- #1
Hypercase
- 62
- 0
hi u guys .
I was hoping u could show me how to calculate the value of a continued fraction.
1/(6+(1/(6+1/...))
I was hoping u could show me how to calculate the value of a continued fraction.
1/(6+(1/(6+1/...))
Originally posted by Hypercase
hi u guys .
I was hoping u could show me how to calculate the value of a continued fraction.
1/(6+(1/(6+1/...))
A continued fraction is a mathematical expression where a number is represented as a sum of fractions, where the denominator of each fraction is added to the next fraction as a whole number. It can be written as [a0; a1, a2, a3...], where a0 is the whole number part and the rest of the numbers are the denominators of the fractions.
The value of a continued fraction can be calculated by starting from the end and working backwards. First, the last fraction is converted into a decimal by dividing the numerator by the denominator. Then, this decimal is added to the fraction before it and the result is used as the numerator for the next fraction. This process is repeated until the first fraction is reached, giving the final value of the continued fraction.
Continued fractions have many applications in mathematics, physics, and engineering. They can be used to approximate irrational numbers with a high degree of accuracy, as well as to solve certain types of equations. They also have connections to other areas of math, such as number theory and dynamical systems.
The convergents of a continued fraction are the fractions obtained by truncating the continued fraction at different points. To find the convergents, the process of calculating the value of the continued fraction can be reversed. Starting from the first fraction, the convergents can be found by simplifying the fractions obtained at each step.
There are many interesting patterns and properties of continued fractions. For example, the convergents of a continued fraction can be used to find the best rational approximations of an irrational number. Also, the continued fraction expansion of a quadratic irrational will eventually repeat in a pattern. Additionally, there are connections between continued fractions and the golden ratio, which has many fascinating properties in math and nature.