419 /www/nrich/html/content/02/06/15plus4/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"><p>Notice the following pattern, written in base 10:</p> <p>$$ \begin{eqnarray} 987654321&amp;=&amp;8\times 123456789 &amp;+&amp;9\\ 98765432&amp;=&amp;8\times 12345678 &amp;+&amp;8 \\ 9876543&amp;=&amp;8\times 1234567 &amp;+&amp;7 \\ 987654&amp;=&amp;8\times 123456 &amp;+&amp;6\\ &amp;\vdots &amp; \\ 9&amp;=&amp;8\times 1&amp;+&amp;1 \end{eqnarray} $$</p> <p>This patterns also holds in bases other than $10$. For example, in base $4$ we have $321 = 2 \times 123 + 3 $, and so on.</p> <p>Why is this the case?</p> <p><br></br>  </p></mdoxml> <mdoxml version="1.0"><br></br> <table cellpadding="2" summary=""> <tbody> <tr> <td style="text-align: center;">$987654321$</td> <td style="text-align: center;">$=$</td> <td style="text-align: center;">$8$</td> <td style="text-align: center;">$\times$</td> <td style="text-align: center;">$123456789$</td> <td style="text-align: center;">$+$</td> <td style="text-align: center;">$9$</td> </tr> <tr> <td style="text-align: center;">$98765432$</td> <td style="text-align: center;">$=$</td> <td style="text-align: center;">$8$</td> <td style="text-align: center;">$\times$</td> <td style="text-align: center;">$12345678$</td> <td style="text-align: center;">$+$</td> <td style="text-align: center;">$8$</td> </tr> <tr> <td style="text-align: center;">$9876543$</td> <td style="text-align: center;">$=$</td> <td style="text-align: center;">$8$</td> <td style="text-align: center;">$\times$</td> <td style="text-align: center;">$1234567$</td> <td style="text-align: center;">$+$</td> <td style="text-align: center;">$7$</td> </tr> <tr> <td style="text-align: center;">$987654$</td> <td style="text-align: center;">$=$</td> <td style="text-align: center;">$8$</td> <td style="text-align: center;">$\times$</td> <td style="text-align: center;">$123456$</td> <td style="text-align: center;">$+$</td> <td style="text-align: center;">$6$</td> </tr> <tr> <td style="text-align: center;">...</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">$9$</td> <td style="text-align: center;">$=$</td> <td style="text-align: center;">$8$</td> <td style="text-align: center;">$\times$</td> <td style="text-align: center;">$1$</td> <td style="text-align: center;">$+$</td> <td style="text-align: center;">$1$</td> </tr> </tbody> </table> <p>Saul Foresta explained as follows why this pattern holds in the decimal system and in other number systems using bases other than base $10$:</p> <p>I generalized the problem for any base $n$ and any number of digits $r$ where $r$ can be anywhere from $1$ to $(n - 1)$.</p> <p>Then after rewriting both sides of the equality given in the problem using sigma notation I arrived at the following:</p> <p>$$ {\sum_{k=1}^r (n-k)n^{r-k}} = {{(n-2)\sum_{k=1}^r kn^{r-k} + r}} $$</p> <p>In each summation $k$ stands for the $k$th digit of the number we're dealing with, reading from left to right. For example, in the number $9876$, $k$ ranges from $1$-$4$, where $9$ is $k=1$, $8$ is $k=2$, and so on.</p> <p>So all I need to do in order to prove that this pattern holds is show that the left side of this equality does indeed equal the right side. Taking the terms like $8 \times123456789$, that is</p> <p>$$ (n-2)\sum_{k=1}^r kn^{r-k} $$</p> <p>over to the left hand side, we will prove that this expression is equal to $r$.</p> <p>[(n-1)n ^r-1 + (n-2)n ^r-2 + (n-3)n ^r-3 + ... + (n-r)] - (n-2)[n ^r-1 + 2n ^r-2 + 3n ^r-3 + ... + r] =<br></br> [n ^r - n ^r-1 + n ^r-1 - 2n ^r-2 + n ^r-2 - 3n ^r-3 + ...+ n - r] - [n ^r - 2n ^r-1 + 2n ^r-1 - 2.2n ^r-2 + ... + (n-2)r]</p> <p>The coefficient of $n^{r-k}$ on this left hand side is $[1-k] - [k+1-2k] = 0$ for</p> <p>$ ( 1 \le k \le r-1) $</p> and the coefficient of $n^r$ is also $0$. <p>The coefficient of $n^0$ is $[-r] - [-2(r)] = r$ and hence this expression is equal to $r$ as required.</p> <br></br></mdoxml> <mdoxml version="1.0"><br></br> <p>All the results can be checked by calculation. For an explanation and proof it is most efficient to use a general method that works for all bases. Moreover algebra reveals the structure that is obscured when using particular numbers to check special cases. Writing $b$ for the base then each expression can be written in terms of $b$ and, by collecting like terms in the expression, or equating the powers of $b$, it can be shown that each expression holds. All that is required is some care over the algebraic manipulation.</p> <br></br></mdoxml> 2 4 0 0 0 0 1 Explain why this pattern holds in other number systems using bases other than base 10. 987654321 = 8 x 123456789 + 9, 98765432 = 8 x 12345678 + 8, 9876543 = 8 x 1234567 + 7, 987654 = 8 x 123456 + 6 ... 9 = 8 x 1 + 1 Using, Applying and Reasoning about Mathematics Mathematical reasoning & proof Numbers and the Number System Number bases Numbers and the Number System Place value