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main/algebra/garners-algorithm.html

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<ul class="metadata page-metadata" data-bi-name="page info" lang="en-us" dir="ltr">
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Last update:
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<span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-date" title="July 26, 2023 08:23:19 UTC">July 26, 2023</span>&emsp;
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<span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-date" title="October 29, 2025 17:23:37 UTC">October 29, 2025</span>&emsp;
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@@ -6727,9 +6737,8 @@ <h2 id="mixed-radix-representation">Mixed Radix Representation<a class="headerli
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<p>We can represent the number <span class="arithmatex">$a$</span> in the <strong>mixed radix</strong> representation:</p>
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<div class="arithmatex">$$a = x_1 + x_2 p_1 + x_3 p_1 p_2 + \ldots + x_k p_1 \cdots p_{k-1} \text{ with }x_i \in [0, p_i)$$</div>
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<p>A mixed radix representation is a positional numeral system, that's a generalization of the typical number systems, like the binary numeral system or the decimal numeral system.
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For instance the decimal numeral system is a positional numeral system with the radix (or base) 10.
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Every a number is represented as a string of digits <span class="arithmatex">$d_1 d_2 d_3 \dots d_n$</span> between <span class="arithmatex">$0$</span> and <span class="arithmatex">$9$</span>, and
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E.g. the string <span class="arithmatex">$415$</span> represents the number <span class="arithmatex">$4 \cdot 10^2 + 1 \cdot 10^1 + 5 \cdot 10^0$</span>.
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For instance, the decimal numeral system is a positional numeral system with the radix (or base) 10.
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Every number is represented as a string of digits <span class="arithmatex">$d_1 d_2 d_3 \dots d_n$</span> between <span class="arithmatex">$0$</span> and <span class="arithmatex">$9$</span>. For example, the string <span class="arithmatex">$415$</span> represents the number <span class="arithmatex">$4 \cdot 10^2 + 1 \cdot 10^1 + 5 \cdot 10^0$</span>.
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In general the string of digits <span class="arithmatex">$d_1 d_2 d_3 \dots d_n$</span> represents the number <span class="arithmatex">$d_1 b^{n-1} + d_2 b^{n-2} + \cdots + d_n b^0$</span> in the positional numeral system with radix <span class="arithmatex">$b$</span>.</p>
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<p>In a mixed radix system, we don't have one radix any more. The base varies from position to position.</p>
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<h2 id="garners-algorithm_1">Garner's algorithm<a class="headerlink" href="#garners-algorithm_1" title="Permanent link">&para;</a></h2>
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<ul class="metadata page-metadata" data-bi-name="page info" lang="en-us" dir="ltr">
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<span class="contributors-text">Contributors:</span>
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<ul class="contributors" data-bi-name="contributors"><li><a href="https://github.com/jakobkogler" title="jakobkogler" data-bi-name="contributorprofile" target="_blank">jakobkogler</a> (97.52%)</li><li><a href="#" title="GILGAMESH" data-bi-name="contributorprofile" target="_blank">GILGAMESH</a> (1.86%)</li><li><a href="https://github.com/hieplpvip" title="hieplpvip" data-bi-name="contributorprofile" target="_blank">hieplpvip</a> (0.62%)</li></ul>
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<ul class="contributors" data-bi-name="contributors"><li><a href="https://github.com/jakobkogler" title="jakobkogler" data-bi-name="contributorprofile" target="_blank">jakobkogler</a> (96.25%)</li><li><a href="#" title="GILGAMESH" data-bi-name="contributorprofile" target="_blank">GILGAMESH</a> (1.88%)</li><li><a href="https://github.com/Aaryan-Dadu" title="Aaryan-Dadu" data-bi-name="contributorprofile" target="_blank">Aaryan-Dadu</a> (1.25%)</li><li><a href="https://github.com/hieplpvip" title="hieplpvip" data-bi-name="contributorprofile" target="_blank">hieplpvip</a> (0.62%)</li></ul>
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</ul>
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