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Simple continued fraction

Simple continued fraction is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Simple continued fraction rather than just read about it. In short: A simple or regular continued fraction is a continued fraction with numerators all equal to one, and denominators built from a sequence { a i } {\displaystyle \{a_{i}\}} of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fraction like a 0 + 1 a 1 + 1 a 2 + 1 ⋱ + 1 a n {\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n…

Simple continued fraction — main illustration
Simple continued fraction — illustration

Key takeaways

  • Simple continued fraction belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Simple continued fraction to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Simple continued fraction from memory before moving on to harder problems.

Reference excerpt

A simple or regular continued fraction is a continued fraction with numerators all equal to one, and denominators built from a sequence { a i } {\displaystyle \{a_{i}\}} of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fraction like

a 0 + 1 a 1 + 1 a 2 + 1 ⋱ + 1 a n {\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}}}

or an infinite continued fraction like

a 0 + 1 a 1 + 1 a 2 + 1 ⋱ . {\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots }}}}}}.}

… excerpt ends here. Continue reading the full article.

Illustrations

Simple continued fraction: Convergents approaching the golden ratio
Convergents approaching the golden ratio
Simple continued fraction: Best rational approximants for irrational numbers π {\displaystyle \pi } (green circle), e {\displaystyle e} (blue diamond), ϕ {\displaystyle \phi } (pink oblong), 3 / 2 {\displaystyle {\sqrt {3}}/2} (grey hexagon), 1 / 2 {\displaystyle 1/{\sqrt {2}}} (red octagon) and 1 / 3 {\displaystyle 1/{\sqrt {3}}} (orange triangle) calculated from their continued fraction expansions, plotted as slopes y / x {\displaystyle y/x} with errors from their true values (black dashes)   .mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:"\a0 · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}vte
Best rational approximants for irrational numbers π {\displaystyle \pi } (green circle), e {\displaystyle e} (blue diamond), ϕ {\displaystyle \phi } (pink oblong), 3 / 2 {\displaystyle {\sqrt {3}}/2} (grey hexagon), 1 / 2 {\displaystyle 1/{\sqrt {2}}} (red octagon) and 1 / 3 {\displaystyle 1/{\sqrt {3}}} (orange triangle) calculated from their continued fraction expansions, plotted as slopes y / x {\displaystyle y/x} with errors from their true values (black dashes)   .mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:"\a0 · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}vte
Simple continued fraction: The following Maple code will generate continued fraction expansions of pi
The following Maple code will generate continued fraction expansions of pi

Worked examples

Example 1 — a first encounter with Simple continued fraction

Start with the simplest possible case. Write down what Simple continued fraction claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Simple continued fraction before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Simple continued fraction ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Simple continued fraction

In research
Simple continued fraction appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Simple continued fraction in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Simple continued fraction is common in secondary-school and first-year university syllabi. It links to neighbouring topics Continued fractions, Mathematical analysis, so understanding it makes those chapters shorter.
In everyday life
Look for Simple continued fraction outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Simple continued fraction in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Simple continued fraction means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Simple continued fraction out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Simple continued fraction in simple terms?

A simple or regular continued fraction is a continued fraction with numerators all equal to one, and denominators built from a sequence { a i } {\displaystyle \{a_{i}\}} of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fraction like a 0 + 1…

Why does Simple continued fraction matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Simple continued fraction?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Simple continued fraction.

Tags

  • Continued fractions
  • Mathematical analysis

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