Continued Fractions Intro
Nested fractions that give best rational approximations of reals.
Key idea
Nested fractions that give best rational approximations of reals.
In depth
A continued fraction writes a number as a + 1/(b + 1/(c + ...)). Truncating gives the best rational approximations. Rational numbers have finite expansions; irrational ones go on forever.
Worked example
3.245 = 3 + 1/(4 + 1/12 + ...). π = [3; 7, 15, 1, ...], and truncating gives 22/7 and 333/106.
Formula
Real-world uses
- Finding good rational approximations of π for engineering
- Calendar cycle and gear ratio design
Common mistake
Thinking the order of terms can be rearranged.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Continued Fractions Intro?
Nested fractions that give best rational approximations of reals. It is a topic at the expert level in Arithmetic & Number Sense.
What is the formula for Continued Fractions Intro?
The key formula is a₀ + 1/(a₁ + 1/(a₂+…)).
Where is Continued Fractions Intro used in real life?
Real-world examples: Finding good rational approximations of π for engineering; Calendar cycle and gear ratio design.
What should I learn before Continued Fractions Intro?
Make sure you are comfortable with Egyptian Fractions, the previous topic in Arithmetic & Number Sense.
What is a common mistake with Continued Fractions Intro?
Thinking the order of terms can be rearranged.
How can I practise Continued Fractions Intro?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Continued Fractions Intro?
Move on to Estimation with Benchmarks next.
Related expert topics in Arithmetic & Number Sense
- Egyptian Fractions – Writing a fraction as a sum of distinct unit fractions.