cp-algorithms
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   Last update:
-  <span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-date">February 17, 2024</span>&emsp;
+  <span class="git-revision-date-localized-plugin git-revision-date-localized-plugin-date">January 6, 2025</span>&emsp;
 
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@@ -6903,7 +6913,7 @@ <h3 id="fast-fourier-transform_1">Fast Fourier Transform<a class="headerlink" hr
 \end{align}$$</div>
 <p>It is easy to see that</p>
 <div class="arithmatex">$$A(x) = A_0(x^2) + x A_1(x^2).$$</div>
-<p>The polynomials <span class="arithmatex">$A_0$</span> and <span class="arithmatex">$A_1$</span> are only half as much coefficients as the polynomial <span class="arithmatex">$A$</span>.
+<p>The polynomials <span class="arithmatex">$A_0$</span> and <span class="arithmatex">$A_1$</span> have only half as many coefficients as the polynomial <span class="arithmatex">$A$</span>.
 If we can compute the <span class="arithmatex">$\text{DFT}(A)$</span> in linear time using <span class="arithmatex">$\text{DFT}(A_0)$</span> and <span class="arithmatex">$\text{DFT}(A_1)$</span>, then we get the recurrence <span class="arithmatex">$T_{\text{DFT}}(n) = 2 T_{\text{DFT}}\left(\frac{n}{2}\right) + O(n)$</span> for the time complexity, which results in <span class="arithmatex">$T_{\text{DFT}}(n) = O(n \log n)$</span> by the <strong>master theorem</strong>.</p>
 <p>Let's learn how we can accomplish that.</p>
 <p>Suppose we have computed the vectors <span class="arithmatex">$\left(y_k^0\right)_{k=0}^{n/2-1} = \text{DFT}(A_0)$</span> and <span class="arithmatex">$\left(y_k^1\right)_{k=0}^{n/2-1} = \text{DFT}(A_1)$</span>.
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