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122 lines
4.1 KiB
HTML
122 lines
4.1 KiB
HTML
<!DOCTYPE html>
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<meta http-equiv="content-type" content="text/html; charset=utf-8">
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<!-- #BeginEditable "doctitle" -->
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<title>Does 0.999... equal 1?</title>
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<h1 class="center">Is this really true?</h1>
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<div class="center"><span class="large"><br></span></div>
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<div class="center"><span class="large">The idea is that 0.9 recurring<br>
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</span><span class="larger">(0.999...
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with the digits going on forever) </span><span class="large"><br>
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is actually equal to 1</span></div>
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<p><i>(Here we write 0.999...
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as notation for 0.9 recurring,
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some people put a little dot above the 9, or a line on top like this: 0.<span style="text-decoration: overline;">9</span>)</i></p>
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<h2>Does 0.999... = 1 ?</h2>
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<p>Let us start by having X = 0.999...</p>
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<p class="so">X = 0.999...</p>
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<p class="so">10X = 9.999...</p>
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<p><i>Subtract X from each side to give us:</i></p>
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<p class="so">9X = 9.999... − X</p>
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<p><i>but we know that X is 0.999..., so: </i></p>
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<p class="so">9X = 9.999... − 0.999...</p>
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<p class="so">9X = 9</p>
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<p><i>Divide both sides by 9:</i></p>
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<p class="so">X = 1</p>
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<p><i>But hang on a moment I thought we said
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X was equal to 0.999... ?<br>
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<br>
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Yes, it does, but from our calculations X is also equal to
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1, so:</i></p>
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<p class="so">X = 0.999... = 1</p>
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<p>And so:</p>
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<p class="so">0.999... = 1</p>
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<div align="left">
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<p> </p>
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<p><i>Does anyone disagree with this? Let me
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know on the <a href="https://www.mathisfunforum.com/index.php">Math is Fun Forum</a>.</i></p>
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<p> </p>
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<a href="numbers/index.html">Numbers Index</a>
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<a href="algebra/index.html">Algebra Index</a>
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Copyright © 2020 MathsIsFun.com
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