Editing Multiplication algorithm (section)

===Russian peasant multiplication===
{{Main|Peasant multiplication}}
The binary method is also known as peasant multiplication, because it has been widely used by people who are classified as peasants and thus have not memorized the [[multiplication table]]s required for long multiplication.<ref>{{Cite web|url=https://www.cut-the-knot.org/Curriculum/Algebra/PeasantMultiplication.shtml|title=Peasant Multiplication|author-link=Alexander Bogomolny|last=Bogomolny|first= Alexander |website=www.cut-the-knot.org|access-date=2017-11-04}}</ref>{{failed verification|date=March 2020}} The algorithm was in use in ancient Egypt.<ref>{{Cite book |first=D. |last=Wells | author-link=David G. Wells | year=1987 |page=44 |title=The Penguin Dictionary of Curious and Interesting Numbers |publisher=Penguin Books |isbn=978-0-14-008029-2}}</ref> Its main advantages are that it can be taught quickly, requires no memorization, and can be performed using tokens, such as [[poker chips]], if paper and pencil aren't available. The disadvantage is that it takes more steps than long multiplication, so it can be unwieldy for large numbers.

====Description====
On paper, write down in one column the numbers you get when you repeatedly halve the multiplier, ignoring the remainder; in a column beside it repeatedly double the multiplicand. Cross out each row in which the last digit of the first number is even, and add the remaining numbers in the second column to obtain the product.

====Examples====
This example uses peasant multiplication to multiply 11 by 3 to arrive at a result of 33.

 Decimal:     Binary:
 11   3       1011  11
 5    6       101  110
 2   <s>12</s>       10  <s>1100</s>
 1   24       1  11000
     ——         ——————
     33         100001

Describing the steps explicitly:

* 11 and 3 are written at the top
* 11 is halved (5.5) and 3 is doubled (6).  The fractional portion is discarded (5.5 becomes 5).
* 5 is halved (2.5) and 6 is doubled (12).  The fractional portion is discarded (2.5 becomes 2).  The figure in the left column (2) is '''even''', so the figure in the right column (12) is discarded.
* 2 is halved (1) and 12 is doubled (24).
* All not-scratched-out values are summed: 3 + 6 + 24 = 33.

The method works because multiplication is [[distributivity|distributive]], so:

: <math>
\begin{align}
3 \times 11 & = 3 \times (1\times 2^0 + 1\times 2^1 + 0\times 2^2 + 1\times 2^3) \\
& = 3 \times (1 + 2 + 8) \\
& = 3 + 6 + 24 \\
& = 33.
\end{align}
</math>

A more complicated example, using the figures from the earlier examples (23,958,233 and 5,830):

 Decimal:             Binary:
 5830  <s>23958233</s>       1011011000110  <s>1011011011001001011011001</s>
 2915  47916466       101101100011  10110110110010010110110010
 1457  95832932       10110110001  101101101100100101101100100
 728  <s>191665864</s>       1011011000  <s>1011011011001001011011001000</s>
 364  <s>383331728</s>       101101100  <s>10110110110010010110110010000</s>
 182  <s>766663456</s>       10110110  <s>101101101100100101101100100000</s>
 91  1533326912       1011011  1011011011001001011011001000000
 45  3066653824       101101  10110110110010010110110010000000
 22  <s>6133307648</s>       10110  <s>101101101100100101101100100000000</s>
 11 12266615296       1011  1011011011001001011011001000000000
 5  24533230592       101  10110110110010010110110010000000000
 2  <s>49066461184</s>       10  <s>101101101100100101101100100000000000</s>
 1  98132922368       1  <u>1011011011001001011011001000000000000</u>
   ————————————          1022143253354344244353353243222210110 (before carry)
   139676498390         10000010000101010111100011100111010110