# Continuous fractions for root 2

• bgwyh_88
In summary, The conversation is about a mathematical proof regarding continued fractions and the question of what \frac{1}{x-1}-1 is. The term is corrected to be "continued" fraction, not "continuous" fraction. The final result of the proof is x=1+\frac{1}{2+\frac{1}{2+...}}, and \frac{1}{x-1}-1=x. The conversation ends with a thank you.
bgwyh_88
Hi all,

Could anyone guide me on the following prove

√2 = 1+1/(2 + 1/(2+ 1/(2+ 1/(2+···))))

Hi bgwyh_88!

Let

$$x=1+\frac{1}{2+\frac{1}{2+...}}$$

Then what is $\frac{1}{x-1}-1$?

By the way, the term in English is "continued" fraction, not "continuous" fraction.

micromass said:
Hi bgwyh_88!

Let

$$x=1+\frac{1}{2+\frac{1}{2+...}}$$

Then what is $\frac{1}{x-1}-1$?

hey micromass,

You will ultimately get

1+ $\frac{1}{2+\frac{1}{2+...}}$

Where did you get

$\frac{1}{x-1}-1$ from?

bgwyh_88 said:
hey micromass,

You will ultimately get

1+ $\frac{1}{2+\frac{1}{2+...}}$

Yes, and that is x. So $\frac{1}{x-1}-1=x$

Where did you get

$\frac{1}{x-1}-1$ from?

You just need to transform x to something that equal x again. It's a standard trick that you had to see once...

micromass said:
Yes, and that is x. So $\frac{1}{x-1}-1=x$

You just need to transform x to something that equal x again. It's a standard trick that you had to see once...

micromass,

Cool. Thanks mate.

## 1. What is a continuous fraction for root 2?

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.

## 2. How do you find the continuous fraction for root 2?

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.

## 3. What is the pattern of a continuous fraction for root 2?

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.

## 4. Why is the continuous fraction for root 2 important?

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.

## 5. Are there any real-life applications of continuous fractions for root 2?

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.

• General Math
Replies
6
Views
1K
• General Math
Replies
12
Views
1K
• General Math
Replies
6
Views
2K
• General Math
Replies
4
Views
1K
• General Math
Replies
1
Views
836
• General Math
Replies
1
Views
945
• General Math
Replies
1
Views
937
• General Math
Replies
1
Views
1K
• General Math
Replies
1
Views
2K
• General Math
Replies
3
Views
680