- #1
bgwyh_88
- 5
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Hi all,
Could anyone guide me on the following prove
√2 = 1+1/(2 + 1/(2+ 1/(2+ 1/(2+···))))
Could anyone guide me on the following prove
√2 = 1+1/(2 + 1/(2+ 1/(2+ 1/(2+···))))
micromass said:Hi bgwyh_88!
Let
[tex]x=1+\frac{1}{2+\frac{1}{2+...}}[/tex]
Then what is [itex]\frac{1}{x-1}-1[/itex]?
bgwyh_88 said:hey micromass,
You will ultimately get
1+ [itex]\frac{1}{2+\frac{1}{2+...}}[/itex]
Where did you get
[itex]\frac{1}{x-1}-1[/itex] from?
micromass said:Yes, and that is x. So [itex]\frac{1}{x-1}-1=x[/itex]
You just need to transform x to something that equal x again. It's a standard trick that you had to see once...
A continuous fraction for root 2 is a representation of the irrational number √2 in the form of a fraction, where the numerator and denominator are both integers and the fraction is written in a specific pattern of nested fractions.
The continuous fraction for root 2 can be found by following a specific algorithm that involves dividing the number 1 by the irrational number √2 and then repeating the process with the reciprocal of the resulting decimal fraction until the decimal fraction becomes periodic.
The pattern of a continuous fraction for root 2 is 1, followed by a nested fraction with the numerator being the denominator of the previous fraction and the denominator being the sum of the numerator and denominator of the previous fraction. This pattern continues until the decimal fraction becomes periodic.
The continuous fraction for root 2 is important because it provides an alternate way of representing the irrational number √2, which has many applications in mathematics, physics, and engineering. It also allows for easier approximation of the value of √2, which is useful in practical calculations.
Yes, there are many real-life applications of continuous fractions for root 2, such as in the design of musical instruments, where the length of the strings can be represented using these fractions. They are also used in the field of number theory and in cryptography for data encryption.