.

are you talking of integer degrees or radians?
 
  • #5
GunnaSix
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Integer radians. Sorry if this was a super obvious question. I guess I was just thinking too much into it.
 
  • #6
Eynstone
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If you mean to ask whether the intervals [2npi,(2n+1)pi) (where sinx >0) contain integers, the answer is yes ( as pi>1).
 
  • #7
ramsey2879
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If you mean to ask whether the intervals [2npi,(2n+1)pi) (where sinx >0) contain integers, the answer is yes ( as pi>1).

SineX = 0 only where X = n*pi radians. But n*pi must be an irrational number or 0. Thus SineX is either greater than or less than 0 for any integer angle n in radians where n>0. I think this is what the poser was asking.
 
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  • #8
HallsofIvy
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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.
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.
 
  • #9
ramsey2879
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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.
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.
 

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