Continued Fractions

We all reach for 22/7 when we need π in a hurry, and astronomers swear by 355/113, which nails π to six decimals. These are not lucky guesses. They fall out of one of the most elegant ideas in number theory: the continued fraction.

The recipe is almost childishly simple. Take any number, write down its whole part, then flip the leftover fraction over and repeat. Do that to π and you get 3, then 7, then 15, then 1, then 292… Stop anywhere and you have a fraction — and that fraction is, in a precise sense, the best possible for its size.

This little peeling process is everywhere it counts: it is why a leap year lands every four years (but not every century), why a clockmaker chooses one gear-tooth count over another, and how mathematicians cracked some of the deepest questions about irrational numbers. Continued fractions are not an open problem or an impossibility result — they are established mathematics, with the best-approximation property proved by Joseph-Louis Lagrange in the 18th century. What makes them worth your time is how much they explain.

Choose a number below — or type your own — and the demo will peel it into a continued fraction. Each row is a convergent: the fraction you get by stopping at that term. Watch the error column shrink dramatically with every step, and notice the famous fractions appearing on their own.

<p class="hint">{{hint}}</p>
<div class="controls">
  <button type="button" data-x="3.141592653589793" data-n="π">π</button>
  <button type="button" data-x="1.618033988749895" data-n="{{golden_ratio_label}}">φ ({{golden}})</button>
  <button type="button" data-x="1.4142135623730951" data-n="√2">√2</button>
  <button type="button" data-x="2.718281828459045" data-n="e">e</button>
  <button type="button" data-x="365.2422" data-n="{{year_label}}">365.2422 ({{year}})</button>
</div>
<div class="custom">
  <input id="num" type="text" value="3.141592653589793" aria-label="{{aria_number}}" />
  <button id="go" type="button">{{expand}}</button>
</div>
<div class="cf" id="cf"></div>
<table class="conv"><thead><tr><th>{{th_term}}</th><th>{{th_convergent}}</th><th>{{th_value}}</th><th>{{th_error}}</th></tr></thead>
<tbody id="rows"></tbody></table>

* { box-sizing: border-box; }
body { font-family: system-ui, sans-serif; color: #222; margin: 0; }
.hint { font-size: .9rem; color: #444; margin: 0 0 .7rem; line-height: 1.45; }
.controls { display: flex; gap: .4rem; flex-wrap: wrap; margin-bottom: .5rem; }
.controls button { font: 600 13px system-ui, sans-serif; padding: .4rem .7rem; border: 1px solid #1d3557;
  background: #eef3f8; color: #1d3557; border-radius: 8px; cursor: pointer; }
.controls button:hover { background: #dce6f0; }
.custom { display: flex; gap: .4rem; margin-bottom: .8rem; }
.custom input { flex: 1; font: 600 14px ui-monospace, monospace; padding: .4rem .6rem;
  border: 1px solid #cdd9e3; border-radius: 8px; min-width: 0; }
.custom button { font: 600 14px system-ui, sans-serif; padding: .4rem .9rem; border: 1px solid #1d3557;
  background: #1d3557; color: #fff; border-radius: 8px; cursor: pointer; }
.cf { font: 600 15px ui-monospace, monospace; color: #1d3557; background: #f3f6fa;
  border: 1px solid #dde6ee; border-radius: 8px; padding: .6rem .7rem; margin-bottom: .8rem;
  word-break: break-word; }
table.conv { border-collapse: collapse; width: 100%; font-size: .9rem; }
table.conv th, table.conv td { text-align: left; padding: .35rem .5rem; border-bottom: 1px solid #e6ecf2; }
table.conv th { color: #5a7088; font-weight: 700; }
table.conv td { font-family: ui-monospace, monospace; }
td.frac { font-weight: 700; color: #1d3557; }
td.err { color: #0a7d33; }
tr.famous td.frac { background: #fff3cd; border-radius: 6px; }

// Code not found

Try π and watch 22/7 and then 355/113 pop out. Try the golden ratio (the hardest number to approximate) and see only 1s — its convergents are ratios of consecutive Fibonacci numbers, and they close in as slowly as any number possibly can. The key fact: no fraction with a smaller denominator ever beats one of these convergents.

What exactly makes these fractions best? Here is the precise picture.

• The algorithm is just division. Write the whole part, take the reciprocal of the remainder, repeat. For a rational number it stops; for an irrational it runs forever.
• Convergents come from a clean recurrence. If the terms are $a_{0}$, $a_{1}$, $a_{2}$, … then numerators and denominators each follow $h_n = a_n h_{n-1} + h_{n-2}$ and $k_n = a_n k_{n-1} + k_{n-2}$. Cheap to compute, exact in integer arithmetic.
• They are provably the best approximations. This is Lagrange's theorem: each convergent $p/q$ is closer to the number than any other fraction with denominator $\le q$. You cannot do better without using bigger numbers.
• A big term means a great shortcut. The 292 in π's expansion is why 355/113 is so spectacularly accurate — a large term means the previous convergent was already almost perfect.
• All 1s is the worst case. The golden ratio $\varphi = [1;\, 1, 1, 1, \ldots]$ is the hardest number to approximate by rationals; that is exactly what makes it the "most irrational" number.

None of this is conjectural. The status is solved, classical mathematics — Euler and Lagrange settled it in the 1700s. Continued fractions sit on the easy side of computation: expanding a number and reading off its best fractions is fast and exact, a refreshing contrast to the hard problems like factoring elsewhere on KipuHub.

"Find the simplest fraction that is close enough" is a question engineers and scientists ask constantly, and continued fractions answer it optimally:

• Calendars. The year is about 365.2422 days. Its convergents give 365 + 1/4 (the Julian rule) and refinements close to the modern 97-leap-days-per-400-years Gregorian rule. Leap years are a continued fraction.
• Gears and clocks. To get an exact ratio with a buildable number of teeth, gearmakers pick a convergent — the best ratio achievable with small wheels.
• Music. Twelve equal semitones approximate a stack of perfect fifths because 7/12 is a great convergent of $\log_{2}(3/2)$.
• Cryptanalysis. The Wiener attack breaks RSA keys with a small secret exponent by reading it off a continued fraction — the same best-approximation property turned into a weapon, related to the hardness of factoring.

The thread is always the same: the most efficient rational stand-in for an awkward real number. For the contrast with problems that have no efficient answer, see P vs NP.

Continued fractions take a messy real number and hand back a ladder of fractions, each one the best you could possibly write at that size. It is why 22/7 and 355/113 feel magical, why the calendar skips a leap year three times every four centuries, and why the golden ratio earns its title as the most irrational number.

Best of all, this is a corner of mathematics that is completely solved — no asterisks, no open conjectures, just a beautiful, exact, fast algorithm. After all the impossibility results and hard problems on KipuHub, like P vs NP, it is worth savoring one idea that simply works, and works perfectly.