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.
[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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