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.
Irrational numbers are real numbers that cannot be expressed as a ratio of two integers. They are numbers that do not terminate or repeat in decimal form, such as pi (3.14159...) or the square root of 2 (1.41421...).
Irrational numbers cannot be written as a fraction of two integers, while rational numbers can. Additionally, irrational numbers have decimal representations that never terminate or repeat, while rational numbers have finite or repeating decimal representations.
In an infinite list of integers, irrational numbers are represented as approximations. For example, pi can be approximated as 3.14, but it cannot be represented exactly as it has an infinite number of decimal places.
No, there is no pattern to the occurrence of irrational numbers in an infinite list of integers. They are randomly distributed and can occur at any point in the list.
Irrational numbers are important in mathematics because they allow us to represent quantities that cannot be expressed as a ratio of two integers. They are also essential in fields such as geometry, where they help us understand and describe the relationship between different shapes and their measurements.