Find the convergents to the fraction

Thread starter Pythagorean12
Start date Oct 15, 2009
Tags

Fraction

In summary, a convergent in fractions is a rational number that results from truncating a continued fraction after a finite number of terms. To find the convergents to a fraction, the fraction must first be converted into a continued fraction and then a recursive algorithm can be used. All fractions can be written as continued fractions, but the number of terms may be infinite for irrational numbers. Finding convergents is significant because it allows us to approximate irrational numbers with rational ones. There are patterns in the convergents of continued fractions, such as the repeating pattern in the convergents of the golden ratio.

Oct 15, 2009

Pythagorean12

Homework Statement

Find the convergents to the fraction 57/44.

Homework Equations

Could anyone provide any good internet resuorces about "finding the convergents to the fraction"?

The Attempt at a Solution

Physics news on Phys.org

Oct 15, 2009

whs

http://en.wikipedia.org/wiki/Convergent_(continued_fraction)

Oct 15, 2009

Pythagorean12

So, the convergents to this fraction are 1,3,3,4?

Related to Find the convergents to the fraction

1. What is the definition of a convergent in fractions?

A convergent in fractions is a rational number that is the result of truncating a continued fraction after a finite number of terms. It is a finite approximation of an irrational number.

2. How do you find the convergents to a fraction?

To find the convergents to a fraction, you must first convert the fraction into a continued fraction. Then, you can use a recursive algorithm to find the convergents by working backwards from the last term of the continued fraction. The numerator and denominator of each convergent can be calculated using the previous convergent and the current term of the continued fraction.

3. Can all fractions be written as a continued fraction?

Yes, all fractions can be written as a continued fraction. However, the continued fraction may have an infinite number of terms if the fraction is irrational.

4. What is the significance of finding the convergents to a fraction?

Finding the convergents to a fraction is significant because it allows us to approximate irrational numbers with rational numbers. This is useful in various mathematical calculations and can provide a better understanding of the properties of irrational numbers.

5. Are there any patterns in the convergents of a continued fraction?

Yes, there are patterns in the convergents of a continued fraction. For example, the convergents of the golden ratio have a repeating pattern of 1, 1, 1, 1, 1... This is because the golden ratio is an irrational number that can be represented by a simple continued fraction of [1; 1, 1, 1, 1, ... ].