Introduction to Rational Number Between √2 and √2 0.0001

In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q, where p and q are integers and q ≠ 0. This article explores how to find a rational number between the square root of 2 and the square root of 2 plus 0.0001. We will cover methods ranging from direct calculation to using continued fractions.

Direct Calculation

To find a rational number between √2 and √2 0.0001, we can start by determining the approximate value of √2. The value of √2 is approximately 1.4142. Therefore, √2 0.0001 is approximately 1.4143.

Now, we need to identify a rational number that lies between 1.4142 and 1.4143. One simple rational number that fits this criterion is 1.41425, which can be expressed as a fraction:

1.41425 141425/100000

Thus, a rational number between √2 and √2 0.0001 is 141425/100000. Alternatively, you could also choose other rational numbers like 1.4143 (14143/10000) or 1.4142 (7071/5000) as long as they fall within the specified range.

General Method Using q and p

To find a rational number between any two real numbers x and y where x , we can use the following method. Let ( q leftlceil frac{2}{y-x} rightrceil ) and ( p leftlfloor x cdot q rightrfloor ). Then, frac{p}{q} is a rational number between x and y.

For x √2 and y √2 0.0001, we have

q frac{2}{0.0001} 20000

p leftlfloor √2 cdot 20000 rightrfloor 28285

Thus, frac{28285}{20000} 1.41425.

Using Continued Fractions

Continued fractions provide a useful method for finding rational approximations of irrational numbers. The continued fraction of √2 is [1; 2, 2, 2, ...], meaning:

√2 1 1/(2 1/(2 1/(2 1/(2 ...))))

The convergents of this continued fraction are fractions that get progressively closer to √2. They are formed by the even and odd terms in the sequence, respectively. For example, the first few convergents are:

3/2, 17/12, 99/70, 577/408, 3363/2378, 19601/13860, ...

Each of these fractions alternates between being just below and just above √2. The ones greater than √2 are the odd terms.

To find a rational number between √2 and √2 0.0001, we can use the fraction 19601/13860, which is one of the convergents of the continued fraction of √2.

In conclusion, there are various methods to find a rational number between √2 and √2 0.0001, ranging from direct calculation to using continued fractions. No matter which method you choose, you can always find a rational number that fits the criteria.