Finding Positive Integers for Irrational Number Interval

Thread starter TJK
Start date Feb 13, 2012
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Integers Interval Irrational Irrational number Positive

In summary, irrational numbers cannot be exactly represented by positive integers due to their infinite decimal places. However, they can be approximated by using a series of positive integers with increasing precision. This involves setting a range for the irrational number and finding a suitable series of positive integers within that range. Although any irrational number can be approximated by positive integers, the accuracy of the approximation depends on the precision of the positive integers used. These approximations have various applications, including improving the efficiency of algorithms in computer science and simplifying complex mathematical problems.

Feb 13, 2012

TJK

someome please help me with this problem:
"Any real numbers x and y with 0 < x < y, there exist positive integers p
and q such that the irrational number s =( p√2)/q is in the interval (x; y)."

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Feb 13, 2012

micromass

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Do you know that the rationals are dense in the reals?? What does that mean??

1. How can irrational numbers be represented by positive integers?

Irrational numbers cannot be exactly represented by positive integers because they have an infinite number of decimal places. However, we can approximate them by using a series of positive integers with increasing precision.

2. What is the process for finding positive integers for an irrational number interval?

The process involves setting a range for the irrational number, then finding a series of positive integers that fall within that range and have increasing precision. This can be done through various mathematical methods such as continued fractions or the Babylonian method.

3. Can any irrational number be approximated by positive integers?

Yes, any irrational number can be approximated by positive integers with increasing precision. However, the larger and more complex the irrational number, the more difficult it may be to find a suitable series of positive integers.

4. How accurate are these approximations of irrational numbers using positive integers?

The accuracy of the approximation depends on the precision of the positive integers used. The more positive integers we use, the closer the approximation will be to the actual value of the irrational number. However, it is not possible to have an exact representation of an irrational number using only positive integers.

5. What are the applications of finding positive integers for irrational number intervals?

One application is in the field of computer science, where approximating irrational numbers using positive integers can help improve the efficiency of algorithms that involve these numbers. It can also be useful in simplifying complex mathematical problems and in understanding the patterns and relationships between different irrational numbers.