Continued Fraction — Definition, Formula & Examples

A continued fraction is a way of expressing a number as a whole number plus a fraction whose denominator itself contains a whole number plus a fraction, and so on. This nested structure can be finite (for rational numbers) or infinite (for irrational numbers).

A continued fraction is an expression of the form $a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cdots}}}$ , where $a_0$ is an integer and $a_1, a_2, a_3, \ldots$ are positive integers called partial quotients. Every real number has a unique continued fraction expansion (with the convention that finite expansions do not end in 1).

Key Formula

x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cdots}}}

Where:

$x$ = The real number being represented
$a_0$ = The integer part of x (can be zero or negative)
$a_1, a_2, a_3, \ldots$ = Positive integers called partial quotients

How It Works

To find the continued fraction of a number, you repeatedly take the integer part and then invert the fractional remainder. Start with your number, write down the integer part as

a_0

, subtract it, and take the reciprocal of what remains. Repeat this process on each new value to get

a_1, a_2, a_3, \ldots

. For a rational number, the process terminates after finitely many steps. For an irrational number like

\sqrt{2}

, the partial quotients continue forever, often with a repeating pattern.

Worked Example

Problem: Express the fraction 43/19 as a continued fraction.

Step 1: Divide 43 by 19. The integer part is 2 with remainder 5, so the fractional part is 5/19.

\frac{43}{19} = 2 + \frac{5}{19}

Step 2: Invert the fractional part to get 19/5. The integer part is 3 with remainder 4, so the fractional part is 4/5.

\frac{19}{5} = 3 + \frac{4}{5}

Step 3: Invert 4/5 to get 5/4. The integer part is 1 with remainder 1.

\frac{5}{4} = 1 + \frac{1}{4}

Step 4: Invert 1/4 to get 4, which is a whole number. The process terminates.

\frac{43}{19} = 2 + \cfrac{1}{3 + \cfrac{1}{1 + \cfrac{1}{4}}}

Answer: The continued fraction representation is [2; 3, 1, 4], meaning the partial quotients are 2, 3, 1, and 4.

Why It Matters

Continued fractions provide the best rational approximations to irrational numbers — this is how ancient astronomers computed accurate calendar ratios. They appear in number theory courses and in algorithms like the Euclidean algorithm for finding greatest common divisors.

Common Mistakes

Mistake: Forgetting to invert the fractional remainder before finding the next partial quotient.

Correction: After subtracting the integer part, always take the reciprocal of the leftover fraction before repeating the process.