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203 lines
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<title>Floor and Ceiling Functions</title>
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<h1 align="center">Floor and Ceiling Functions</h1>
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<p>The floor and ceiling functions give us the <b>nearest <a href="../whole-numbers.html">integer</a></b> up or down.</p>
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<div class="example">
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<h3>Example: What is the floor and ceiling of 2.31?</h3>
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<p align="center"><img src="images/function-floor-ceiling.svg" alt="Floor and Ceiling function" /></p>
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<p align="center" class="larger">The Floor of 2.31 is <b>2</b> <br />
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The Ceiling of 2.31 is <b>3</b></p>
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</div>
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<h3>Floor and Ceiling of Integers</h3>
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<p>What if we want the floor or ceiling of a number that is already an integer?</p>
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<p align="center" class="larger">That's easy: no change!</p>
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<div class="example">
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<h3>Example: What is the floor and ceiling of 5?</h3>
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<p align="center" class="larger">The Floor of 5 is <b>5</b> <br />
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The Ceiling of 5 is <b>5</b></p>
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</div>
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<p>Here are some example values for you:</p>
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<table border="1" align="center">
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<tr>
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<th width="60" align="center">x</th>
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<th width="80" align="center">Floor</th>
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<th width="80" align="center">Ceiling</th>
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</tr>
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<tr>
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<td width="50" align="center">−1.1</td>
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<td width="80" align="center">−2</td>
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<td width="80" align="center">−1</td>
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</tr>
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<tr>
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<td width="50" align="center">0</td>
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<td width="80" align="center">0</td>
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<td width="80" align="center">0</td>
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</tr>
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<tr>
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<td width="50" align="center">1.01</td>
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<td width="80" align="center">1</td>
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<td width="80" align="center">2</td>
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</tr>
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<tr>
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<td width="50" align="center">2.9</td>
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<td width="80" align="center">2</td>
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<td width="80" align="center">3</td>
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</tr>
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<tr>
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<td width="50" align="center">3</td>
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<td width="80" align="center">3</td>
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<td width="80" align="center">3</td>
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</tr>
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</table>
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<h2>Symbols</h2>
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<p>The symbols for floor and ceiling are like the square brackets <span class="large">[ ]</span> with the top or bottom part missing:</p>
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<p align="center"><img src="images/function-floor-ceiling-symbol.svg" alt="Floor and Ceiling function symbols" /></p>
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<p>But I prefer to use the word form: <b>floor</b>(x) and <b>ceil</b>(x)</p>
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<h2> Definitions</h2>
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<p>How do we give this a formal definition? </p>
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<div class="example">
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<h3>Example: How do we define the floor of 2.31?</h3>
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<p>Well, it has to be an integer ...</p>
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<p align="center">... and it has to be <b>less than</b> (or maybe equal to) 2.31, right?</p>
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<ul>
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<li><b>2</b> is less than 2.31 ... </li>
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<li>but <b>1</b> is also less than 2.31, </li>
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<li>and so is <b>0</b>, and <b>-1, -2, -3, etc. </b></li>
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</ul>
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<p align="center" class="larger">Oh no! There are lots of integers less than 2.31.</p>
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<p> </p>
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<p>So which one do we choose? </p>
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<p align="center">Choose the <b>greatest</b> one (which is <b>2</b> in this case)</p>
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<p>So we get:</p>
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<p align="center" class="larger">The <b>greatest</b> integer that is <b>less than</b> (or equal to) 2.31 is <b>2</b></p>
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</div>
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<p>Which leads to our definition: </p>
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<div class="def">
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<p>Floor Function: the greatest integer that is less than or equal to <b>x</b></p>
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</div>
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<p>Likewise for Ceiling:</p>
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<div class="def">
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<p>Ceiling Function: the least integer that is greater than or equal to <b>x</b></p>
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</div>
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<h2>As A Graph</h2>
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<p>The Floor Function is this curious "step" function (like an infinite staircase):</p>
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<p align="center"><img src="images/function-floor-graph.svg" alt="Floor function graph" /></p>
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<p align="center" class="larger">The Floor Function</p>
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<p>A solid dot means "including" and an open dot means "not including". </p>
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<div class="example">
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<h3>Example: at <b>x=2</b> we meet:</h3>
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<ul>
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<li> an <b>open dot</b> at y=1 (so it does not include x=2), </li>
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<li>and a <b>solid dot</b> at y=2 (which <i>does</i> include x=2) </li>
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</ul>
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<p>so the answer is <b>y=2</b></p>
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</div>
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<p>And this is the Ceiling Function:</p>
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<p align="center"><img src="images/function-ceiling-graph.svg" alt="Ceiling function graph" /></p>
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<p align="center"><span class="larger">The Ceiling Function</span></p>
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<h2> The "Int" Function</h2>
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<p>The "Int" function (short for "integer") is like the "Floor" function, BUT some calculators and computer programs show different results when given negative numbers:</p>
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<ul>
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<li>Some say <span class="larger">int(−3.65) = <b>−4</b></span> (the same as the Floor function)</li>
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<li>Others say <span class="larger">int(−3.65) = <b>−3</b></span> (the neighbouring integer <b>closest to zero, </b>or "just throw away the .65")</li>
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</ul>
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<p>So be careful with this function!</p>
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<h2>The "Frac" Function</h2>
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<p> With the Floor Function, we "throw away" the fractional part. That part is called the "frac" or "fractional part" function: </p>
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<p align="center" class="large">frac(x) = x − floor(x)</p>
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<p>It looks like a sawtooth:</p>
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<p align="center"><img src="images/function-frac.svg" alt="Frac function graph" /></p>
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<p align="center"><span class="larger">The Frac Function</span></p>
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<div class="example">
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<h3>Example: what is frac(3.65)?</h3>
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<p><b> frac(x) = x − floor(x) </b></p>
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<p>So: frac(3.65) = 3.65 − floor(3.65) = 3.65 − 3 = <b>0.65</b></p>
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</div>
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<div class="example">
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<h3>Example: what is frac(−3.65)?</h3>
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<p><b>frac(x) = x − floor(x) </b></p>
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<p>So: frac(−3.65) = (−3.65) − floor(−3.65) = (−3.65) − (−4) = −3.65 + 4 = <b>0.35</b></p>
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<p align="center" class="larger"> </p>
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<p><b>BUT</b> many calculators and computer programs use <b>frac(x) = x − int(x)</b>, and so their result depends on how they calculate <b>int(x)</b>:</p>
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<ul>
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<li>Some say frac(−3.65) = <b>0.35</b> i.e −3.65 − (−4)</li>
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<li>Others say frac(−3.65) = <b>−0.65</b> i.e. −3.65 − (−3)</li>
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</ul>
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<p>So be careful using this function with negative values.</p>
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</div>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(2434, 2435, 2436, 2437, 2438, 2439);</script>
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</div>
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<div class="related"><a href="functions-common.html">Common Functions Reference</a><a href="index.html">Sets Index</a></div>
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