What are continued fractions used for?
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 the method of continued fractions?
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.
Is Pi a continued fraction?
The continued fraction expansion for pi. And the first few convergents are: 3 (duh), 22/7 (Pi Approximation Day), 333/106, 355/113, and 103,993/33,102. A continued fraction reciting contest is a much better way of celebrating pi than reciting its decimal digits.
Who published a paper on continued fractions?
Leonhard Euler
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.
Why are continued fractions the best approximations?
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. By increasing the denominators of our fractions, we can get as close as we want.
How was Pi discovered?
The Egyptians calculated the area of a circle by a formula that gave the approximate value of 3.1605 for π. The first calculation of π was done by Archimedes of Syracuse (287–212 BC), one of the greatest mathematicians of the ancient world. In this way, Archimedes showed that π is between 3 1/7 and 3 10/71.
Can pi be in a fraction?
Value of pi Pi is an irrational number, which means that it is a real number that cannot be expressed by a simple fraction. That’s because pi is what mathematicians call an “infinite decimal” — after the decimal point, the digits go on forever and ever.
Did ancient Egyptians use fractions?
Numbers and basic computation appeared in Ancient Egypt as early as 2700 BCE. But you might not know that Ancient Egyptians demanded that every fraction have 1 in the numerator. They wanted to write any rational between 0 and 1 as a sum of such “unit” fractions.
Who discovered Egyptian fractions?
One of the earliest publications of Paul Erdős proved that it is not possible for a harmonic progression to form an Egyptian fraction representation of an integer. The reason is that, necessarily, at least one denominator of the progression will be divisible by a prime number that does not divide any other denominator.