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<html data-require="math subhints">
<head>
<title>Estimating square roots</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
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<div class="exercise">
<div class="problems">
<div id="square">
<div class="vars">
<var id="N">randRange( 2, 11 )</var>
<var id="Q">randRange( N * N + 1, (N + 1) * (N + 1) - 1 )</var>
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<p class="question">The value of <code>\sqrt{<var>Q</var>}</code> lies between which two <a href="#" class="show-definition" data-definition="consecutive-integers">consecutive integers</a>?</p>
<div class="definition" id="consecutive-integers">
Integers that appear in order when counting, for example 2 and 3.
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<div class="solution" data-type="multiple">
<p><span class="sol short35" data-forms="integer"><var>N</var></span> <code>&lt; \sqrt{<var>Q</var>} &lt;</code> <span class="sol short35" data-forms="integer"><var>N + 1</var></span></p>
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<div class="hints">
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<p>Consider the perfect squares near <code><var>Q</var></code>.
[<a href="#" class="show-subhint" data-subhint="explain-perfect-square">What are perfect squares?</a>]
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<div class="subhint" id="explain-perfect-square">
<p>
Perfect squares are integers which can be obtained by squaring an integer.
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<p>
The first 13 perfect squares are:
</p>
<code>\qquad 1,4,9,16,25,36,49,64,81,100,121,144,169</code>
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<p><code><var>N * N</var></code> is the nearest perfect square less than <code><var>Q</var></code>.</p>
<p><code><var>(N + 1) * (N + 1)</var></code> is the nearest perfect square more than <code><var>Q</var></code>.</p>
<p>So, <code><var>N * N</var> &lt; <var>Q</var> &lt; <var>(N + 1) * (N + 1)</var></code>.</p>
<div>
<p><code>\sqrt{<var>N * N</var>} &lt; \sqrt{<var>Q</var>} &lt; \sqrt{<var>(N + 1)*(N + 1)</var>}</code></p>
<p><code><var>N</var> &lt; \sqrt{<var>Q</var>} &lt; <var>N + 1</var></code></p>
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