§2. 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: .

§3. Usage of the continued fractions in other spheres

In 1862 жылы Holland scholar Christian Huygens constructed one of the first mechanic planetariums. To count the movement of the planets he used the theory of the continued fractions. In nature most of phenomena change according to the laws of the spiral curve. For example, lieves of tree branches are placed in the kind of a spiral or a screw. He found the spiral in the placing of sunflower seeds, pinecons, pineapple and cactus thorns. Collaborate work of the botanists and mathematicians explained the reason of such wonderful phenomena of nature. Placement of the branch life, sunflower seed, pinecone show Fibonacci numbers, i.e. this is based on the law of golden ratio. Spider’s net and windstorms are as a spiral. When North roes are scared they run away like a spiral. DNA molecule is covered with two spiral. For example, three spirals of a pinecon are directed to one side and five spirals are directed to another side. Every alternate number is equal to the sum of the two numbers first from the third one:

0,1,1,22,3,5,8,13,21,34,55,89,144,233,…

For example, three spirals of a pinecon are directed to one side and five spirals are directed to another side. In pinecone there is a spiral -5 and 8, and ___ has a spiral -8 and 13. Now if we consider the ratio between adjacent numbers of Feboccini, then they are divided into the continued fractions in this way: 53 = 1;1,1,1 85 = 1;1,1,1,1,2113=1;1,1,1,1,1,1,1,8955 = [1;1,1,1,1,1,1,1,1,1,1]. If we consider a convergent fraction called in mathematics as a golden ratio: 1+52 = 1+11+11+11+11+…=…5.

Placement of a leave in such a spiral form makes it possible for a plant to take a lot of sun rays.

B.C. times Babylonians found out that the sun and the moon eclipse repeated in 18 years and 10 days. But why they could not explain this repetition. The reason of repetition of the eclipse phenomenon was defined much later. To define it a theory of the continued fractions was used.

Continued fractions are widely used in botany, music, mechanics, physiscs and architecture.

Conclusion

Having investigated this topic we get known one type of the fractions – continued fractions. Due to our scientific project in addition to the continued fractions we learned their features, their convergent fractions, their usage in finding the roots of the equations and the way of converting common fractions into the continued fractions. Also we studied the ways of the usage in mathematics and other natural sciences.

Moreover, the role of the continued fractions in making a calendar is great. We have considered in this project accuracy and correlations among the continued fractions of calendars like of Julian, Persian (Omar Khayyam), Gregorian, Midler and also the differences among these calendars.

If to consider the usage in physics when using these fractions in making an electric chain, it was found that is possible to save much voltage and some resistors. Also we have given proofs to the usage of the continued fractions in botany, mechanics and architecture. For example, lieves of tree branches are placed in the kind of a spiral or a screw. the spiral was found in the placing of sunflower seeds, pinecons, pineapple and cactus thorns.