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Grid method multiplication
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<h2 id="calculations">Calculations</h2> <h3>Introductory motivation</h3> <p>The grid method can be introduced by thinking about how to add up the number of points in a regular array, for example the number of squares of chocolate in a chocolate bar. As the size of the calculation becomes larger, it becomes easier to start counting in tens; and to represent the calculation as a box which can be sub-divided, rather than drawing a multitude of dots.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p>At the simplest level, pupils might be asked to apply the method to a calculation like 3 × 17. Breaking up ("partitioning") the 17 as (10 + 7), this unfamiliar multiplication can be worked out as the sum of two simple multiplications: </p> <table><tbody><tr><th scope="col">×</th><th scope="col">10</th><th scope="col">7</th></tr><tr><th scope="row">3</th><td>30</td><td>21</td></tr></tbody></table> <p>so 3 × 17 = 30 + 21 = 51. </p><p>This is the "grid" or "boxes" structure which gives the multiplication method its name. </p><p>Faced with a slightly larger multiplication, such as 34 × 13, pupils may initially be encouraged to also break this into tens. So, expanding 34 as 10 + 10 + 10 + 4 and 13 as 10 + 3, the product 34 × 13 might be represented: </p> <table><tbody><tr><th scope="col">×</th><th scope="col">10</th><th scope="col">10</th><th scope="col">10</th><th scope="col">4</th></tr><tr><th scope="row">10</th><td>100</td><td>100</td><td>100</td><td>40</td></tr><tr><th scope="row">3</th><td>30</td><td>30</td><td>30</td><td>12</td></tr></tbody></table> <p>Totalling the contents of each row, it is apparent that the final result of the calculation is (100 + 100 + 100 + 40) + (30 + 30 + 30 + 12) = 340 + 102 = 442. </p> <h3>Standard blocks</h3> <p>Once pupils have become comfortable with the idea of splitting the whole product into contributions from separate boxes, it is a natural step to group the tens together, so that the calculation 34 × 13 becomes </p> <table><tbody><tr><th scope="col">×</th><th scope="col">30</th><th scope="col">4</th></tr><tr><th scope="row">10</th><td>300</td><td>40</td></tr><tr><th scope="row">3</th><td>90</td><td>12</td></tr></tbody></table> <p>giving the addition </p> <table><tbody><tr><td> 300 40 90 + 12 ———— 442</td></tr></tbody></table> <p>so 34 × 13 = 442. </p><p>This is the most usual form for a grid calculation. In countries such as the UK where teaching of the grid method is usual, pupils may spend a considerable period of time regularly setting out calculations like the above, until the method is entirely comfortable and familiar. </p> <h3>Larger numbers</h3> <p>The grid method extends straightforwardly to calculations involving larger numbers. </p><p>For example, to calculate 345 × 28, the student could construct the grid with six easy multiplications </p> <table><tbody><tr><th scope="col">×</th><th scope="col">300</th><th scope="col">40</th><th scope="col">5</th></tr><tr><th scope="row">20</th><td>6000</td><td>800</td><td>100</td></tr><tr><th scope="row">8</th><td>2400</td><td>320</td><td>40</td></tr></tbody></table> <p>to find the answer 6900 + 2760 = 9660. </p><p>However, by this stage (at least in standard current UK teaching practice) pupils may be starting to be encouraged to set out such a calculation using the traditional long multiplication form without having to draw up a grid. </p><p>Traditional long multiplication can be related to a grid multiplication in which only one of the numbers is broken into tens and units parts to be multiplied separately: </p> <table><tbody><tr><th scope="col">×</th><th scope="col">345</th></tr><tr><th scope="row">20</th><td>6900</td></tr><tr><th scope="row">8</th><td>2760</td></tr></tbody></table> <p>The traditional method is ultimately faster and much more compact; but it requires two significantly more difficult multiplications which pupils may at first struggle with . Compared to the grid method, traditional long multiplication may also be more abstract and less manifestly clear , so some pupils find it harder to remember what is to be done at each stage and why . Pupils may therefore be encouraged for quite a period to use the simpler grid method alongside the more efficient traditional long multiplication method, as a check and a fall-back. </p> <h2 id="other-applications">Other applications</h2> <h3>Fractions</h3> <p>While not normally taught as a standard method for <a href="/facts/Fractions/ghnQE9B3">multiplying fractions</a>, the grid method can readily be applied to simple cases where it is easier to find a product by breaking it down. </p><p>For example, the calculation 21/2 × 11/2 can be set out using the grid method </p> <table><tbody><tr><th scope="col">×</th><th scope="col">2</th><th scope="col">1/2</th></tr><tr><th scope="row">1</th><td>2</td><td>1/2</td></tr><tr><th scope="row">1/2</th><td>1</td><td>1/4</td></tr></tbody></table> <p>to find that the resulting product is 2 + 1/2 + 1 + 1/4 = 33/4 </p> <h3>Algebra</h3> <p>The grid method can also be used to illustrate the multiplying out of a product of <a href="/facts/Binomial_(polynomial)/cfFcsR8h">binomials</a>, such as (<i>a</i> + 3)(<i>b</i> + 2), a standard topic in <a href="/facts/Elementary_algebra/973nufT7">elementary algebra</a> (although one not usually met until <a href="/facts/Secondary_school/GjEqd0fa">secondary school</a>): </p> <table><tbody><tr><th scope="col">×</th><th scope="col"><i>a</i></th><th scope="col">3</th></tr><tr><th scope="row"><i>b</i></th><td><i>ab</i></td><td>3<i>b</i></td></tr><tr><th scope="row">2</th><td>2<i>a</i></td><td>6</td></tr></tbody></table> <p>Thus (<i>a</i> + 3)(<i>b</i> + 2) = <i>ab</i> + 3<i>b</i> + 2<i>a</i> + 6. </p> <h3>Computing</h3> <p>32-bit CPUs usually lack an instruction to multiply two 64-bit integers. However, most CPUs support a "multiply with overflow" instruction, which takes two 32-bit operands, multiplies them, and puts the 32-bit result in one register and the overflow in another, resulting in a carry. For example, these include the umull instruction added in the <a href="/facts/ARM_architecture/SALsYA79">ARMv4t instruction set</a> or the pmuludq instruction added in <a href="/facts/SSE2/cheNDPSK">SSE2</a> which operates on the lower 32 bits of an <a href="/facts/SIMD/gQHWQSpo">SIMD</a> register containing two 64-bit lanes. </p><p>On platforms that support these instructions, a slightly modified version of the grid method is used. The differences are: </p> <ol><li>Instead of operating on multiples of 10, they are operated on 32-bit integers.</li> <li>Instead of higher bits being multiplied by ten, they are multiplied by 0x100000000. This is usually done by either shifting to the left by 32 or putting the value into a specific register that represents the higher 32 bits.</li> <li>Any values that lie above the 64th bit are truncated. This means that multiplying the highest bits is not required, because the result will be shifted out of the 64-bit range. This also means that only a 32-bit multiply is required for the higher multiples.</li></ol> <table><tbody><tr><th scope="col">×</th><th scope="col"><i>b</i></th><th scope="col"><i>a</i></th></tr><tr><th scope="row"><i>d</i></th><td>-</td><td><i>ad</i></td></tr><tr><th scope="row"><i>c</i></th><td><i>bc</i></td><td><i>ac</i></td></tr></tbody></table> <p>This would be the routine in C: </p> #include <stdint.h> uint64_t multiply(uint64_t ab, uint64_t cd) { /* These shifts and masks are usually implicit, as 64-bit integers * are often passed as 2 32-bit registers. */ uint32_t b = ab >> 32, a = ab & 0xFFFFFFFF; uint32_t d = cd >> 32, c = cd & 0xFFFFFFFF; /* multiply with overflow */ uint64_t ac = (uint64_t)a * (uint64_t)c; uint32_t high = ac >> 32; /* overflow */ uint32_t low = ac & 0xFFFFFFFF; /* 32-bit multiply and add to high bits */ high += (a * d); /* add ad */ high += (b * c); /* add bc */ /* multiply by 0x100000000 (via left shift) and add to the low bits with a binary or. */ return ((uint64_t)high << 32) | low; } <p>This would be the routine in ARM assembly: </p> multiply: @ a = r0 @ b = r1 @ c = r2 @ d = r3 push {r4, lr} @ backup r4 and lr to the stack umull r12, lr, r2, r0 @ multiply r2 and r0, store the result in r12 and the overflow in lr mla r4, r2, r1, lr @ multiply r2 and r1, add lr, and store in r4 mla r1, r3, r0, r4 @ multiply r3 and r0, add r4, and store in r1 @ The value is shifted left implicitly because the @ high bits of a 64-bit integer are returned in r1. mov r0, r12 @ Set the low bits of the return value to r12 (ac) pop {r4, lr} @ restore r4 and lr from the stack bx lr @ return the low and high bits in r0 and r1 respectively <h2 id="mathematics">Mathematics</h2> <p>Mathematically, the ability to break up a multiplication in this way is known as the <a href="/facts/Distributive_law/0LNBrVJl">distributive law</a>, which can be expressed in algebra as the property that <i>a</i>(<i>b</i>+<i>c</i>) = <i>ab</i> + <i>ac</i>. The grid method uses the distributive property twice to expand the product, once for the horizontal factor, and once for the vertical factor. </p><p>Historically the grid calculation (tweaked slightly) was the basis of a method called <a href="/facts/Lattice_multiplication/14qW5Na2">lattice multiplication</a>, which was the standard method of multiple-digit multiplication developed in medieval Arabic and Hindu mathematics. Lattice multiplication was introduced into Europe by <a href="/facts/Fibonacci/d5rA6p8c">Fibonacci</a> at the start of the thirteenth century along with Arabic numerals themselves; although, like the numerals also, the ways he suggested to calculate with them were initially slow to catch on. <a href="/facts/Napier%2527s_bones/uc1DHGLA">Napier's bones</a> were a calculating help introduced by the Scot <a href="/facts/John_Napier/Wg0HAmq5">John Napier</a> in 1617 to assist lattice-method calculations. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Multiplication_algorithm/68xf6EdK">Multiplication algorithm</a></li> <li><a href="/facts/Multiplication_Table/LG1He6qH">Multiplication Table</a></li></ul> <ul><li><a href="/facts/Rob_Eastaway/FuMB3tPb">Rob Eastaway</a> and Mike Askew, <i>Maths for Mums and Dads</i>, Square Peg, 2010. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-224-08635-6. pp. 140–153.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.mathsonline.org/pages/boxmult.html?56&551">Long multiplication − The Box method</a>, <i>Maths online</i>.</li> <li><a href="https://www.bbc.co.uk/schools/gcsebitesize/maths/number/multiplicationdivisionrev1.shtml">Long multiplication and division</a>, BBC GCSE Bitesize</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>https://tspiteri.gitlab.io/gmp-mpfr-sys/gmp/Algorithms.html#Multiplication-Algorithms [dead link] <a href="https://tspiteri.gitlab.io/gmp-mpfr-sys/gmp/Algorithms.html#Multiplication-Algorithms" target="_blank">https://tspiteri.gitlab.io/gmp-mpfr-sys/gmp/Algorithms.html#Multiplication-Algorithms</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Long multiplication − The Box method <a href="http://www.mathsonline.org/pages/boxmult.html?56&551" target="_blank">http://www.mathsonline.org/pages/boxmult.html?56&551</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Long multiplication and division <a href="https://www.bbc.co.uk/schools/gcsebitesize/maths/number/multiplicationdivisionrev1.shtml" target="_blank">https://www.bbc.co.uk/schools/gcsebitesize/maths/number/multiplicationdivisionrev1.shtml</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>