Where Continued Fractions Matter

Introduction

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.

Try It: Watch the Convergents Close In

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">Pick a number or type your own, then peel it into a continued fraction.
    Each row is a <b>convergent</b> — the fraction you get by stopping there. Watch the
    <b>error</b> column collapse, and the famous fractions appear on their own.</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 φ">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="days in a year">365.2422 (year)</button>
</div>
<div class="custom">
  <input id="num" type="text" value="3.141592653589793" aria-label="number" />
  <button id="go" type="button">Expand</button>
</div>
<div class="cf" id="cf"></div>
<table class="conv"><thead><tr><th>term</th><th>convergent</th><th>value</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; }

// Continued fraction expansion: write the whole part, take the reciprocal of the
// leftover, repeat. Convergents h_n/k_n use the recurrence
//   h_n = a_n*h_{n-1} + h_{n-2},  k_n = a_n*k_{n-1} + k_{n-2}.
// Lagrange's theorem (18th c.): each convergent is the BEST rational approximation
// among all fractions with denominator <= its own. Classical, solved mathematics.
const FAMOUS = new Set(['22/7','355/113','3/2','7/5','17/12','1/4','365/1']);

function expand(x){
  const terms=[]; const conv=[];
  let v=x, hm1=1,hm2=0,km1=0,km2=1;
  for(let i=0;i<12;i++){
    const a=Math.floor(v); terms.push(a);
    const h=a*hm1+hm2, k=a*km1+km2;
    conv.push([h,k]);
    hm2=hm1; hm1=h; km2=km1; km1=k;
    const frac=v-a;
    if(frac < 1e-10) break;   // rational / numerical floor reached
    v=1/frac;
    if(k > 1e9) break;        // keep denominators readable
  }
  return {terms, conv};
}

function fmtCF(terms){
  if(terms.length===1) return '[' + terms[0] + ']';
  return '[' + terms[0] + '; ' + terms.slice(1).join(', ') + ']';
}

function show(x, name){
  const {terms, conv} = expand(x);
  document.getElementById('cf').textContent =
    (name?name+'  =  ':'') + fmtCF(terms);
  const tb=document.getElementById('rows'); tb.innerHTML='';
  conv.forEach(([h,k],i)=>{
    const val=h/k, err=Math.abs(x-val);
    const tr=document.createElement('tr');
    const key=h+'/'+k;
    if(FAMOUS.has(key)) tr.className='famous';
    tr.innerHTML =
      '<td>'+terms[i]+'</td>'+
      '<td class="frac">'+key+'</td>'+
      '<td>'+val.toFixed(8)+'</td>'+
      '<td class="err">'+(err===0?'exact':err.toExponential(2))+'</td>';
    tb.appendChild(tr);
  });
  console.log(name||x, '->', fmtCF(terms),
    '| best convergent', conv[conv.length-1].join('/'));
}

document.querySelectorAll('.controls button').forEach(b=>{
  b.addEventListener('click',()=>{
    document.getElementById('num').value=b.dataset.x;
    show(parseFloat(b.dataset.x), b.dataset.n);
  });
});
document.getElementById('go').addEventListener('click',()=>{
  const x=parseFloat(document.getElementById('num').value);
  if(isFinite(x)) show(x, '');
  else document.getElementById('cf').textContent='Enter a valid number.';
});
show(Math.PI, 'π');

<p class="hint">Pick a number or type your own, then peel it into a continued fraction.
    Each row is a <b>convergent</b> — the fraction you get by stopping there. Watch the
    <b>error</b> column collapse, and the famous fractions appear on their own.</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 φ">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="days in a year">365.2422 (year)</button>
</div>
<div class="custom">
  <input id="num" type="text" value="3.141592653589793" aria-label="number" />
  <button id="go" type="button">Expand</button>
</div>
<div class="cf" id="cf"></div>
<table class="conv"><thead><tr><th>term</th><th>convergent</th><th>value</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; }

// Continued fraction expansion: write the whole part, take the reciprocal of the
// leftover, repeat. Convergents h_n/k_n use the recurrence
//   h_n = a_n*h_{n-1} + h_{n-2},  k_n = a_n*k_{n-1} + k_{n-2}.
// Lagrange's theorem (18th c.): each convergent is the BEST rational approximation
// among all fractions with denominator <= its own. Classical, solved mathematics.
const FAMOUS = new Set(['22/7','355/113','3/2','7/5','17/12','1/4','365/1']);

function expand(x){
  const terms=[]; const conv=[];
  let v=x, hm1=1,hm2=0,km1=0,km2=1;
  for(let i=0;i<12;i++){
    const a=Math.floor(v); terms.push(a);
    const h=a*hm1+hm2, k=a*km1+km2;
    conv.push([h,k]);
    hm2=hm1; hm1=h; km2=km1; km1=k;
    const frac=v-a;
    if(frac < 1e-10) break;   // rational / numerical floor reached
    v=1/frac;
    if(k > 1e9) break;        // keep denominators readable
  }
  return {terms, conv};
}

function fmtCF(terms){
  if(terms.length===1) return '[' + terms[0] + ']';
  return '[' + terms[0] + '; ' + terms.slice(1).join(', ') + ']';
}

function show(x, name){
  const {terms, conv} = expand(x);
  document.getElementById('cf').textContent =
    (name?name+'  =  ':'') + fmtCF(terms);
  const tb=document.getElementById('rows'); tb.innerHTML='';
  conv.forEach(([h,k],i)=>{
    const val=h/k, err=Math.abs(x-val);
    const tr=document.createElement('tr');
    const key=h+'/'+k;
    if(FAMOUS.has(key)) tr.className='famous';
    tr.innerHTML =
      '<td>'+terms[i]+'</td>'+
      '<td class="frac">'+key+'</td>'+
      '<td>'+val.toFixed(8)+'</td>'+
      '<td class="err">'+(err===0?'exact':err.toExponential(2))+'</td>';
    tb.appendChild(tr);
  });
  console.log(name||x, '->', fmtCF(terms),
    '| best convergent', conv[conv.length-1].join('/'));
}

document.querySelectorAll('.controls button').forEach(b=>{
  b.addEventListener('click',()=>{
    document.getElementById('num').value=b.dataset.x;
    show(parseFloat(b.dataset.x), b.dataset.n);
  });
});
document.getElementById('go').addEventListener('click',()=>{
  const x=parseFloat(document.getElementById('num').value);
  if(isFinite(x)) show(x, '');
  else document.getElementById('cf').textContent='Enter a valid number.';
});
show(Math.PI, 'π');

Result

Console

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.

The Real Math

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.

Where It Matters

"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.

Conclusion

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.

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Continued Fractions

The cleanest fractions hiding inside every real number