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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikibooks, open books for an open world</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><div class="subpages">< <bdi dir="ltr"><a href="/wiki/Traditional_Abacus_and_Bead_Arithmetic" title="Traditional Abacus and Bead Arithmetic">Traditional Abacus and Bead Arithmetic</a></bdi> | <bdi dir="ltr"><a href="/wiki/Traditional_Abacus_and_Bead_Arithmetic/Division" title="Traditional Abacus and Bead Arithmetic/Division">Division</a></bdi></div><div id="mw-fr-revision-messages"><div class="cdx-message mw-fr-message-box cdx-message--block cdx-message--notice mw-fr-basic mw-fr-stable-unreviewed plainlinks noprint"><span class="cdx-message__icon"></span><div class="cdx-message__content">This page may need to be <a href="/wiki/Wikibooks:REVIEW" 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class="mw-editsection-bracket">]</span></span></div> <p>The division table contains 45 rules, including the 9 diagonal elements for multi-digit divisors. </p> <table class="wikitable"> <caption>Division Table (八算, <i>Hassan</i>, <i>Bā suàn</i>) </caption> <tbody><tr> <td>1/9>1+1 </td> <td>2/9>2+2 </td> <td>3/9>3+3 </td> <td>4/9>4+4 </td> <td>5/9>5+5 </td> <td>6/9>6+6 </td> <td>7/9>7+7 </td> <td>8/9>8+8 </td> <td style="background: lightgrey;">9/9>9+9 </td></tr> <tr> <td>1/8>1+2 </td> <td>2/8>2+4 </td> <td>3/8>3+6 </td> <td>4/8>5+0 </td> <td>5/8>6+2 </td> <td>6/8>7+4 </td> <td>7/8>8+6 </td> <td style="background: lightgrey;">8/8>9+8 </td> <td> </td></tr> <tr> <td>1/7>1+3 </td> <td>2/7>2+6 </td> <td>3/7>4+2 </td> <td>4/7>5+5 </td> <td>5/7>7+1 </td> <td>6/7>8+4 </td> <td style="background: lightgrey;">7/7>9+7 </td> <td> </td> <td> </td></tr> <tr> <td>1/6>1+4 </td> <td>2/6>3+2 </td> <td>3/6>5+0 </td> <td>4/6>6+4 </td> <td>5/6>8+2 </td> <td style="background: lightgrey;">6/6>9+6 </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td>1/5>2+0 </td> <td>2/5>4+0 </td> <td>3/5>6+0 </td> <td>4/5>8+0 </td> <td style="background: lightgrey;">5/5>9+5 </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td>1/4>2+2 </td> <td>2/4>5+0 </td> <td>3/4>7+2 </td> <td style="background: lightgrey;">4/4>9+4 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td>1/3>3+1 </td> <td>2/3>6+2 </td> <td style="background: lightgrey;">3/3>9+3 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td>1/2>5+0 </td> <td style="background: lightgrey;">2/2>9+2 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td style="background: lightgrey;">1/1>9+1 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr></tbody></table> <p>The same number of independent elements that we find in the multiplication table (given the commutativity of this operation) whose memorization was one of the feats of our childhood in school. Memorizing the division table is therefore a similar task to learning the multiplication table. </p><p>These rules: </p> <ul><li>From an operational point of view, these rules should be read or interpreted slightly differently depending on whether we use the traditional (<b>TDA</b>) or the modern (<b>MDA</b>) division arrangement. <ul><li>when using <b>MDA</b>, the rule <b>a/b>q+r</b> must be read: <i>“write <b>q</b> as interim quotient digit to the left, clear <b>a</b> and add <b>r</b> to the right”</i></li> <li>When using <b>TDA</b>, the rule <b>a/b>q+r</b> must be read: <i>“change <b>a</b> into <b>q</b> as interim quotient digit and add <b>r</b> to the right”</i></li></ul></li> <li>From a theoretical point of view, each rule expresses the result of a Euclidean division: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (10\times a)/b=q,r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>10</mn> <mo>×</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> <mo>=</mo> <mi>q</mi> <mo>,</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (10\times a)/b=q,r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5bd50f69fe1f963ffedd1d78ee971fad4f47468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.615ex; height:2.843ex;" alt="{\textstyle (10\times a)/b=q,r}"></span> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>: quotient, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>: remainder, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"></span> digits from 1 to 9) or, equivalently <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 10a=q\cdot b+r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>10</mn> <mi>a</mi> <mo>=</mo> <mi>q</mi> <mo>⋅</mo> <mi>b</mi> <mo>+</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 10a=q\cdot b+r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a87369a8fa0fd425157edb15aa86ac9faa158ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.288ex; height:2.509ex;" alt="{\textstyle 10a=q\cdot b+r}"></span></li></ul> <p>If we think about this last point, in fact there is no need to memorize the division rules since we can obtain them in situ, when we need them, by a simple mental process. But then we would be making a mental effort similar to that required with the modern method of division and we would be moving away from the philosophy of the traditional method. There is no doubt, the efficiency and goodness of the traditional method is only achieved by memorizing the rules and we should only resort to the aforementioned mental process during the learning phase, when some rule resists coming to memory. </p><p>Fortunately, a series of patterns that appear in the division table come to our aid making it easier for us to learn it, leaving only 14 <i>hard rules</i> out of a total of 45. </p> <div class="mw-heading mw-heading2"><h2 id="Easy_rules">Easy rules</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Traditional_Abacus_and_Bead_Arithmetic/Division/Learning_the_division_table&veaction=edit&section=2" title="Edit section: Easy rules" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Traditional_Abacus_and_Bead_Arithmetic/Division/Learning_the_division_table&action=edit&section=2" title="Edit section's source code: Easy rules"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the chapter: <a href="/wiki/Traditional_Abacus_and_Bead_Arithmetic/Division/Guide_to_traditional_division_(%E5%B8%B0%E9%99%A4%E6%B3%95)" title="Traditional Abacus and Bead Arithmetic/Division/Guide to traditional division (帰除法)">Guide to traditional division (帰除法)</a> we already mentioned that the division rules by 9, 5 and 2, as well as the diagonal rules, have a particularly simple structure that allows almost immediate memorization. </p> <table class="wikitable"> <caption>Easy rules </caption> <tbody><tr> <th>Diagonal </th> <th>Divide by 9 </th> <th>Divide by 5 </th> <th>Divide by 2 </th></tr> <tr> <td>1/1>9+1 </td> <td>1/9>1+1 </td> <td>1/5>2+0 </td> <td>1/2>5+0 </td></tr> <tr> <td>2/2>9+2 </td> <td>2/9>2+2 </td> <td>2/5>4+0 </td> <td> </td></tr> <tr> <td>3/3>9+3 </td> <td>3/9>3+3 </td> <td>3/5>6+0 </td> <td> </td></tr> <tr> <td>4/4>9+4 </td> <td>4/9>4+4 </td> <td>4/5>8+0 </td> <td> </td></tr> <tr> <td>5/5>9+5 </td> <td>5/9>5+5 </td> <td> </td> <td> </td></tr> <tr> <td>6/6>9+6 </td> <td>6/9>6+6 </td> <td> </td> <td> </td></tr> <tr> <td>7/7>9+7 </td> <td>7/9>7+7 </td> <td> </td> <td> </td></tr> <tr> <td>8/8>9+8 </td> <td>8/9>8+8 </td> <td> </td> <td> </td></tr> <tr> <td>9/9>9+9 </td> <td> </td> <td> </td> <td> </td></tr></tbody></table> <p>For this reason, the examples presented in that chapter only made use of divisors starting with 2,5 and 9. If you practice several examples with such divisors, it will not be difficult for you to memorize these 22 rules (almost half of the total!); which is a drastic reduction in the work to be done and not the only one. </p> <div class="mw-heading mw-heading2"><h2 id="Division_by_8">Division by 8</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Traditional_Abacus_and_Bead_Arithmetic/Division/Learning_the_division_table&veaction=edit&section=3" title="Edit section: Division by 8" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Traditional_Abacus_and_Bead_Arithmetic/Division/Learning_the_division_table&action=edit&section=3" title="Edit section's source code: Division by 8"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Of the remaining rules, the division by 8 series is the longest but not the most difficult, since it has an internal structure: </p> <table class="wikitable" style="text-align:center"> <caption>Division by 8 rules </caption> <tbody><tr> <td>1/8>1+2 </td> <td>5/8>6+2 </td></tr> <tr> <td>2/8>2+4 </td> <td>6/8>7+4 </td></tr> <tr> <td>3/8>3+6 </td> <td>7/8>8+6 </td></tr> <tr> <td colspan="2">4/8>5+0 </td></tr></tbody></table> <p>Leaving aside 4/8>5+0 (think of this as 8x5 = 40), the two sub-series 1, 2, 3 and 5, 6, 7 have the same remainders and the quotients are as simple as 1, 2, 3 and 6, 7, 8; so, without a doubt, this will not be the series that will be the most difficult for you to learn. </p> <div class="mw-heading mw-heading2"><h2 id="Subdiagonal_rules">Subdiagonal rules</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Traditional_Abacus_and_Bead_Arithmetic/Division/Learning_the_division_table&veaction=edit&section=4" title="Edit section: Subdiagonal rules" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Traditional_Abacus_and_Bead_Arithmetic/Division/Learning_the_division_table&action=edit&section=4" title="Edit section's source code: Subdiagonal rules"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Finally, as a last resort for learning, note the following series of terms adjacent to the diagonal of the table. </p> <table class="wikitable" style="text-align:center"> <caption>Subdiagonal rules </caption> <tbody><tr> <td>4/5>8+0 </td></tr> <tr> <td>5/6>8+2 </td></tr> <tr> <td>6/7>8+4 </td></tr> <tr> <td>7/8>8+6 </td></tr> <tr> <td>8/9>8+8 </td></tr></tbody></table> <p>There are really only two new rules here, but grasping the structure of the table above will also help you memorize the rules for divisors 5, 8, and 9. </p> <div class="mw-heading mw-heading2"><h2 id="Hard_rules">Hard rules</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Traditional_Abacus_and_Bead_Arithmetic/Division/Learning_the_division_table&veaction=edit&section=5" title="Edit section: Hard rules" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Traditional_Abacus_and_Bead_Arithmetic/Division/Learning_the_division_table&action=edit&section=5" title="Edit section's source code: Hard rules"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In summary, of the 45 rules included in the division table, 31 fall within one of the previous patterns (grayed) </p> <table class="wikitable"> <caption>Division Table (八算, <i>Hassan</i>, <i>Bā suàn</i>) </caption> <tbody><tr> <td style="background: lightgrey;">1/9>1+1 </td> <td style="background: lightgrey;">2/9>2+2 </td> <td style="background: lightgrey;">3/9>3+3 </td> <td style="background: lightgrey;">4/9>4+4 </td> <td style="background: lightgrey;">5/9>5+5 </td> <td style="background: lightgrey;">6/9>6+6 </td> <td style="background: lightgrey;">7/9>7+7 </td> <td style="background: lightgrey;">8/9>8+8 </td> <td style="background: lightgrey;">9/9>9+9 </td></tr> <tr> <td style="background: lightgrey;">1/8>1+2 </td> <td style="background: lightgrey;">2/8>2+4 </td> <td style="background: lightgrey;">3/8>3+6 </td> <td style="background: lightgrey;">4/8>5+0 </td> <td style="background: lightgrey;">5/8>6+2 </td> <td style="background: lightgrey;">6/8>7+4 </td> <td style="background: lightgrey;">7/8>8+6 </td> <td style="background: lightgrey;">8/8>9+8 </td> <td> </td></tr> <tr> <td>1/7>1+3 </td> <td>2/7>2+6 </td> <td>3/7>4+2 </td> <td>4/7>5+5 </td> <td>5/7>7+1 </td> <td style="background: lightgrey;">6/7>8+4 </td> <td style="background: lightgrey;">7/7>9+7 </td> <td> </td> <td> </td></tr> <tr> <td>1/6>1+4 </td> <td>2/6>3+2 </td> <td>3/6>5+0 </td> <td>4/6>6+4 </td> <td style="background: lightgrey;">5/6>8+2 </td> <td style="background: lightgrey;">6/6>9+6 </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td style="background: lightgrey;">1/5>2+0 </td> <td style="background: lightgrey;">2/5>4+0 </td> <td style="background: lightgrey;">3/5>6+0 </td> <td style="background: lightgrey;">4/5>8+0 </td> <td style="background: lightgrey;">5/5>9+5 </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td>1/4>2+2 </td> <td>2/4>5+0 </td> <td>3/4>7+2 </td> <td style="background: lightgrey;">4/4>9+4 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td>1/3>3+1 </td> <td>2/3>6+2 </td> <td style="background: lightgrey;">3/3>9+3 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td style="background: lightgrey;">1/2>5+0 </td> <td style="background: lightgrey;">2/2>9+2 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <td style="background: lightgrey;">1/1>9+1 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr></tbody></table> <p>and we are left with only 14 "hard" rules to memorize with no other help. This is no longer a huge job. Cheer up and don't give up! with some effort and practice, the greatest of the arcane mysteries of Traditional Bead Arithmetic will be yours! </p> <div class="mw-heading mw-heading2"><h2 id="The_combined_multiplication-division_table">The combined multiplication-division table</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Traditional_Abacus_and_Bead_Arithmetic/Division/Learning_the_division_table&veaction=edit&section=6" title="Edit section: The combined multiplication-division table" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Traditional_Abacus_and_Bead_Arithmetic/Division/Learning_the_division_table&action=edit&section=6" title="Edit section's source code: The combined multiplication-division table"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>What follows is a simple historical note with little or no practical relevance. </p><p>The multiplication table in the English language contains all the 81 two-digit products in any order; that is, it includes both 8x9 = 72 and 9x8 = 72, which is unnecessary given the commutativity of the multiplication. On the contrary, in Chinese it only contained one of the terms of these pairs 8x9 = 72; always with the first factor less than or equal to the second<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>. On the other hand, the division rules were enunciated by giving first the divisor that is always greater than the dividend, with the exception of the rules that we have called diagonals in which it is equal. This allows a combined multiplication-division table to be conceived that covers the entire "space" of pairs of digits as operands: </p><p><br /> </p> <table class="wikitable"> <caption>Combined table of multiplication and division rules </caption> <tbody><tr> <td>9✕9 81 </td> <td>9\8 8+8 </td> <td>9\7 7+7 </td> <td>9\6 6+6 </td> <td>9\5 5+5 </td> <td>9\4 4+4 </td> <td>9\3 3+3 </td> <td>9\2 2+2 </td> <td>9\1 1+1 </td></tr> <tr> <td>8✕9 72 </td> <td>8✕8 64 </td> <td>8\7 8+6 </td> <td>8\6 7+4 </td> <td>8\5 6+2 </td> <td>8\4 5+0 </td> <td>8\3 3+6 </td> <td>8\2 2+4 </td> <td>8\1 1+2 </td></tr> <tr> <td>7✕9 63 </td> <td>7✕8 56 </td> <td>7✕7 49 </td> <td>7\6 8+4 </td> <td>7\5 7+1 </td> <td>7\4 5+5 </td> <td>7\3 4+2 </td> <td>7\2 2+6 </td> <td>7\1 1+3 </td></tr> <tr> <td>6✕9 54 </td> <td>6✕8 48 </td> <td>6✕7 42 </td> <td>6✕6 36 </td> <td>6\5 8+2 </td> <td>6\4 6+4 </td> <td>6\3 5+0 </td> <td>6\2 3+2 </td> <td>6\1 1+4 </td></tr> <tr> <td>5✕9 45 </td> <td>5✕8 40 </td> <td>5✕7 35 </td> <td>5✕6 30 </td> <td>5✕5 25 </td> <td>5\4 8+0 </td> <td>5\3 6+0 </td> <td>5\2 4+0 </td> <td>5\1 2+0 </td></tr> <tr> <td>4✕9 36 </td> <td>4✕8 32 </td> <td>4✕7 28 </td> <td>4✕6 24 </td> <td>4✕5 20 </td> <td>4✕4 16 </td> <td>4\3 7+2 </td> <td>4\2 5+0 </td> <td>4\1 2+2 </td></tr> <tr> <td>3✕9 27 </td> <td>3✕8 24 </td> <td>3✕7 21 </td> <td>3✕6 18 </td> <td>3✕5 15 </td> <td>3✕4 12 </td> <td>3✕3  9 </td> <td>3\2 2+6 </td> <td>3\1 3+1 </td></tr> <tr> <td>2✕9 18 </td> <td>2✕8 16 </td> <td>2✕7 14 </td> <td>2✕6 12 </td> <td>2✕5 10 </td> <td>2✕4  8 </td> <td>2✕3  6 </td> <td>2✕2  4 </td> <td>2\1 5+0 </td></tr> <tr> <td>1✕9  9 </td> <td>1✕8  8 </td> <td>1✕7  7 </td> <td>1✕6  6 </td> <td>1✕5  5 </td> <td>1✕4  4 </td> <td>1✕3  3 </td> <td>1✕2  2 </td> <td>1✕1  1 </td></tr></tbody></table> <p>Where we have altered the writing of our division rules to adapt them to the order of arguments used in Chinese. To highlight this fact we have replaced "/" by "\", so that the division rules as they appear in the above table must be interpreted in the form: Read <b>a\b c+d</b>:  as: <b>a</b> divide into <b>b0</b> <b>c</b> times leaving <b>d</b> as remainder. </p><p>The combined table has 81 elements or rules, to which we must add the diagonal rules </p> <table class="wikitable"> <tbody><tr> <td>Diagonal </td></tr> <tr> <td>1/1>9+1 </td></tr> <tr> <td>2/2>9+2 </td></tr> <tr> <td>3/3>9+3 </td></tr> <tr> <td>4/4>9+4 </td></tr> <tr> <td>5/5>9+5 </td></tr> <tr> <td>6/6>9+6 </td></tr> <tr> <td>7/7>9+7 </td></tr> <tr> <td>8/8>9+8 </td></tr> <tr> <td>9/9>9+9 </td></tr></tbody></table> <p>and the rules for revising down given in the previous chapter. </p> <table class="wikitable"> <caption>Rules to revise down (two-digit divisors) </caption> <tbody><tr> <th>While dividing by </th> <th>Revise q to </th> <th>Add to remainder </th></tr> <tr> <td>1 </td> <td>q-1 </td> <td>+1 </td></tr> <tr> <td>2 </td> <td>q-1 </td> <td>+2 </td></tr> <tr> <td>3 </td> <td>q-1 </td> <td>+3 </td></tr> <tr> <td>4 </td> <td>q-1 </td> <td>+4 </td></tr> <tr> <td>5 </td> <td>q-1 </td> <td>+5 </td></tr> <tr> <td>6 </td> <td>q-1 </td> <td>+6 </td></tr> <tr> <td>7 </td> <td>q-1 </td> <td>+7 </td></tr> <tr> <td>8 </td> <td>q-1 </td> <td>+8 </td></tr> <tr> <td>9 </td> <td>q-1 </td> <td>+9 </td></tr></tbody></table> <p>that were studied separately. This adds up to a total of 99 rules to which we can add the approximately 50 addition and subtraction rules. The traditional learning of the abacus consisted fundamentally of the memorization and practice of these 150 rules. </p> <div class="mw-heading mw-heading2"><h2 id="Statistical_rules">Statistical rules</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Traditional_Abacus_and_Bead_Arithmetic/Division/Learning_the_division_table&veaction=edit&section=7" title="Edit section: Statistical rules" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Traditional_Abacus_and_Bead_Arithmetic/Division/Learning_the_division_table&action=edit&section=7" title="Edit section's source code: Statistical rules"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>What follows is a matter that arises from practice, not from any book in the past. The diagonal rules for divisors 1 and 2 </p> <table class="wikitable"> <caption>Diagonal rules for 1 and 2 </caption> <tbody><tr> <td>2/2>9+2 </td></tr> <tr> <td>1/1>9+1 </td></tr></tbody></table> <p>are excessive in the sense that we are often forced to revise up the divisor several times. In practice the following two <i>statistical</i> rules (to give them a name) behave better allowing a faster calculation. </p> <table class="wikitable"> <caption>Statistical rules </caption> <tbody><tr> <td>2/2>7+6 </td></tr> <tr> <td>1/1>7+3 </td></tr></tbody></table> <p>Please try them sometime during your practice! </p><p><br /> </p> <div class="noprint" style="border-top: 5px double DarkGray; font-weight: 500; margin: 2rem 3vw; padding: 1rem 3vw;"><i>Next Page: <a href="/wiki/Traditional_Abacus_and_Bead_Arithmetic/Division/Dealing_with_overflow" title="Traditional Abacus and Bead Arithmetic/Division/Dealing with overflow">Dealing with overflow</a></i> | <small>Previous Page: <a href="/wiki/Traditional_Abacus_and_Bead_Arithmetic/Division/Guide_to_traditional_division_(%E5%B8%B0%E9%99%A4%E6%B3%95)" title="Traditional Abacus and Bead Arithmetic/Division/Guide to traditional division (帰除法)">Guide to traditional division (帰除法)</a></small><br /><span style="font-size: small; margin-left: 1.5em; line-height: 2.5;">Home: <a href="/wiki/Traditional_Abacus_and_Bead_Arithmetic/Division" title="Traditional Abacus and Bead Arithmetic/Division">Traditional Abacus and Bead Arithmetic/Division</a></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r4271529">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFChéng_Dàwèi_(程大位)1993" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/w/index.php?title=Cheng_Dawei&action=edit&redlink=1" class="new" title="Cheng Dawei (does not exist)">Chéng Dàwèi (程大位)</a> (1993) [1592]. <i>Suànfǎ Tǒngzōng (算法統宗)</i> (in Chinese). Zhōngguó kēxué jìshù diǎnjí tōng huì (中國科學技術典籍通彙).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Su%C3%A0nf%C7%8E+T%C7%92ngz%C5%8Dng+%28%E7%AE%97%E6%B3%95%E7%B5%B1%E5%AE%97%29&rft.pub=Zh%C5%8Dnggu%C3%B3+k%C4%93xu%C3%A9+j%C3%ACsh%C3%B9+di%C7%8Enj%C3%AD+t%C5%8Dng+hu%C3%AC+%28%E4%B8%AD%E5%9C%8B%E7%A7%91%E5%AD%B8%E6%8A%80%E8%A1%93%E5%85%B8%E7%B1%8D%E9%80%9A%E5%BD%99%29&rft.date=1993&rft.au=Ch%C3%A9ng+D%C3%A0w%C3%A8i+%28%E7%A8%8B%E5%A4%A7%E4%BD%8D%29&rfr_id=info%3Asid%2Fen.wikibooks.org%3ATraditional+Abacus+and+Bead+Arithmetic%2FDivision%2FLearning+the+division+table" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Unknown parameter <code class="cs1-code">|trans_title=</code> ignored (<code class="cs1-code">|trans-title=</code> suggested) (<a href="/wiki/Help:CS1_errors#parameter_ignored_suggest" title="Help:CS1 errors">help</a>)</span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4271529"><cite id="CITEREFChen2013" class="citation book cs1 cs1-prop-foreign-lang-source">Chen, Yifu (2013). <a rel="nofollow" class="external text" href="http://www.theses.fr/2013PA070061"><i>L’étude des Différents Modes de Déplacement des Boules du Boulier et de l’Invention de la Méthode de Multiplication Kongpan Qianchengfa et son Lien avec le Calcul Mental</i></a> (PhD thesis) (in French). 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