ÿþ<!DOCTYPE html > <html style="DIRECTION: ltr" lang="en"> <head> <title>Pascal&#39;s Extended Triangle</title> <meta content="text/html; charset=unicode" http-equiv="Content-Type"/> <link rel="stylesheet" type="text/css" href="mathematics.css"/> <meta name="author" content="Ken Ward"/> <meta name="description" content="Mathematical Series"/> <script src="1mathSeries.js"> <!--// //--> </script> <!--files for css menu--> <script src="../1mathMenuForAll.js"> <!--// //--> </script> <link rel="stylesheet" href="../1mathMenu.css" type="text/css"> <!--[if IE]> <style type="text/css" media="screen"> #menu ul li {float: left; width: 100%;} </style> <![endif]--> <!--[if lt IE 7]> <style type="text/css" media="screen"> body { behavior: url(csshover.htc); font-size: 100%; } #menu ul li a {height: 1%;} #menu a, #menu h2 { font: bold 0.7em/1.4em arial, helvetica, sans-serif; } .style1 { background-color: #996633; } .style2 { text-align: center; border-style: solid; border-width: 2px; } .style4 { border-style:solid; border-width: 1px; border-collapse:collapse; } .style5 { border-style: solid; border-width: 1px; } .style6 { vertical-align: middle; } .style7 { font-family: "Times New Roman"; } </style> <![endif]--> <!--end of css menu stuff--> </head> <body onload="writeMenu()"> <!--menu requires onload="writeMenu(), and the div with id below--> <div id="menu"> <!--start of menu--> </div> <!--end of menu--> <a href="multiNomialExpansion.htm#up"> <img style="BORDER-BOTTOM: 0px solid; BORDER-LEFT: 0px solid; BORDER-TOP: 0px solid; BORDER-RIGHT: 0px solid" alt="Navigation" src="nav.gif" width="100" height="25"/> </a> <img style="BORDER-BOTTOM: 0px solid; BORDER-LEFT: 0px solid; WIDTH: 75px; HEIGHT: 75px; BORDER-TOP: 0px solid; BORDER-RIGHT: 0px solid" alt="mathematics" src="clear.gif"/> <h1 style="TEXT-ALIGN: center"> Ken Ward&#39;s Mathematics Pages</h1><h2> Series <span lang="en-gb">P<span class="style1">ascal&#39;s Extended Triangle</span></span></h2> <p> <span lang="en-gb">On this page we are mainly concerned with Pascal&#39;s Triangle expanded into negative integers. Other expansions include fractions. </span> </p> <p> <span lang="en-gb">On this page, I sometimes write <img alt="" class="style6" src="img76.gif"> as B(n,k)</span></p> See also <a href="pascalsTriangle.htm"><span lang="en-gb">P</span>ascal<span lang="en-gb">&#39;</span>s<span lang="en-gb"> </span>Triangle</a><h3>Page Contents</h3> <ol> <li><a href="#Binomial_Expansions_for_Negative_n"> Binomial Expansions for Negative n</a></li> <li><a href="#List_of_Expansions_of_Binomials_with_Negative_n"> List of Expansions of Binomials with Negative n</a></li> <li><a href="#Pascals_Extended_Triangle_for_Negative_n"> Pascals Extended Triangle for Negative n</a></li> <li><a href="#Upper_Negation_or_Negating_the_Upper_Index">Upper Negation or Negating the Upper Index</a></li> <li><a href="#Proof_of_Upper_Negation_or_Negating_the_Upper_Index">Proof of Upper Negation or Negating the Upper Index</a></li> </ol> <p>&nbsp;</p> <p>&nbsp;</p> <h2><span lang="en-gb"><a name="Binomial_Expansions_for_Negative_n">Binomial Expansions for Negative n</a></span></h2> <p><span lang="en-gb">Of course, we rememeber that by binomial we refer to (a+b)<sup>n</sup>, and the like. We usually expand these using the binomial <strong> theorem</strong>. If we did not know the binomial theorem, then we would expand these binomials in some other way (such as straight division). On this page, we are interested in binomials of the form (1+x)<sup><span class="superscript">"</span></sup><span class="superscript">n </span>&nbsp;where n is a positive integer or zero and |x|&lt;1. Unless |x|&lt;1 the binomial does not converge for negative n. (See <a href="binomialProofNegativeIntegers.htm">binomialProofNegativeIntegers</a>). We explore some of these expansions which we discover by straight division (because we have forgotten the binomial theorem, and also because we do not want to assume what we seek to prove). </span></p> <h2><span lang="en-gb"> <a name="List_of_Expansions_of_Binomials_with_Negative_n">List of Expansions of Binomials with Negative n</a></span></h2> <p><span lang="en-gb">The series below are expansions for various binomials, to eleven terms and for eleven values of n:</span></p> <p><img alt="" src="listOfBinsNeg.gif" height="350" width="1007"><span lang="en-gb"> [1.01]</span></p> <p><span lang="en-gb">All these (except when n=0) are infinite series that converge when |x|&lt;1. </span></p> <p>&nbsp;</p> <h2><span lang="en-gb"><a name="Pascals_Extended_Triangle_for_Negative_n">Pascal&#39;s Extended Triangle for Negative n</a></span></h2> <p><span lang="en-gb">Using the information from the expansions above [1.01], derived by straight division, we can make a table as below:</span></p> <table border="1" cellpadding="0" cellspacing="0" class="style4" style="width: 576pt" width="768"> <colgroup> <col span="12" style="width:48pt" width="64"> </colgroup> <tr height="19" style="height:14.4pt"> <th class="style2" colspan="12" height="19">Pascal&#39;s Extended Triangle</th> </tr> <tr height="19" style="height:14.4pt"> <td class="style5" height="19"></td> <th class="style2" colspan="11">k</th> </tr> <tr height="19" style="height:14.4pt"> <th class="style2" height="19">n</th> <th align="right" class="style5">0</th> <th align="right" class="style5">1</th> <th align="right" class="style5">2</th> <th align="right" class="style5">3</th> <th align="right" class="style5">4</th> <th align="right" class="style5">5</th> <th align="right" class="style5">6</th> <th align="right" class="style5">7</th> <th align="right" class="style5">8</th> <th align="right" class="style5">9</th> <th align="right" class="style5">10</th> </tr> <tr height="19" style="height:14.4pt"> <th class="style5" height="19">-10</th> <td align="right" class="style5">1</td> <td align="right" class="style5">-10</td> <td align="right" class="style5">55</td> <td align="right" class="style5">-220</td> <td align="right" class="style5">715</td> <td align="right" class="style5">-2002</td> <td align="right" class="style5">5005</td> <td align="right" class="style5">-11440</td> <td align="right" class="style5">24310</td> <td align="right" class="style5">-48620</td> <td align="right" class="style5">92378</td> </tr> <tr height="19" style="height:14.4pt"> <th class="style5" height="19">-9</th> <td align="right" class="style5">1</td> <td align="right" class="style5">-9</td> <td align="right" class="style5">45</td> <td align="right" class="style5">-165</td> <td align="right" class="style5">495</td> <td align="right" class="style5">-1287</td> <td align="right" class="style5">3003</td> <td align="right" class="style5">-6435</td> <td align="right" class="style5">12870</td> <td align="right" class="style5">-24310</td> <td align="right" class="style5">43758</td> </tr> <tr height="19" style="height:14.4pt"> <th class="style5" height="19">-8</th> <td align="right" class="style5">1</td> <td align="right" class="style5">-8</td> <td align="right" class="style5">36</td> <td align="right" class="style5">-120</td> <td align="right" class="style5">330</td> <td align="right" class="style5">-792</td> <td align="right" class="style5">1716</td> <td align="right" class="style5">-3432</td> <td align="right" class="style5">6435</td> <td align="right" class="style5">-11440</td> <td align="right" class="style5">19448</td> </tr> <tr height="19" style="height:14.4pt"> <th class="style5" height="19">-7</th> <td align="right" class="style5">1</td> <td align="right" class="style5">-7</td> <td align="right" class="style5">28</td> <td align="right" class="style5">-84</td> <td align="right" class="style5">210</td> <td align="right" class="style5">-462</td> <td align="right" class="style5">924</td> <td align="right" class="style5">-1716</td> <td align="right" class="style5">3003</td> <td align="right" class="style5">-5005</td> <td align="right" class="style5">8008</td> </tr> <tr height="19" style="height:14.4pt"> <th class="style5" height="19">-6</th> <td align="right" class="style5">1</td> <td align="right" class="style5">-6</td> <td align="right" class="style5">21</td> <td align="right" class="style5">-56</td> <td align="right" class="style5">126</td> <td align="right" class="style5">-252</td> <td align="right" class="style5">462</td> <td align="right" class="style5">-792</td> <td align="right" class="style5">1287</td> <td align="right" class="style5">-2002</td> <td align="right" class="style5">3003</td> </tr> <tr height="19" style="height:14.4pt"> <th class="style5" height="19">-5</th> <td align="right" class="style5">1</td> <td align="right" class="style5">-5</td> <td align="right" class="style5">15</td> <td align="right" class="style5">-35</td> <td align="right" class="style5">70</td> <td align="right" class="style5">-126</td> <td align="right" class="style5">210</td> <td align="right" class="style5">-330</td> <td align="right" class="style5">495</td> <td align="right" class="style5">-715</td> <td align="right" class="style5">1001</td> </tr> <tr height="19" style="height:14.4pt"> <th class="style5" height="19">-4</th> <td align="right" class="style5">1</td> <td align="right" class="style5">-4</td> <td align="right" class="style5">10</td> <td align="right" class="style5">-20</td> <td align="right" class="style5">35</td> <td align="right" class="style5">-56</td> <td align="right" class="style5">84</td> <td align="right" class="style5">-120</td> <td align="right" class="style5">165</td> <td align="right" class="style5">-220</td> <td align="right" class="style5">286</td> </tr> <tr height="19" style="height:14.4pt"> <th class="style5" height="19">-3</th> <td align="right" class="style5">1</td> <td align="right" class="style5">-3</td> <td align="right" class="style5">6</td> <td align="right" class="style5">-10</td> <td align="right" class="style5">15</td> <td align="right" class="style5">-21</td> <td align="right" class="style5">28</td> <td align="right" class="style5">-36</td> <td align="right" class="style5">45</td> <td align="right" class="style5">-55</td> <td align="right" class="style5">66</td> </tr> <tr height="19" style="height:14.4pt"> <th class="style5" height="19">-2</th> <td align="right" class="style5">1</td> <td align="right" class="style5">-2</td> <td align="right" class="style5">3</td> <td align="right" class="style5">-4</td> <td align="right" class="style5">5</td> <td align="right" class="style5">-6</td> <td align="right" class="style5">7</td> <td align="right" class="style5">-8</td> <td align="right" class="style5">9</td> <td align="right" class="style5">-10</td> <td align="right" class="style5">11</td> </tr> <tr height="19" style="height:14.4pt"> <th class="style5" height="19">-1</th> <td align="right" class="style5">1</td> <td align="right" class="style5">-1</td> <td align="right" class="style5">1</td> <td align="right" class="style5">-1</td> <td align="right" class="style5">1</td> <td align="right" class="style5">-1</td> <td align="right" class="style5">1</td> <td align="right" class="style5">-1</td> <td align="right" class="style5">1</td> <td align="right" class="style5">-1</td> <td align="right" class="style5">1</td> </tr> <tr height="19" style="height:14.4pt"> <th class="style5" height="19">0</th> <td align="right" class="style5">1</td> <td align="right" class="style5">0</td> <td align="right" class="style5">0</td> <td align="right" class="style5">0</td> <td align="right" class="style5">0</td> <td align="right" class="style5">0</td> <td align="right" class="style5">0</td> <td align="right" class="style5">0</td> <td align="right" class="style5">0</td> <td align="right" class="style5">0</td> <td align="right" class="style5">0</td> </tr> </table> <br><span lang="en-gb">The table above is Pascal&#39;s Extended Triangle for negative numbers. Other extensions involve fractions, etc. <br>We note that whenever k&lt;n, then B(n,k)=0, that is:<br>&nbsp;<img alt="" src="img3C1.gif"> [2.01<br> So if we consider B(0,0), which is 1 in the table, by applying the Addition Law, we note:<br><br><img alt="" src="img70.gif"> [2.02]<br>We note that the left-hand-side is 1, and B(-1,0) is one, so B(-1,-1) must be 0, as indicated in the general definition [2.01]<br><br>We can verify that the addition law seems to apply for some negative n. <br></span>&nbsp;<h2> <a name="Addition_Law,_Proof_for_all_n">Addition Law<span lang="en-gb">, Proof for all n</span></a></h2> <p><span lang="en-gb">Just like Pascal&#39;s Triangle, we can discover any number from this table from the numbers in earlier rows, using the relationship:<br> <img style="width: 198px; height: 43px;" alt="additionLaw.gif" src="additionLaw.gif"> from&nbsp; <a href="pascalsTriangle.htm#1.2,_the_Addition_Law">pascal&#39;s triangle</a></span></p> <p><span lang="en-gb">First we claim to prove this formula for n as a positive integer and for n equal to zero.</span></p> <p><span lang="en-gb">If n is a positive integer then we can interpret the first binomial, n over k, as a combination, n choose k. Suppose we have n marbles, one of them is blue, and the rest are red. We can pick k marbles (containing a blue one, or not) in <img alt="" class="style6" src="img4D.gif"> ways. We can select k marbles with all of them red, in <img alt="" class="style6" src="img4B.gif">ways. That is we remove the blue marble (so it cannot be selected), and pick out k red marble at random. This is all the combinations except those containing exactly one blue marble. We can select these remaining combinations by picking out the blue marble and selecting (k-1) marbles from the (n-1) red marbles to make up the k required in <img alt="" class="style6" src="img71.gif"> ways. Clearly, the total selections <img alt="" class="style6" src="img4D.gif"> are:</span></p> <p><span lang="en-gb"> <img style="width: 198px; height: 43px;" alt="additionLaw.gif" src="additionLaw.gif" class="style6"></span></p> <p><span lang="en-gb">which is the addition law. Hence the addition law is true for all positive n. (It is also true at zero, because then all the terms are zero). We also note that both sides of the equation are polynomials of degree k. By the </span> <a href="../polynomials/polyNomialsIdntical.htm#Polynomial_Argument"> <span lang="en-gb">polynomial argument, </span></a>t<span lang="en-gb">wo polynomials of degree k can agree at most at k points, otherwise they are the same polynomial. The equation agrees at an infinite number of points (when n is a positive integer) hence the polynomials are identical and therefore true for all real (and complex values). </span></p> <p><span lang="en-gb">The addition law is, therefore, true for all values of n. Hence, we ought to write n as r to indicate we are referring to all numbers:</span></p> <p><img alt="" src="img5C1.gif"><span lang="en-gb"><span class="style7"> %</span>&nbsp; [3.01]</span></p> <h2><span lang="en-gb"><a name="Upper_Negation_or_Negating_the_Upper_Index">Upper Negation or Negating the Upper Index</a></span></h2> <p><span lang="en-gb">Upper Negation is also called negating the upper index.</span></p> <p><span lang="en-gb">The diagonals in <a href="pascalsTriangle.htm#Table Pascals Triangle">Pascal&#39;s Triangle</a> seem to contain the same numbers as the rows in Pascal&#39;s Extended Triangle (for Negative Integers) above. </span></p> <p><span lang="en-gb">There is a rule for the relationship:</span></p> <p><img alt="upper negation" src="img72.gif"><span lang="en-gb">&nbsp; [4.01]</span></p> <p><span lang="en-gb">Where r is a real number (also complex) and k is an integer. If k&lt;0, then both sides are zero. </span></p> <p><span lang="en-gb">This is called Upper Negation, because we compose the right-hand-side binomial (after writing (-1)<span class="superscript">k</span>) by writing down k (in the upper and lower indexes) and take the upper index, r, of the left-hand-side, negate it, (-r) and add it to the k in the upper index and finally subtract 1. </span></p> <p><span lang="en-gb">This formula is useful in solving problems involving Binomials, with no restrictions on r and for k as an integer. For instance, starting with the right-hand-side, it enables us to remove the k from the upper index, sometimes simplifying the expression. </span></p> <h2><span lang="en-gb"> <a name="Proof_of_Upper_Negation_or_Negating_the_Upper_Index">Proof of Upper Negation or Negating the Upper Index</a></span></h2> <p><span lang="en-gb">We wish to prove Upper Negation:</span></p> <p><img alt="upper negation" src="img72.gif"><span lang="en-gb">&nbsp; [4.01, repeated]</span></p> <p><span lang="en-gb">To begin with we take r as a positive integer (or zero) and expand as a combination. </span></p> <p><span lang="en-gb">We expand the left-hand-side:</span></p> <p><img alt="" src="imgF1.gif"><span lang="en-gb"> [5.01]</span></p> <p><span lang="en-gb">Now multiply every factor by (-1), k times:</span></p> <p><img alt="" src="img73.gif"><span lang="en-gb"> [5.02]</span></p> <p><span lang="en-gb">Reverse the order of each of the factors:</span></p> <p><img alt="" src="img74.gif"><span lang="en-gb"> [5.03]</span></p> <p><span lang="en-gb">We notice that the factors are the binomial coefficient, <img alt="" class="style6" src="img1E.gif">, so we can write:</span></p> <p><img alt="" src="img75.gif"><span lang="en-gb"><span class="style7"> %</span> [5.04]</span></p> <p><span lang="en-gb">The two sides are the same for all r as a positive integer (or zero) and for k as an integer. If k&lt;0, then the expressions are zero by definition. As with the addition law, we can note that both sides of the equation are polynomials of degree r. Such polynomials can agree at at most r points, but as these agree at an infinite number (the positive integers) then the two sides are identical and are true for all values of r. (k must be an integer). </span></p> <p>&nbsp;</p> <p>&nbsp;</p><br><br/><br/><br/><br/><a name="up"></a><br/><a href="../index.html"> Ken Ward&#39;s Mathematics Pages</a><br/><hr/><script type="text/javascript"> <!--// NextPreviousWriter(); MenuWriter(); CopyRighter(); //--> </script><noscript> <p>No script follows:</p> </noscript><hr/></body> </html>