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  182.   <section id="module-fractions">

  183. <span id="fractions-rational-numbers"></span><h1><a class="reference internal" href="#module-fractions" title="fractions: Rational numbers."><code class="xref py py-mod docutils literal notranslate"><span class="pre">fractions</span></code></a> — Rational numbers<a class="headerlink" href="#module-fractions" title="Link to this heading">¶</a></h1>

  184. <p><strong>Source code:</strong> <a class="reference external" href="https://github.com/python/cpython/tree/3.12/Lib/fractions.py">Lib/fractions.py</a></p>

  185. <hr class="docutils" />

  186. <p>The <a class="reference internal" href="#module-fractions" title="fractions: Rational numbers."><code class="xref py py-mod docutils literal notranslate"><span class="pre">fractions</span></code></a> module provides support for rational number arithmetic.</p>

  187. <p>A Fraction instance can be constructed from a pair of integers, from

  188. another rational number, or from a string.</p>

  189. <dl class="py class">

  190. <dt class="sig sig-object py" id="fractions.Fraction">

  191. <em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">fractions.</span></span><span class="sig-name descname"><span class="pre">Fraction</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">numerator</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">denominator</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction" title="Link to this definition">¶</a></dt>

  192. <dt class="sig sig-object py">

  193. <em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">fractions.</span></span><span class="sig-name descname"><span class="pre">Fraction</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">other_fraction</span></span></em><span class="sig-paren">)</span></dt>

  194. <dt class="sig sig-object py">

  195. <em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">fractions.</span></span><span class="sig-name descname"><span class="pre">Fraction</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">float</span></span></em><span class="sig-paren">)</span></dt>

  196. <dt class="sig sig-object py">

  197. <em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">fractions.</span></span><span class="sig-name descname"><span class="pre">Fraction</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">decimal</span></span></em><span class="sig-paren">)</span></dt>

  198. <dt class="sig sig-object py">

  199. <em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">fractions.</span></span><span class="sig-name descname"><span class="pre">Fraction</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">string</span></span></em><span class="sig-paren">)</span></dt>

  200. <dd><p>The first version requires that <em>numerator</em> and <em>denominator</em> are instances

  201. of <a class="reference internal" href="numbers.html#numbers.Rational" title="numbers.Rational"><code class="xref py py-class docutils literal notranslate"><span class="pre">numbers.Rational</span></code></a> and returns a new <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instance

  202. with value <code class="docutils literal notranslate"><span class="pre">numerator/denominator</span></code>. If <em>denominator</em> is <code class="docutils literal notranslate"><span class="pre">0</span></code>, it

  203. raises a <a class="reference internal" href="exceptions.html#ZeroDivisionError" title="ZeroDivisionError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ZeroDivisionError</span></code></a>. The second version requires that

  204. <em>other_fraction</em> is an instance of <a class="reference internal" href="numbers.html#numbers.Rational" title="numbers.Rational"><code class="xref py py-class docutils literal notranslate"><span class="pre">numbers.Rational</span></code></a> and returns a

  205. <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instance with the same value.  The next two versions accept

  206. either a <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> or a <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate"><span class="pre">decimal.Decimal</span></code></a> instance, and return a

  207. <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instance with exactly the same value.  Note that due to the

  208. usual issues with binary floating-point (see <a class="reference internal" href="../tutorial/floatingpoint.html#tut-fp-issues"><span class="std std-ref">Floating Point Arithmetic:  Issues and Limitations</span></a>), the

  209. argument to <code class="docutils literal notranslate"><span class="pre">Fraction(1.1)</span></code> is not exactly equal to 11/10, and so

  210. <code class="docutils literal notranslate"><span class="pre">Fraction(1.1)</span></code> does <em>not</em> return <code class="docutils literal notranslate"><span class="pre">Fraction(11,</span> <span class="pre">10)</span></code> as one might expect.

  211. (But see the documentation for the <a class="reference internal" href="#fractions.Fraction.limit_denominator" title="fractions.Fraction.limit_denominator"><code class="xref py py-meth docutils literal notranslate"><span class="pre">limit_denominator()</span></code></a> method below.)

  212. The last version of the constructor expects a string or unicode instance.

  213. The usual form for this instance is:</p>

  214. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="n">sign</span><span class="p">]</span> <span class="n">numerator</span> <span class="p">[</span><span class="s1">&#39;/&#39;</span> <span class="n">denominator</span><span class="p">]</span>

  215. </pre></div>

  216. </div>

  217. <p>where the optional <code class="docutils literal notranslate"><span class="pre">sign</span></code> may be either ‘+’ or ‘-’ and

  218. <code class="docutils literal notranslate"><span class="pre">numerator</span></code> and <code class="docutils literal notranslate"><span class="pre">denominator</span></code> (if present) are strings of

  219. decimal digits (underscores may be used to delimit digits as with

  220. integral literals in code).  In addition, any string that represents a finite

  221. value and is accepted by the <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> constructor is also

  222. accepted by the <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> constructor.  In either form the

  223. input string may also have leading and/or trailing whitespace.

  224. Here are some examples:</p>

  225. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="kn">import</span> <span class="n">Fraction</span>

  226. <span class="gp">&gt;&gt;&gt; </span><span class="n">Fraction</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="p">)</span>

  227. <span class="go">Fraction(-8, 5)</span>

  228. <span class="gp">&gt;&gt;&gt; </span><span class="n">Fraction</span><span class="p">(</span><span class="mi">123</span><span class="p">)</span>

  229. <span class="go">Fraction(123, 1)</span>

  230. <span class="gp">&gt;&gt;&gt; </span><span class="n">Fraction</span><span class="p">()</span>

  231. <span class="go">Fraction(0, 1)</span>

  232. <span class="gp">&gt;&gt;&gt; </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">&#39;3/7&#39;</span><span class="p">)</span>

  233. <span class="go">Fraction(3, 7)</span>

  234. <span class="gp">&gt;&gt;&gt; </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">&#39; -3/7 &#39;</span><span class="p">)</span>

  235. <span class="go">Fraction(-3, 7)</span>

  236. <span class="gp">&gt;&gt;&gt; </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">&#39;1.414213 </span><span class="se">\t\n</span><span class="s1">&#39;</span><span class="p">)</span>

  237. <span class="go">Fraction(1414213, 1000000)</span>

  238. <span class="gp">&gt;&gt;&gt; </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">&#39;-.125&#39;</span><span class="p">)</span>

  239. <span class="go">Fraction(-1, 8)</span>

  240. <span class="gp">&gt;&gt;&gt; </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">&#39;7e-6&#39;</span><span class="p">)</span>

  241. <span class="go">Fraction(7, 1000000)</span>

  242. <span class="gp">&gt;&gt;&gt; </span><span class="n">Fraction</span><span class="p">(</span><span class="mf">2.25</span><span class="p">)</span>

  243. <span class="go">Fraction(9, 4)</span>

  244. <span class="gp">&gt;&gt;&gt; </span><span class="n">Fraction</span><span class="p">(</span><span class="mf">1.1</span><span class="p">)</span>

  245. <span class="go">Fraction(2476979795053773, 2251799813685248)</span>

  246. <span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">decimal</span> <span class="kn">import</span> <span class="n">Decimal</span>

  247. <span class="gp">&gt;&gt;&gt; </span><span class="n">Fraction</span><span class="p">(</span><span class="n">Decimal</span><span class="p">(</span><span class="s1">&#39;1.1&#39;</span><span class="p">))</span>

  248. <span class="go">Fraction(11, 10)</span>

  249. </pre></div>

  250. </div>

  251. <p>The <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> class inherits from the abstract base class

  252. <a class="reference internal" href="numbers.html#numbers.Rational" title="numbers.Rational"><code class="xref py py-class docutils literal notranslate"><span class="pre">numbers.Rational</span></code></a>, and implements all of the methods and

  253. operations from that class.  <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instances are <a class="reference internal" href="../glossary.html#term-hashable"><span class="xref std std-term">hashable</span></a>,

  254. and should be treated as immutable.  In addition,

  255. <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> has the following properties and methods:</p>

  256. <div class="versionchanged">

  257. <p><span class="versionmodified changed">Changed in version 3.2: </span>The <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> constructor now accepts <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> and

  258. <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate"><span class="pre">decimal.Decimal</span></code></a> instances.</p>

  259. </div>

  260. <div class="versionchanged">

  261. <p><span class="versionmodified changed">Changed in version 3.9: </span>The <a class="reference internal" href="math.html#math.gcd" title="math.gcd"><code class="xref py py-func docutils literal notranslate"><span class="pre">math.gcd()</span></code></a> function is now used to normalize the <em>numerator</em>

  262. and <em>denominator</em>. <a class="reference internal" href="math.html#math.gcd" title="math.gcd"><code class="xref py py-func docutils literal notranslate"><span class="pre">math.gcd()</span></code></a> always return a <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate"><span class="pre">int</span></code></a> type.

  263. Previously, the GCD type depended on <em>numerator</em> and <em>denominator</em>.</p>

  264. </div>

  265. <div class="versionchanged">

  266. <p><span class="versionmodified changed">Changed in version 3.11: </span>Underscores are now permitted when creating a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instance

  267. from a string, following <span class="target" id="index-0"></span><a class="pep reference external" href="https://peps.python.org/pep-0515/"><strong>PEP 515</strong></a> rules.</p>

  268. </div>

  269. <div class="versionchanged">

  270. <p><span class="versionmodified changed">Changed in version 3.11: </span><a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> implements <code class="docutils literal notranslate"><span class="pre">__int__</span></code> now to satisfy

  271. <code class="docutils literal notranslate"><span class="pre">typing.SupportsInt</span></code> instance checks.</p>

  272. </div>

  273. <div class="versionchanged">

  274. <p><span class="versionmodified changed">Changed in version 3.12: </span>Space is allowed around the slash for string inputs: <code class="docutils literal notranslate"><span class="pre">Fraction('2</span> <span class="pre">/</span> <span class="pre">3')</span></code>.</p>

  275. </div>

  276. <div class="versionchanged">

  277. <p><span class="versionmodified changed">Changed in version 3.12: </span><a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instances now support float-style formatting, with

  278. presentation types <code class="docutils literal notranslate"><span class="pre">&quot;e&quot;</span></code>, <code class="docutils literal notranslate"><span class="pre">&quot;E&quot;</span></code>, <code class="docutils literal notranslate"><span class="pre">&quot;f&quot;</span></code>, <code class="docutils literal notranslate"><span class="pre">&quot;F&quot;</span></code>, <code class="docutils literal notranslate"><span class="pre">&quot;g&quot;</span></code>, <code class="docutils literal notranslate"><span class="pre">&quot;G&quot;</span></code>

  279. and <code class="docutils literal notranslate"><span class="pre">&quot;%&quot;&quot;</span></code>.</p>

  280. </div>

  281. <dl class="py attribute">

  282. <dt class="sig sig-object py" id="fractions.Fraction.numerator">

  283. <span class="sig-name descname"><span class="pre">numerator</span></span><a class="headerlink" href="#fractions.Fraction.numerator" title="Link to this definition">¶</a></dt>

  284. <dd><p>Numerator of the Fraction in lowest term.</p>

  285. </dd></dl>

  286. 

  287. <dl class="py attribute">

  288. <dt class="sig sig-object py" id="fractions.Fraction.denominator">

  289. <span class="sig-name descname"><span class="pre">denominator</span></span><a class="headerlink" href="#fractions.Fraction.denominator" title="Link to this definition">¶</a></dt>

  290. <dd><p>Denominator of the Fraction in lowest term.</p>

  291. </dd></dl>

  292. 

  293. <dl class="py method">

  294. <dt class="sig sig-object py" id="fractions.Fraction.as_integer_ratio">

  295. <span class="sig-name descname"><span class="pre">as_integer_ratio</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.as_integer_ratio" title="Link to this definition">¶</a></dt>

  296. <dd><p>Return a tuple of two integers, whose ratio is equal

  297. to the original Fraction.  The ratio is in lowest terms

  298. and has a positive denominator.</p>

  299. <div class="versionadded">

  300. <p><span class="versionmodified added">New in version 3.8.</span></p>

  301. </div>

  302. </dd></dl>

  303. 

  304. <dl class="py method">

  305. <dt class="sig sig-object py" id="fractions.Fraction.is_integer">

  306. <span class="sig-name descname"><span class="pre">is_integer</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.is_integer" title="Link to this definition">¶</a></dt>

  307. <dd><p>Return <code class="docutils literal notranslate"><span class="pre">True</span></code> if the Fraction is an integer.</p>

  308. <div class="versionadded">

  309. <p><span class="versionmodified added">New in version 3.12.</span></p>

  310. </div>

  311. </dd></dl>

  312. 

  313. <dl class="py method">

  314. <dt class="sig sig-object py" id="fractions.Fraction.from_float">

  315. <em class="property"><span class="pre">classmethod</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">from_float</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">flt</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.from_float" title="Link to this definition">¶</a></dt>

  316. <dd><p>Alternative constructor which only accepts instances of

  317. <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> or <a class="reference internal" href="numbers.html#numbers.Integral" title="numbers.Integral"><code class="xref py py-class docutils literal notranslate"><span class="pre">numbers.Integral</span></code></a>. Beware that

  318. <code class="docutils literal notranslate"><span class="pre">Fraction.from_float(0.3)</span></code> is not the same value as <code class="docutils literal notranslate"><span class="pre">Fraction(3,</span> <span class="pre">10)</span></code>.</p>

  319. <div class="admonition note">

  320. <p class="admonition-title">Note</p>

  321. <p>From Python 3.2 onwards, you can also construct a

  322. <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instance directly from a <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a>.</p>

  323. </div>

  324. </dd></dl>

  325. 

  326. <dl class="py method">

  327. <dt class="sig sig-object py" id="fractions.Fraction.from_decimal">

  328. <em class="property"><span class="pre">classmethod</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">from_decimal</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">dec</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.from_decimal" title="Link to this definition">¶</a></dt>

  329. <dd><p>Alternative constructor which only accepts instances of

  330. <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate"><span class="pre">decimal.Decimal</span></code></a> or <a class="reference internal" href="numbers.html#numbers.Integral" title="numbers.Integral"><code class="xref py py-class docutils literal notranslate"><span class="pre">numbers.Integral</span></code></a>.</p>

  331. <div class="admonition note">

  332. <p class="admonition-title">Note</p>

  333. <p>From Python 3.2 onwards, you can also construct a

  334. <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instance directly from a <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate"><span class="pre">decimal.Decimal</span></code></a>

  335. instance.</p>

  336. </div>

  337. </dd></dl>

  338. 

  339. <dl class="py method">

  340. <dt class="sig sig-object py" id="fractions.Fraction.limit_denominator">

  341. <span class="sig-name descname"><span class="pre">limit_denominator</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">max_denominator</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1000000</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.limit_denominator" title="Link to this definition">¶</a></dt>

  342. <dd><p>Finds and returns the closest <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> to <code class="docutils literal notranslate"><span class="pre">self</span></code> that has

  343. denominator at most max_denominator.  This method is useful for finding

  344. rational approximations to a given floating-point number:</p>

  345. <div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="kn">import</span> <span class="n">Fraction</span>

  346. <span class="gp">&gt;&gt;&gt; </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">&#39;3.1415926535897932&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">limit_denominator</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>

  347. <span class="go">Fraction(355, 113)</span>

  348. </pre></div>

  349. </div>

  350. <p>or for recovering a rational number that’s represented as a float:</p>

  351. <div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">cos</span>

  352. <span class="gp">&gt;&gt;&gt; </span><span class="n">Fraction</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">))</span>

  353. <span class="go">Fraction(4503599627370497, 9007199254740992)</span>

  354. <span class="gp">&gt;&gt;&gt; </span><span class="n">Fraction</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">limit_denominator</span><span class="p">()</span>

  355. <span class="go">Fraction(1, 2)</span>

  356. <span class="gp">&gt;&gt;&gt; </span><span class="n">Fraction</span><span class="p">(</span><span class="mf">1.1</span><span class="p">)</span><span class="o">.</span><span class="n">limit_denominator</span><span class="p">()</span>

  357. <span class="go">Fraction(11, 10)</span>

  358. </pre></div>

  359. </div>

  360. </dd></dl>

  361. 

  362. <dl class="py method">

  363. <dt class="sig sig-object py" id="fractions.Fraction.__floor__">

  364. <span class="sig-name descname"><span class="pre">__floor__</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.__floor__" title="Link to this definition">¶</a></dt>

  365. <dd><p>Returns the greatest <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate"><span class="pre">int</span></code></a> <code class="docutils literal notranslate"><span class="pre">&lt;=</span> <span class="pre">self</span></code>.  This method can

  366. also be accessed through the <a class="reference internal" href="math.html#math.floor" title="math.floor"><code class="xref py py-func docutils literal notranslate"><span class="pre">math.floor()</span></code></a> function:</p>

  367. <div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">floor</span>

  368. <span class="gp">&gt;&gt;&gt; </span><span class="n">floor</span><span class="p">(</span><span class="n">Fraction</span><span class="p">(</span><span class="mi">355</span><span class="p">,</span> <span class="mi">113</span><span class="p">))</span>

  369. <span class="go">3</span>

  370. </pre></div>

  371. </div>

  372. </dd></dl>

  373. 

  374. <dl class="py method">

  375. <dt class="sig sig-object py" id="fractions.Fraction.__ceil__">

  376. <span class="sig-name descname"><span class="pre">__ceil__</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.__ceil__" title="Link to this definition">¶</a></dt>

  377. <dd><p>Returns the least <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate"><span class="pre">int</span></code></a> <code class="docutils literal notranslate"><span class="pre">&gt;=</span> <span class="pre">self</span></code>.  This method can

  378. also be accessed through the <a class="reference internal" href="math.html#math.ceil" title="math.ceil"><code class="xref py py-func docutils literal notranslate"><span class="pre">math.ceil()</span></code></a> function.</p>

  379. </dd></dl>

  380. 

  381. <dl class="py method">

  382. <dt class="sig sig-object py" id="fractions.Fraction.__round__">

  383. <span class="sig-name descname"><span class="pre">__round__</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.__round__" title="Link to this definition">¶</a></dt>

  384. <dt class="sig sig-object py">

  385. <span class="sig-name descname"><span class="pre">__round__</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">ndigits</span></span></em><span class="sig-paren">)</span></dt>

  386. <dd><p>The first version returns the nearest <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate"><span class="pre">int</span></code></a> to <code class="docutils literal notranslate"><span class="pre">self</span></code>,

  387. rounding half to even. The second version rounds <code class="docutils literal notranslate"><span class="pre">self</span></code> to the

  388. nearest multiple of <code class="docutils literal notranslate"><span class="pre">Fraction(1,</span> <span class="pre">10**ndigits)</span></code> (logically, if

  389. <code class="docutils literal notranslate"><span class="pre">ndigits</span></code> is negative), again rounding half toward even.  This

  390. method can also be accessed through the <a class="reference internal" href="functions.html#round" title="round"><code class="xref py py-func docutils literal notranslate"><span class="pre">round()</span></code></a> function.</p>

  391. </dd></dl>

  392. 

  393. <dl class="py method">

  394. <dt class="sig sig-object py" id="fractions.Fraction.__format__">

  395. <span class="sig-name descname"><span class="pre">__format__</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">format_spec</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">/</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.__format__" title="Link to this definition">¶</a></dt>

  396. <dd><p>Provides support for float-style formatting of <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a>

  397. instances via the <a class="reference internal" href="stdtypes.html#str.format" title="str.format"><code class="xref py py-meth docutils literal notranslate"><span class="pre">str.format()</span></code></a> method, the <a class="reference internal" href="functions.html#format" title="format"><code class="xref py py-func docutils literal notranslate"><span class="pre">format()</span></code></a> built-in

  398. function, or <a class="reference internal" href="../reference/lexical_analysis.html#f-strings"><span class="std std-ref">Formatted string literals</span></a>. The

  399. presentation types <code class="docutils literal notranslate"><span class="pre">&quot;e&quot;</span></code>, <code class="docutils literal notranslate"><span class="pre">&quot;E&quot;</span></code>, <code class="docutils literal notranslate"><span class="pre">&quot;f&quot;</span></code>, <code class="docutils literal notranslate"><span class="pre">&quot;F&quot;</span></code>, <code class="docutils literal notranslate"><span class="pre">&quot;g&quot;</span></code>, <code class="docutils literal notranslate"><span class="pre">&quot;G&quot;</span></code>

  400. and <code class="docutils literal notranslate"><span class="pre">&quot;%&quot;</span></code> are supported. For these presentation types, formatting for a

  401. <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> object <code class="docutils literal notranslate"><span class="pre">x</span></code> follows the rules outlined for

  402. the <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> type in the <a class="reference internal" href="string.html#formatspec"><span class="std std-ref">Format Specification Mini-Language</span></a> section.</p>

  403. <p>Here are some examples:</p>

  404. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="kn">import</span> <span class="n">Fraction</span>

  405. <span class="gp">&gt;&gt;&gt; </span><span class="nb">format</span><span class="p">(</span><span class="n">Fraction</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">),</span> <span class="s1">&#39;.40g&#39;</span><span class="p">)</span>

  406. <span class="go">&#39;0.1428571428571428571428571428571428571429&#39;</span>

  407. <span class="gp">&gt;&gt;&gt; </span><span class="nb">format</span><span class="p">(</span><span class="n">Fraction</span><span class="p">(</span><span class="s1">&#39;1234567.855&#39;</span><span class="p">),</span> <span class="s1">&#39;_.2f&#39;</span><span class="p">)</span>

  408. <span class="go">&#39;1_234_567.86&#39;</span>

  409. <span class="gp">&gt;&gt;&gt; </span><span class="sa">f</span><span class="s2">&quot;</span><span class="si">{</span><span class="n">Fraction</span><span class="p">(</span><span class="mi">355</span><span class="p">,</span><span class="w"> </span><span class="mi">113</span><span class="p">)</span><span class="si">:</span><span class="s2">*&gt;20.6e</span><span class="si">}</span><span class="s2">&quot;</span>

  410. <span class="go">&#39;********3.141593e+00&#39;</span>

  411. <span class="gp">&gt;&gt;&gt; </span><span class="n">old_price</span><span class="p">,</span> <span class="n">new_price</span> <span class="o">=</span> <span class="mi">499</span><span class="p">,</span> <span class="mi">672</span>

  412. <span class="gp">&gt;&gt;&gt; </span><span class="s2">&quot;</span><span class="si">{:.2%}</span><span class="s2"> price increase&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">Fraction</span><span class="p">(</span><span class="n">new_price</span><span class="p">,</span> <span class="n">old_price</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>

  413. <span class="go">&#39;34.67% price increase&#39;</span>

  414. </pre></div>

  415. </div>

  416. </dd></dl>

  417. 

  418. </dd></dl>

  419. 

  420. <div class="admonition seealso">

  421. <p class="admonition-title">See also</p>

  422. <dl class="simple">

  423. <dt>Module <a class="reference internal" href="numbers.html#module-numbers" title="numbers: Numeric abstract base classes (Complex, Real, Integral, etc.)."><code class="xref py py-mod docutils literal notranslate"><span class="pre">numbers</span></code></a></dt><dd><p>The abstract base classes making up the numeric tower.</p>

  424. </dd>

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