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Pythagorean12
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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"?
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.
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.
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.
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.
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, ... ].