Russian Peasant Multiplication

Paul Hunt has let us know about this method of multiplication created by the Russians (Paul Hunt's Hint). 


What a little wonder!


These Russians are realistically creative.


From the Math Forum @ Drexel (Math Forum, Drexel), we read:

What is Russian peasant multiplication? How do I use it?

The way most people learn to multiply large numbers looks something like this:

    86
  x 57
 ------
   602
+ 4300
 ------
  4902

If you know your multiplication facts, this "long multiplication" is quick and relatively simple. However, there are many other ways to multiply. One of these methods is often called the Russian peasant algorithm. You don't need multiplication facts to use the Russian peasant algorithm; you only need to double numbers, cut them in half, and add them up. Here are the rules:

• Write each number at the head of a column.
• Double the number in the first column, and halve the number in the second column.
     If the number in the second column is odd, divide it by two and drop the remainder.
• If the number in the second column is even, cross out that entire row.
• Keep doubling, halving, and crossing out until the number in the second column is 1.
• Add up the remaining numbers in the first column. The total is the product of your original numbers.

Let's multiply 57 by 86 as an example:

Write each number at the head of a column.


57

86

Double the number in the first column, and halve the number in the second column.


57	86
114	43

If the number in the second column is even, cross out that entire row.


57	86
114	43

Keep doubling, halving, and crossing out until the number in the second column is 1.


57	86
114	43
228	21
456	10
912	5
1824	2
3648	1

Add up the remaining numbers in the first column.


57	86
114	43
228	21
456	10
912	5
1824	2
+ 3648	1
4902

Real Russian peasants may have tracked their doublings with bowls of pebbles, instead of columns of numbers. (They probably weren't interested in problems as large as our example, though; four thousand pebbles would be hard to work with!) Russian peasants weren't the only ones to use this method of multiplication. The ancient Egyptians invented a similar process thousands of years earlier, and computers are still using related methods today.

Our explanation for such a phenomenon is:

Suppose we are multiplying constant A by constant B.

We then have A x B.

Now suppose the result is C, another constant, so that we now have  A x B = C.

We know that if we multiply A by a constant, say t, and we divide B by the same constant, the result remains unaltered. 

Let's prove that:

At x 1/t B = (t x 1/t) x A x B  = A x B.

If we keep on applying their recipe, we then have always the same result, C, that we had at the beginning of the proposal, when doing A x B.

Notice that if we have even numbers that factor nicely, we simply pick the result to the left and that is already our C:

4    16 

8     8

16   4

32   2

64   1

4 x 16 = 64.

Notice that we can always double the first number with no difficulties but we frequently will have difficulties in halving the second number, since lots of times it will be an odd number.

Notice that any number times two will give us an even number, so that we will always have even numbers to the left, apart from at most the first, at the very top of the column.

If the second number factors nicely we get only even numbers on the second column. In this case, we should be getting the C from the first column, last line in the process.

The process resumes to the following:

a) Even/odd number, even number that perfectly divides by two all the way: We do not strike any and get C from the last line, left.

b) Even/odd number, even number that does not perfectly divide by two all the way/odd number: Strike the even number row, that is the row where there is an even number to the right in the process and add up what is not struck.

The process becomes a bit more complex when we have an odd number or an even number that does not perfectly divide by two all the way to the right side then.

That is obviously because we are halving something that does not split nicely into two (there is a remainder).


In mathematical writing, we could translate what we do to the numbers involved in the Peasant Multiplication in the following way:

Suppose we have  X * Y as our original proposal.

We then do  ((X*2)*2)*2  to the left side by the fourth line of our calculation. Thus, X₃ = ((X*2)*2)*2  and X₀ = X.

(We do |_ |_(|_(Y/2)_| )/2_| )/2_|  to the right side by the fourth line of our calculation. Thus, Y₃ = |_ |_(|_(Y/2)_| )/2_| )/2_|  and Y₀ = Y.

|_(Y/2)_| = (Y-1)/2 if Y = 2n + 1

and

|_(Y/2)_| = Y/2 if Y = 2n

If we do left number times right number, we get  2X_n-1 (Y_n-1-1)/2, that is, X_n-1 (Y_n-1-1), that is, X_n-1 Y_n-1- X_n-1 whenever the right number is not an even number.

This just means that we have to add X_n-1 to recover our original calculation (notice that, when we translate the index, we divide and multiply by two an equal number of times).

When our right result is 1, all we have to do is then adding up all Xs, which are then going to appear only on the lines where we do not have an even number to the right.

This is a really interesting system, right?

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On the Cardinality of Sets

On (Good Math, 2014):

How about this? Take the set of all natural numbers. Divide it into two sets: the set of even naturals, and the set of odd naturals. Now you have two infinite sets,
the set {0, 2, 4, 6, 8, ...}, and the set {1, 3, 5, 7, 9, ...}. The size of both of those sets is the ω - which is also the size of the original set you started with.

Size for mathematicians, in terms of numerical sets, is the cardinality of those sets, which is measured from a human perspective: If we cannot count by hand or by any instrument that we have invented, in its totality, then it is all infinite, therefore of the same size… . 

We have written about that before (Pinheiro, 2009): They are mixing up things.

Basically, when we enter the World of Mathematics, we have to use a special language developed to deal with it, so that our brain adapts. 

This special language cannot be formed from terms that point to more than one world object, since we need maximum objectivity in Mathematics: All is supposed to be abstract and perfect.

In this way, we cannot simply say that the cardinality is infinity and we then have the same size of set when we have even natural numbers or the whole of the natural numbers.

The cardinality is infinity in what regards our capability of counting, of seeing, etc.

Both situations create the same effect on our eyes and in our heads but they are situations that should be very different if the perspective of the World of Mathematics is considered.

How do we deal with the straight line world?

On (Good Math, 2013):

Take a line segment. How many points are in it? It's infinite. So, from that infinite set, remove an infinite set of points. How many points are left? It's still infinite.

Apparently, in the same way that we deal with the sets of numbers!

There is obviously only one solution to this problem: Segment all.

Lebesgue, with his infinitely perspicacious mind, had already noticed that somehow. See (Rochford, 2013):

The idea behind the Lebesgue measure is that the size of the interval (a,b) ought to be equal to its length, b−a. The construction of the Lebesgue measure generalizes this notion to a much larger class of subsets of the real numbers.

Let m denote the Lebesgue measure. From the previous discussion, we define m((a,b))=b−a. In order to extend the definition of the Lebesgue measure to a slightly larger class of sets, let us first consider the size of the set (0,1)∪(3,5). It seems quite reasonable to define

m((0,1)∪(3,5))=(1−0)+(5−3)=1+2=3.

We see that this idea readily generalizes to finite unions of pairwise disjoint intervals (recall that two sets are disjoint if their intersection is empty, that is, if they do not overlap). In that case, we define the Lebesgue measure of the union to be the sum of the Lebesgue measures of each individual interval. 

If we segment the naturals, then we get that the set of the natural numbers is bigger than the set of even/odd numbers and this idea is not only intuitive but is obviously also true.

Take the naturals from 1 to 5.

When we consider them as a whole, we have {1, 2, 3, 4, 5} in this 'interval'.

When we consider just the even ones, we have {2, 4} instead.

When we compare numerical sets, we must compare them by parts and comparing one part is equivalent to comparing the whole if they are regular things.

This way, 2N is smaller than N because when we consider the accumulation criterion {x belongs to N/1 ≤ x ≤ 5} we get fewer elements in 2N and that is a reasonable sample (of size that is acceptable).

There seems to realistically be no mystery in any of it and we are simply uttering wrong things when we say that the cardinality of the reals is the same as the cardinality of the rational numbers.

The cardinality of the reals is greater than the cardinality of the rational numbers, quite trivially, since we add numbers to each small interval.

Marcia’s Measure would then be the slices of the sets under analysis that correspond to an interval of Lebesgue measure five and would be suitable to determine how big an ordered numerical set is when compared to another, provided that the elements of each one of the sets under analysis be equally distributed along the real line.

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DO YOU WANT TO PLAY LOVE GAMES? GET THE BOOK HERE!

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References:

Rochford, A. (2013). There are Almost no Rational Numbers. Retrieved January 1^st 2014 from http://www.austinrochford.com/posts/2013-12-31-almost-no-rationals.html?utm_content=bufferc9d48&utm_source=buffer&utm_medium=twitter&utm_campaign=Buffer#! 

Good Math, Bad Math. (2014). The Banach-Tarski non-Paradox. Retrieved January 1^st 2014 from http://scientopia.org/blogs/goodmath/2012/01/06/the-banach-tarski-non-paradox/

 

Pinheiro. (2009). Infinis. Retrieved January 1^st 2014 from http://www.protosociology.de/Download/Pinheiro-Infinis.pdf