Understanding Continued Fractions

What is a Continued Fraction?

A continued fraction is a representation of a real number using a sequence of integers and reciprocals. It's an alternative way to express numbers, especially useful for representing irrational numbers.

Formula and Notation

A continued fraction is typically written as:

\[a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \ddots}}}\]

Where \(a_0\) is an integer, and \(a_1, a_2, a_3, \ldots\) are positive integers.

It's often abbreviated as \([a_0; a_1, a_2, a_3, \ldots]\).

Calculation Steps

To convert a number \(x\) to a continued fraction:

1. Take the integer part of \(x\): \(a_0 = \lfloor x \rfloor\)
2. Calculate the fractional part: \(r = x - a_0\)
3. If \(r = 0\), stop. Otherwise, set \(x = \frac{1}{r}\) and go back to step 1

Example

Let's convert 3.245 to a continued fraction:

1. \(a_0 = \lfloor 3.245 \rfloor = 3\), \(r = 3.245 - 3 = 0.245\)
2. \(x = \frac{1}{0.245} = 4.0816...\)
3. \(a_1 = \lfloor 4.0816... \rfloor = 4\), \(r = 4.0816... - 4 = 0.0816...\)
4. \(x = \frac{1}{0.0816...} = 12.25\)
5. \(a_2 = \lfloor 12.25 \rfloor = 12\), \(r = 12.25 - 12 = 0.25\)
6. \(x = \frac{1}{0.25} = 4\)
7. \(a_3 = 4\)

Therefore, 3.245 ≈ [3; 4, 12, 4]

Visual Representation

This diagram visually represents the continued fraction for 3.245.