Continued fractions express numbers by nesting fractions inside fractions, creating a compact staircase of denominators. They are useful because they often give excellent rational approximations to irrational numbers using surprisingly small integers. This makes them important in number theory, computation, measurement, and any situation where a decimal must be replaced by a fraction.
They also reveal hidden structure in numbers that ordinary decimal notation can hide.
A simple continued fraction has the form a0 + 1/(a1 + 1/(a2 + 1/(a3 + ...))), where a0 is an integer and the later terms are positive integers. Cutting off the fraction after a certain number of terms gives a convergent, which is a rational approximation to the original number. For example, π begins [3; 7, 15, 1, ...], and the convergent [3; 7] equals 22/7.
Infinite continued fractions can represent irrational numbers exactly, while finite continued fractions always represent rational numbers.
Key Facts
- Simple continued fraction form: x = a0 + 1/(a1 + 1/(a2 + 1/(a3 + ...))).
- Bracket notation: [a0; a1, a2, a3] means a0 + 1/(a1 + 1/(a2 + 1/a3)).
- Every finite simple continued fraction represents a rational number.
- Every irrational number has an infinite, nonrepeating continued fraction expansion.
- Convergents are found by truncating: [a0], [a0; a1], [a0; a1, a2], and so on.
- The golden ratio has continued fraction φ = [1; 1, 1, 1, ...] and satisfies φ = 1 + 1/φ.
Vocabulary
- Continued fraction
- A continued fraction is an expression where a number is written using a sequence of nested fractions.
- Simple continued fraction
- A simple continued fraction is a continued fraction whose numerators are all 1 and whose partial quotients are integers.
- Partial quotient
- A partial quotient is one of the integer entries a0, a1, a2, and so on in a continued fraction.
- Convergent
- A convergent is a rational approximation made by stopping a continued fraction after a finite number of terms.
- Irrational number
- An irrational number is a number that cannot be written as a ratio of two integers and has an infinite nonrepeating decimal expansion.
Common Mistakes to Avoid
- Reading [3; 7, 15] as 3/7/15 is wrong because bracket notation means nested addition and reciprocals, not repeated division.
- Forgetting the reciprocal step is wrong because each new partial quotient appears in a denominator under 1, so [2; 3] equals 2 + 1/3, not 2 + 3.
- Assuming more decimal places always give better fractions is wrong because continued fraction convergents often give the best small-denominator approximations.
- Treating a finite continued fraction as irrational is wrong because any finite nesting of integer operations and reciprocals simplifies to a rational number.
Practice Questions
- 1 Evaluate the continued fraction [2; 3, 4] as a single fraction in lowest terms.
- 2 Find the first three convergents of sqrt(2) using sqrt(2) = [1; 2, 2, 2, ...].
- 3 Explain why [1; 1, 1, 1, ...] represents a number greater than 1 but less than 2, without converting it to a decimal.