Continued Fractions Reference Cheat Sheet

Continued fractions are expressions that represent numbers through repeated integer parts and reciprocals. They help students approximate irrational numbers, analyze rational numbers, and understand why some fractions give especially accurate approximations. This cheat sheet gives the notation, algorithms, and main formulas needed for advanced algebra, number theory, and pre-calculus work.

Key Facts

A simple continued fraction has the form $[a_0; a_1, a_2, a_3, \ldots] = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \cdots}}}$ , where $a_0$ is an integer and $a_i$ for $i \ge 1$ are positive integers.
To convert a rational number to a finite continued fraction, repeatedly divide using the Euclidean algorithm: $x = a_0 + \frac{1}{x_1}$ , then $x_1 = a_1 + \frac{1}{x_2}$ , and continue until the remainder is $0$ .
The $n$ th convergent of $[a_0; a_1, \ldots, a_n]$ is $\frac{p_n}{q_n}$ , a rational approximation formed by stopping the continued fraction after $a_n$ .
The convergent numerators and denominators satisfy $p_n = a_n p_{n-1} + p_{n-2}$ and $q_n = a_n q_{n-1} + q_{n-2}$ .
Useful starting values for convergents are $p_{-2} = 0$ , $p_{-1} = 1$ , $q_{-2} = 1$ , and $q_{-1} = 0$ .
Consecutive convergents obey $p_n q_{n-1} - p_{n-1} q_n = (-1)^{n-1}$ , so each convergent $\frac{p_n}{q_n}$ is already in lowest terms.
For an irrational number $x$ , the convergents usually alternate around $x$ , and the error satisfies $\left|x - \frac{p_n}{q_n}\right| < \frac{1}{q_n q_{n+1}}$ .
A quadratic irrational number has an eventually periodic continued fraction, such as $\sqrt{2} = [1; \overline{2}]$ .

Vocabulary

Continued fraction: A continued fraction is a number representation built from an integer plus the reciprocal of another expression of the same type.
Simple continued fraction: A simple continued fraction is a continued fraction $[a_0; a_1, a_2, \ldots]$ whose terms after $a_0$ are positive integers.
Partial quotient: A partial quotient is one of the integers $a_0, a_1, a_2, \ldots$ in a continued fraction.
Convergent: A convergent is a rational approximation $\frac{p_n}{q_n}$ found by stopping a continued fraction after finitely many partial quotients.
Periodic continued fraction: A periodic continued fraction is a continued fraction whose partial quotients repeat in a pattern, written with an overline such as $[1; \overline{2}]$ .
Euclidean algorithm: The Euclidean algorithm is the repeated division process used to find greatest common divisors and finite continued fractions for rational numbers.

Common Mistakes to Avoid

Dropping the reciprocal step is wrong because $[a_0; a_1, a_2]$ means $a_0 + \frac{1}{a_1 + \frac{1}{a_2}}$ , not $a_0 + a_1 + a_2$ .
Using zero or negative values for $a_i$ when $i \ge 1$ is wrong for simple continued fractions because those partial quotients must be positive integers.
Forgetting the initial recurrence values is wrong because $p_n = a_n p_{n-1} + p_{n-2}$ and $q_n = a_n q_{n-1} + q_{n-2}$ need $p_{-2} = 0$ , $p_{-1} = 1$ , $q_{-2} = 1$ , and $q_{-1} = 0$ .
Rounding too early is wrong because continued fractions rely on exact integer parts and exact reciprocals, so decimal rounding can change later partial quotients.
Assuming every infinite continued fraction is periodic is wrong because periodic continued fractions correspond to quadratic irrational numbers, not all irrational numbers.

Practice Questions

1 Write $\frac{43}{19}$ as a simple continued fraction.
2 Find the first four convergents of $[1; 2, 2, 2, \ldots]$ .
3 Use the recurrence formulas to find $\frac{p_3}{q_3}$ for $[2; 1, 3, 4]$ .
4 Explain why the continued fraction for a rational number must end, but the continued fraction for an irrational number does not.

Understanding Continued Fractions Reference

The division process behind a continued fraction is the same process used to find the greatest common divisor of two whole numbers. Each whole-number quotient becomes one entry in the expansion. This explains why a rational number eventually stops.

Its remainders become smaller nonnegative whole numbers, so one remainder must reach zero. For an irrational number, there is no final exact remainder, and the entries continue forever. When working by hand, keep every reciprocal step separate.

A common error is to take the reciprocal before removing the whole-number part. Another common error is forgetting that entries after the first one must be positive in a simple continued fraction.

Convergents are useful because they improve an estimate in a very organized way. Their denominators often grow quickly, while the error becomes very small. This makes them much better than choosing a nearby fraction at random.

More importantly, a convergent is often the best possible approximation among fractions whose denominator is no larger than its own denominator. That idea matters whenever a quantity must be represented with limited whole-number parts.

A gear ratio, a musical tuning ratio, a screen refresh relationship, or a repeating schedule may need a simple fraction rather than a long decimal. Continued fractions show which simple fractions preserve the most accuracy.

The recurrence rule is efficient because it avoids rebuilding every nested fraction from the beginning. To find the next numerator or denominator, multiply the latest value by the new entry, then add the value before it. The initial values are not arbitrary bookkeeping.

They make the same rule work from the first convergent onward. The relationship between neighboring convergents has another important consequence. Their cross-product difference has absolute value one.

Therefore no whole number greater than one can divide both parts of a convergent. It also means neighboring convergents are extremely close together in a precise fraction sense. This helps explain why the target value usually lies between successive estimates.

Large entries in a continued fraction deserve special attention. A large entry produces an unusually strong approximation at the step just before it. For example, if a decimal is very close to a fraction with a modest denominator, its continued fraction often has a large later entry.

This is a useful pattern in data analysis, measurement, and numerical work, though measured values can contain rounding error. Periodic patterns are important for square roots that are not whole numbers. Their repetition comes from a finite set of possible remainder states during the calculation.

This fact connects continued fractions to quadratic equations and to Pell equations, which seek whole-number solutions to equations involving squares. Students should distinguish a repeating decimal from a periodic continued fraction. The repeating blocks follow different rules, even when they describe the same number.

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Continued Fractions Reference Cheat Sheet

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