Multiplication Methods and Casting out Nines

Per Repiego

Also known as “by parts”, is a method of division. Break the divisor into factors and then proceed to divide the dividend sequentially with the factors[1]. The first factor will begin to divide and then divide the sum with the second factor and so on. For example, 7650/90 would have 90 broken into three factors: 5, 2, 9 (or 3, 3). Divide 7650 by 5 and the answer would be 1530; divide 1530 by 2 and the answer is 765; divide 765 by 9 and the answer is 85 which is the final answer. 

The history behind this method seems obscure. In the West, the method appears to be accredited to the Italians who made the method mainstream[2]. The pros behind this method are that it simplifies long division and could reduce the amount of papyrus used by mathematicians back in the day. Another pro is that all numbers are able to be broken down into small enough factors for long division. The cons however are that it makes long division take a long time and space to write down all the factors used for the division, but the process is easy enough to not make a mistake. The major conceptual idea behind this method could be that large numbers are a sum of its parts. The factors of a large number can be multiplied into that large number. Another concept here is that numbers do not have to be large in order to divide against another much larger number to get an answer.

Prosthaphaeresis

            This method is used to multiply, divide, add and subtract trigonometric identities. Adding trig identities like cos(a) cos(b) would be added with the formula cos (a + b) + cos (a – b) / 2; similarly, subtracting sin(a) sin(b) would use the formula cos (a – b) – cos (a + b) / 2. Multiplication requires one to find angles a and b which looks like x = cos(a) and y = cos(b). Compute cos (a + b) and cos (a – b) and divide the product by two. A table of cosines is required to translate the answer into angles. Add or subtract what is left.

Logarithms are dated back to Babylonia, but the Greeks like Archimedes used logarithms much like our own[3]. The best example of its usage may used in star-measurement or astronomy. The pros here are that this method makes it possible to add, subtract, multiply, and divide trig identities. The cons are that it requires a table of trig identities and can only be applied with trig identities. The conceptual idea behind this may be that even trig identities can be added, multiplied, subtracted, and divided.

Doubling and Halving

Doubling and Halving is a method used for making multiplication problems easier. You double one factor and halve another. So, 5 x 68 would become 10 x 34. Russian peasants, Ancient Egyptians, and Ethiopians are said to use or have used this method[4]. Pros of this method are that it makes multiplication easier and do-able without pen and paper. This can be used for matrices as well. Cons are that this method does not always prove useful and requires pen and paper for large factors.

Comparisons

            Per Repiego cannot quite be compared to Prosthaphaeresis because one deals with logarithms and trig identities. Per Repiego and the doubling and halving method both attempt to make larger digits into smaller ones, thus making the problem more manageable. However, Per Repiego makes the digits of the dividend smaller but breaking it into factors. Doubling and halving does not do this. Instead, one number being multiplied is halved and that may result in a factor, but it is not guaranteed. Doubling and halving is also not really comparable to Prosthapheresis other than the two break problems into smaller pieces like Per Repiego does as well.  

Part II

Casting out nines[5] is a method that can be used to check an answer. It relates with basic arithmetic as the method only requires the user to be able to add and multiply. When given an addition problem like 8,413 + 752 + 5,362 = 14,527, casting out nines requires you to cross out any 9s or digits that up to 9 (e.g., 7 and 2). 8,413 has its 8 and 1 crossed out resulting in a leftover 4 + 3 = 7. Next, 752 can cast out one 9, leaving 5 alone. 14,527 has two 9s to be crossed out (5 + 4 and 7 + 2), leaving 1 behind. The next step is to add what is left of all the addends (7, 5, 7) and to try to cast out the nines; 7 + 5 + 7 = 19, cast nines out 1 + 9 = 10, cast nines again 1 + 0 = 1. Now we check the sum and see that what was left behind—1. What was left of the sum is exactly what the addends resulted in after casting out the nines which means the problem was done correctly.

            The concept behind the casting out nines is very similar to how modular arithmetic works. For instance, 8,413 + 752 + 5,362 (mod 9) results in the same way casting out nines did in the previous example except for how the leftovers are used. After taking out the digits from each addend that sums to 9, we are left with 7 + 5 + 7 (mod 9) or 19 (mod 9). Here we can remove the 9 from 19 and we are left with 1 (mod 9). The sum of 8,413 + 752 + 5,362 can also be used here. 14,527 (mod 9) leaves us with 1 (mod 9) just like the addends before. Further, if one concept is similar to modular arithmetic, then they are also similar as to how check digits work as well except we would have to cast out 10s, 11s, or 13s ((mod 10) (mod 11) or (mod 13) respectively). A few other rules apply depending on the check digit algorithm (ISBN 10, UPC, ISBN 13 all have different check digits). Like modular arithmetic, casting out nines has limitations. The method is most useful for just checking your adding, subtracting, dividing, or multiplying result and even then, it is possible that you make an error in the basic arithmetic and the casting out nines tells you did it right. So, other than checking some answers, casting out nines is not too useful.

Comparing the casting of nines to the doubling and halving method above works so long as the latter works. Halving prime digits does not work. However, for the instances that you can double and half in a multiplication problem, you can also cast out nines to check the answer. For example, 5 x 24 after doubling and halving is 10 x 12 = 120. After casting out nines, we are left with (1 + 0) x (1 + 2) = (1 + 2) or 1 x 3 = 3. Both sides are equal, so casting out nines worked, and our problem is correct.

             

Resources

1)      Bellew, Pat. "Origins of Some Arithmetic Terms." Origins of Some Arithmetic Terms. Accessed September 23, 2018. http://www.pballew.net/arithme1.html#divide.

2)      Jackson, Lambert Lincoln. "The Educational Significance of Sixteenth Century Arithmetic from the Point of View of the Present Time." Google Books. Accessed September 23, 2018. https://books.google.com/books?id=ZSBRAAAAYAAJ&pg=PA62&lpg=PA62&dq=per repiego:.edu&source=bl&ots=AwNeJF066M&sig=0VLWlfK7mFLtC3dro6EGFxqEC-A&hl=en&sa=X&ved=2ahUKEwi5wISS_8_dAhUp4IMKHXq3DrAQ6AEwAnoECAgQAQ#v=onepage&q=per repiego:.edu&f=false.

3)      Pierce, R. C. "A Brief History of Logarithms." The Two-Year College Mathematics Journal 8, no. 1 (1977): 22-26. doi:10.2307/3026878.

4)      Wright, Colin. "Russian Peasant Multiplication." RSS. Accessed September 23, 2018. http://www.solipsys.co.uk/new/RussianPeasantMultiplication.html.

5)      Meredith, David. "CHECK YOUR MATH: CASTING OUT NINES." October 25, 2010. Accessed October 7, 2018. http://online.sfsu.edu/meredith/165/Casting.

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[1] Bellew, Pat. "Origins of Some Arithmetic Terms." Origins of Some Arithmetic Terms. Accessed September 23, 2018. http://www.pballew.net/arithme1.html#divide.

[2] Jackson, Lambert Lincoln. "The Educational Significance of Sixteenth Century Arithmetic from the Point of View of the Present Time." Google Books. Accessed September 23, 2018. https://books.google.com/books?id=ZSBRAAAAYAAJ&pg=PA62&lpg=PA62&dq=per repiego:.edu&source=bl&ots=AwNeJF066M&sig=0VLWlfK7mFLtC3dro6EGFxqEC-A&hl=en&sa=X&ved=2ahUKEwi5wISS_8_dAhUp4IMKHXq3DrAQ6AEwAnoECAgQAQ#v=onepage&q=per repiego:.edu&f=false.

[3] Pierce, R. C. "A Brief History of Logarithms." The Two-Year College Mathematics Journal 8, no. 1 (1977): 22-26. doi:10.2307/3026878.

[4] Wright, Colin. "Russian Peasant Multiplication." RSS. Accessed September 23, 2018. http://www.solipsys.co.uk/new/RussianPeasantMultiplication.html.

[5] Meredith, David. "CHECK YOUR MATH: CASTING OUT NINES." October 25, 2010. Accessed October 7, 2018. http://online.sfsu.edu/meredith/165/Casting.

Snippet on Globalism

The economic, political, and social interconnectedness and interdependence among societies in the world we are faced with is not new, but its effects are still being understood. Global trade and intercommunication between countries far apart from one another is now seen as normal. More recently, the discussion revolving around the offshoring of jobs has been brought towards the forefront of American politics. However, due to how globalization has made it easier and more rewarding for companies to offshore, outsource, and insource jobs, the offshoring, outsourcing, and insourcing of jobs is likely to stick around unless viable alternatives are made. Further, in the global scale of things, we find surrogacy work is also a job that can be offshored as a cheaper alternative to domestic surrogacy work. So too has the human reproductive process become a business.

From a business perspective, the world can be seen as a pool of resources. The people in various countries are a sort of global workforce or a potential workforce. By offshoring jobs, companies can recruit from this workforce pool or pool of human capital in a way that is more affordable than recruiting in, say, America. As Skipper (2007) states, technically any job that impersonally delivers can be offshored. Certain service and IT jobs, for example, can easily be offshored because the product they deliver can be impersonally delivered unlike the work of a therapist which must be personally delivered. Due to America’s immigration policy as well as labor laws, the jobs that are offshored are done so because recruiting people who are more likely to work offshored jobs for lower pay in America is difficult. Recruiting illegal immigrants to work below minimum wage is also illegal. Workers that would likely work for less are also difficult to recruit from the pool of immigrants coming into America.

The preference system in application process for immigration deters low-skilled workers from entering America legally. First preference visas are given to highly skilled/educated individuals who are likely professors and researchers (O’Sullivan, 2012). Those that qualify for the first preference are able to be granted visas without much delay. Second or third preference visas require labor certification. This certification includes a qualifier that prevents or deters individuals applying for a visa hoping to work a job that can be filled by “eligible and qualified American workers” (O’Sullivan, 2012). However, to apply for the labor certification, one must also fulfill the Program Electronic Review Management which has other qualifiers that deter or prevent the typical low-skilled laborer from meeting the requirements. To be granted a visa under the third preference can take up to six years as well. These among other reasons are why it is difficult to recruit from immigrant labor under the second preference visas. This could partially explain as to why offshoring occurs: low-skilled workers can be found elsewhere.

One example of low-skilled work that can be offshored are surrogate mothers. People from India and across the others are able to rent the womb of Indian women to provide them a child. The cost of surrogacy is about half of what it is in India when compared to America (Rudrappa, 2015). This price is likely the reason why people look towards India to find surrogate mothers, effectively offshoring the work of what some women in America are willing to do. The surrogate mothers get paid about the same amount of money they would earn by working three years in a garment factory but with better living conditions. The work in the garment factory is quite often coupled with health issues and harassment from supervisors. Meanwhile, surrogate mothers are given a dorm, supervision, and other necessities in order to make sure the birthing process goes well. This is what makes being a surrogate mother seem appealing as opposed to working in a garment factory. Nonetheless, the work associated with being a surrogate mother is not without its downsides. C-sections, birthing complications, and other side-effects are made possible by being a surrogate mother. Some of the surrogate mothers are not able to separate their emotional investment in the children they eventually birth, and this causes emotional pain.

In a globalist world, even the process of birthing can be offshored because the task of giving birth is not necessarily one that must be personally delivered. America’s restrictive immigration only makes offshoring look more appealing to companies able to offshore. However, the workers in America lose jobs because of offshoring and the laborers that benefit from the offshoring also are negatively affected by the poor working conditions often associated with labor that can outsourced. Even in the case of being a surrogate mother for an American couple, there are benefits and negatives associated with taking such a task.