Continued fraction and calendar
As we know 1 year 1 = 365 days 5 hours 48 minutes 46 seconds = 365,242199...days. Days are measured as a circadian period and fractions of the sun years conform to its length. It is possible to make a calendar of 365, 242199…=365;4,7,1,3,5,…days.
The first convergent fraction 36514 conforms to Julian calendar: every four years it is a leap year. Let’s find the difference among the followings,
365,25-365,242199 = 0,007801 days = 0,187224 hours = 11,23344 minutes = 11 minutes 14 seconds, i.e. the average length of the year is greater than that period for 11 minutes 14 seconds. The third convergent fraction 365;4,7,1=365833 is a basis of the Persian calendar. It was suggested in 1079 by mathematician, astronomer and poet Omar Khayyam. Here if we find the difference as 365833-365,242199 = 0,000225242 days = 0,005406 hours = 0,324348 minutes = 19,5 seconds, then our error will be 19 seconds in one year, but if we divide years into a 33-year cycle, the cycle inside the leap year coming after every four years is counted seven times and the eighth is not convenient in the fifth one, i.e. ІІІ-ІІІ-ІІІ-ІІІ-ІІІ-ІІІ-ІІІ-ІІІ-ІІІ-ІІІ-ІІІ (here sticks express years and dashes express leap years).
In 1864 Russian astronomer Johann Heinrich Medler suggested that the fourth convergent fraction 365, 24199…=36531128 gave one more calendar. Meddler suggested this in XXth century in Russia. For this it was necessary to pass 1 leap year every 128 years, because according to Julian calendar there are 32 leap years in 128 years. This calendar was accepted, because number 128 is not round. What is our error here?
36531128-365,242199 = -0,0000115 days = -0,000276 hours = -0,01656 minutes = -0,99 seconds = -1second!
Therefore this calendar has an error of 1 second.
In 1582 Pope Gregory ХІІІ made reformation correcting inaccuracy of Julian calendar. Also the the sequence of common and leap years, if the number is ended with two zeroes and the number is divided into four, then it is a leap year.
The error has become 10 days (years 1700, 1800, 1900) since the Christ’s new year. According to Julian calendar 13 days does not fit. And now let’s count the length of Gregorian calendar. According to Julian calendar if 100 years of 400 years are leap and according to Gregorian calendar they are 97, that is why the average length of Gregorian year is 36597400 days = 365,2425 days = 365 days 5 hours 49 minutes 12 seconds, it is greater 26 seconds than the current period. There are different types of moon-and-sun calendars in states of South-Eastern Asia, Iran and Israel.
Usage in physics.
Let’s consider the following examples of the continued fractions usage in physics.
Example1. Making resistances.
The following information is clear to us from the physics textbooks. If we connect resistors with resistance in chain, then the total resistance of the chain will be equal to the number . And if we connect these resistors parallel, then the total resistance will be equal to this number:
Now let’s take many resistors with resistance equal to 1. Can we make an electric chain with the resistors’ resistance ?
The solution of this problem seems to be simple, because in order to make an electric chain with the resistance it is necessary to connect resistors with resistance equal to 1 parallel, then we take a chain with the resistance , after this it is possible to connect such chains times in chain. When making this chain we will need a resistor . Therefore, in order to make a chain with the resistance we will need a resistor . The number of resistors is less than 14, but can we make a chain with the resistance ?
For this purpose we are going to use the method of the continued fractions. If we connect 2 resistors with the resistance equal to 1 parallel, then the chain’s resistance will be . Now if we connect three resistors equal to 1 in chain, then the total resistance will be .
Hereof to make a chain with the resistance we need 5 resistors.
Example 2. It is necessary to make a chain with the resistance .
To make this chain we can connect 7 resistors parallel and connect such 10 chains in chain, then the resistance will be . For it we need a resistor . Can we use resistors less than 70 to make a chain with the resistance ?
To answer this question we are going to group into continued fractions:
For this we make 2 resistors with the resistance equal to 1 and 1 block of 3 resistors connected in chain. Then the resistance of this block will be:
And now we connect this block to 1 resistor with the resistance equal to 1 in chain. Then the resistance of the whole chain will be: .
Usage in music
Since the time of Bach evenly tempered scale containing 12 semitones in each octave is used in music. If the string length l ( for a given tension ) produces a sound "do" the first octave , corresponding to the frequency f, equal to 512 vibrations per second, the string length ( stringed instruments , this length is obtained by pressing a finger in the appropriate place ) makes a sound , having frequency ( original fifth) , and the string length produces a sound having a frequency 2f ( octave) . Our ear when comparing two sounds captures not the attitude of their frequencies , and the logarithm of this ratio . Naturally just take the binary logarithm to one octave interval was measured as a unit :
Why is there the division of octave to 12 intervals? To make octave and natural quint fit into the same equal temperament ( the division of the octave into equal intervals by ear ) , octave should be divided into so many parts to make number of close to the selected fraction denominator . Suitable fractions among
will be shot
Approximation 1 and 1/ 2 is too rough . Approximation 3/5 corresponds to the pentatonic that existed among the peoples of the East, and the approach of 7/12 - the most suitable . Error
audibly indistinguishable .