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Usage in physics.

Continued fractions and their usage | Scheme of the continued fractions | Convergent fractions | Some ways to use the continued fractions | Finding a square root | Finding one pair roots of the equation performed through the continued fractions | Usage of the continued fractions in other spheres |


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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: .

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