Find the denominator n of the rational number kn (k,n are both integers) such that ∣Root 2 - k/n∣ is the smallest possible, out of all rational numbers with denominator less than 1000?
To find the denominator n of the rational number kn (k, n are both integers) such that |√2 - k/n| is the smallest possible, out of all rational numbers with a denominator less than 1000, we can use a technique called continued fractions.
Continued fractions are a way to represent a real number as an infinite sequence of fractions. The idea is to iteratively take the reciprocal of the fractional part of the real number and continue the process until we get a whole number. In this case, we will apply continued fractions to √2.
The algorithm for finding the continued fraction representation of a real number involves the following steps:
1. Start with the given real number. In this case, the real number is √2.
2. Take the integer part of the real number as your first fraction. For √2, the integer part is 1, so the first fraction is 1/1.
3. Subtract the fractional part from the real number and take its reciprocal. For √2, the fractional part is √2 - 1. The reciprocal of this is 1/(√2 - 1), which can be simplified using rationalization of the denominator to get (√2 + 1)/1.
4. Repeat steps 2 and 3 with the new fraction. The process is as follows:
- Integer part: 1 (from 1/1)
- Fractional part: √2 + 1 (from (√2 + 1)/1)
- Reciprocal: 1/(√2 + 1) = (√2 - 1)/(-1) = 1 - √2
- Next fraction: 1/1 + 1/(1 - √2) = 2 + √2
Continuing this process, we get the following sequence of fractions:
1, 2 + √2, 2 + 2√2, 5 + 2√2, 7 + 5√2, 19 + 9√2, 26 + 19√2, 71 + 33√2, ...
We stop this process when we reach a fraction whose denominator exceeds 1000. In this case, we can see that the denominators of the fractions will alternate between odd and even values. So, the smallest possible denominator less than 1000 will be an odd number.
Thus, the denominator n of the rational number kn that minimizes |√2 - k/n| is the largest odd denominator in the continued fraction representation of √2 that is less than 1000.
By calculating the continued fractions until we reach a denominator larger than 1000, we find that the largest odd denominator is 961.
Therefore, the denominator n of the rational number kn (k, n are both integers) such that |√2 - k/n| is the smallest possible, out of all rational numbers with a denominator less than 1000, is 961.