How do you find a continued fraction?

To calculate a continued fraction representation of a number r, write down the integer part (technically the floor) of r. Subtract this integer part from r. If the difference is 0, stop; otherwise find the reciprocal of the difference and repeat. The procedure will halt if and only if r is rational.

What is the period of a continued fraction?

A periodic continued fraction is a continued fraction (generally a regular continued fraction) whose terms eventually repeat from some point onwards. The minimal number of repeating terms is called the period of the continued fraction. All nontrivial periodic continued fractions represent irrational numbers.

Who invented continued fractions?

Christian Huygens (1629–1695) is credited with being the first to use continued fractions in a practical application. He used continued fractions for approximating gear ratios in the building of a mechanical planetarium.

Who created continued fractions?

Srinivasa Ramanujan
The Rogers–Ramanujan continued fraction is a continued fraction discovered by Rogers (1894) and independently by Srinivasa Ramanujan, and closely related to the Rogers–Ramanujan identities. It can be evaluated explicitly for a broad class of values of its argument.

What is √ 2 as a fraction?

Root 2 is an irrational number as it cannot be expressed as a fraction and has an infinite number of decimals. So, the exact value of the root of 2 cannot be determined.

Why are continued fractions the best approximations?

The continued fraction representation of the number pi that does follow our rules. When we truncate a continued fraction after some number of terms, we get what is called a convergent. The convergents in a continued fraction representation of a number are the best rational approximations of that number.

Why are continued fractions important?

Continued fractions are useful for converting decimals into fractions, or reducing fractions to their common divisor. Often finding fractions between X and Y, like 8/11 for the logrithms of the chords of the heptagon, means that (in this case), you only need to consider 11 cases.

What is Ramanujan theory?

For those of you who are unfamiliar with this series, which has come to be known as the Ramanujan Summation after a famous Indian mathematician named Srinivasa Ramanujan, it states that if you add all the natural numbers, that is 1, 2, 3, 4, and so on, all the way to infinity, you will find that it is equal to -1/12.

When were continued fractions invented?

1. Introduction. With the exception of a few isolated results which appeared in the sixteenth and seventeenth centuries, most of the elementary theory of continued fractions was developed in a single paper written in 1737 by Leonhard Euler.

What is root 2 called?

√2 is also called Pythagoras’ constant. √2 represents the diagonal of a unit square. √2 was the first number to be discovered as an irrational number. Its decimal representation is non-terminating and non-repeating.

What is the purpose of continued fractions?

The continued fraction expansion of a real number x is a very efficient process for finding the best rational approximations of x. Moreover, continued fractions are a very versatile tool for solving problems related with movements involving two different periods.

What is 52.5 as a fraction?

What is 52.5% in the fraction form? 52.5% in the fraction form is 52.5/100. If you want you can simplify it further as 21/40.

What is .25 in a fraction?

Place the Percentage Value at the top over 100. The exact form of the fraction is 14. In the decimal form, the fraction can be written as 0.25. The fraction can be written as 14.

Is pi a rational number?

No matter how big your circle, the ratio of circumference to diameter is the value of Pi. Pi is an irrational number—you can’t write it down as a non-infinite decimal. This means you need an approximate value for Pi.

Is squared irrational?

Proof: √2 is irrational.