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.
Understanding Math: Continued Fractions
One way to create the terms is to use repeated division, the same idea behind the Euclidean algorithm for finding a greatest common divisor. Start with a fraction such as thirteen fifths. The whole-number part is two, leaving three fifths.
Turn the remaining part upside down to get five thirds. Its whole-number part is one, leaving two thirds. Repeating this process gives two, one, one, two.
The process stops because the remainders get smaller at each division. This explains why rational numbers produce finite lists of terms.
There is an efficient pattern for calculating each convergent without rebuilding the entire nested expression. Keep track of two earlier numerators and two earlier denominators. The new numerator equals the current term times the previous numerator plus the numerator before that.
The new denominator follows the same rule. This pattern is useful in computer programs because it uses only whole-number arithmetic.
It also shows that neighboring convergents are closely linked. Their cross-products differ by one, so they cannot be reduced to a simpler fraction in the usual way.
The approximations from continued fractions have a special strength. A convergent is usually much closer to the target number than a random fraction with a similar-sized denominator. The next term gives an important clue.
When that term is large, the current convergent is especially accurate. For pi, the fraction twenty-two sevenths works well partly because the next relevant term is fifteen.
This matters when a measurement, a gear ratio, or a repeating schedule needs small whole numbers. A denominator of seven is often easier to use than a denominator of several hundred, even when the larger fraction is more accurate.
Some infinite patterns reveal a number type immediately. Square roots of whole numbers that are not perfect squares have repeating continued fraction terms after the first term. For example, the square root of two begins with one followed by twos forever.
This repeating behavior connects continued fractions to quadratic equations and to Pell equations, which ask for whole-number solutions to equations involving squares. Such equations appear in number theory and in old problems about nearly matching lengths or areas with whole-number measurements.
When working by hand, start at the innermost fraction and move outward. A common mistake is to add a term before taking the reciprocal of the part below it. Another mistake is to round decimals too early, since a small rounding change can alter later terms.
Finite expansions can have two valid endings because a final term greater than one can be replaced by one less followed by a final one. Using the version whose last term is greater than one keeps answers consistent. Check each convergent with a calculator only after finding it exactly as a fraction.
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.