Editing Multiplication algorithm (section)

===Lattice multiplication===
{{main|Lattice multiplication}}
[[File:Hindu lattice.svg|thumb|right|First, set up the grid by marking its rows and columns with the numbers to be multiplied. Then, fill in the boxes with tens digits in the top triangles and units digits on the bottom.]]
[[File:Hindu lattice 2.svg|thumb|right|Finally, sum along the diagonal tracts and carry as needed to get the answer]]

Lattice, or sieve, multiplication is algorithmically equivalent to long multiplication. It requires the preparation of a lattice (a grid drawn on paper) which guides the calculation and separates all the multiplications from the [[addition]]s. It was introduced to Europe in 1202 in [[Fibonacci]]'s [[Liber Abaci]]. Fibonacci described the operation as mental, using his right and left hands to carry the intermediate calculations. [[Matrakçı Nasuh]] presented 6 different variants of this method in this 16th-century book, Umdet-ul Hisab. It was widely used in [[Enderun]] schools across the Ottoman Empire.<ref>{{cite journal |last1=Corlu |first1=M.S. |last2=Burlbaw |first2=L.M. |last3=Capraro |first3=R.M. |last4=Corlu |first4=M.A. |last5=Han |first5=S. |title=The Ottoman Palace School Enderun and the Man with Multiple Talents, Matrakçı Nasuh |journal=Journal of the Korea Society of Mathematical Education Series D: Research in Mathematical Education |volume=14 |issue=1 |pages=19–31 |date=2010 |doi= |url=https://koreascience.kr/article/JAKO201017337333137.page}}</ref> [[Napier's bones]], or [[Napier's rods]] also used this method, as published by Napier in 1617, the year of his death.

As shown in the example, the multiplicand and multiplier are written above and to the right of a lattice, or a sieve. It is found in [[Muhammad ibn Musa al-Khwarizmi]]'s "Arithmetic", one of Leonardo's sources mentioned by Sigler, author of "Fibonacci's Liber Abaci", 2002.{{citation needed|date=January 2016}}

*During the multiplication phase, the lattice is filled in with two-digit products of the corresponding digits labeling each row and column: the tens digit goes in the top-left corner.
*During the addition phase, the lattice is summed on the diagonals.
* Finally, if a carry phase is necessary, the answer as shown along the left and bottom sides of the lattice is converted to normal form by carrying ten's digits as in long addition or multiplication.

====Example====
The pictures on the right show how to calculate 345 × 12 using lattice multiplication. As a more complicated example, consider the picture below displaying the computation of 23,958,233 multiplied by 5,830 (multiplier); the result is 139,676,498,390.  Notice 23,958,233 is along the top of the lattice and 5,830 is along the right side.  The products fill the lattice and the sum of those products (on the diagonal) are along the left and bottom sides.  Then those sums are totaled as shown.
{|
| rowspan="2" |
<pre>
     2   3   9   5   8   2   3   3
   +---+---+---+---+---+---+---+---+-
   |1 /|1 /|4 /|2 /|4 /|1 /|1 /|1 /|
   | / | / | / | / | / | / | / | / | 5
 01|/ 0|/ 5|/ 5|/ 5|/ 0|/ 0|/ 5|/ 5|
   +---+---+---+---+---+---+---+---+-
   |1 /|2 /|7 /|4 /|6 /|1 /|2 /|2 /|
   | / | / | / | / | / | / | / | / | 8
 02|/ 6|/ 4|/ 2|/ 0|/ 4|/ 6|/ 4|/ 4|
   +---+---+---+---+---+---+---+---+-
   |0 /|0 /|2 /|1 /|2 /|0 /|0 /|0 /|
   | / | / | / | / | / | / | / | / | 3
 17|/ 6|/ 9|/ 7|/ 5|/ 4|/ 6|/ 9|/ 9|
   +---+---+---+---+---+---+---+---+-
   |0 /|0 /|0 /|0 /|0 /|0 /|0 /|0 /|
   | / | / | / | / | / | / | / | / | 0
 24|/ 0|/ 0|/ 0|/ 0|/ 0|/ 0|/ 0|/ 0|
   +---+---+---+---+---+---+---+---+-
     26  15  13  18  17  13  09  00</pre>
||
<pre>
 01
 002
 0017
 00024
 000026
 0000015
 00000013
 000000018
 0000000017
 00000000013
 000000000009
 0000000000000
 —————————————
  139676498390
</pre>
|-
||
 = 139,676,498,390
|}