Download Group C essays

Howtofindyoursoulmate?
Oneday,PlatoaskedhisteacherSocrates,“Whatislove?HowcanIfindit?”Socratesanswered,“Thereisa
vastwheatfield.Walkforwardwithoutturningback,andpickonlyonestalk”Platowalkedforward,andbefore
long,hereturnedwithemptyhands.Hesaid“Ididfindthemostmagnificentstalk,butdidn’tknowiftherewere
anybetteronesahead,soIdidnotpickit.AsIwalkedfurther,thestalksthatIsawwerenotasgoodastheearlier
one,soIdidn’tpickanyintheend.”Socratesthensaid,“ThatisLOVE.”
Similarly,whiledatingwedon’tknowifthereisabetterpersoninthefuture,norcanwedateeverysingle
personandcomebackforthebestone.Thushowtofindtherightpersonbecomesaproblem.Whynotsolveitin
amathematicalway?Firstletusmakesomerules:SaythereareNtotalcandidates.Assumewecanonlydateone
personatatimeforanytimeperiod.Attheendoftherelationship,youmustdecidewhetherornotyouwantto
bewithhim/her.Ifyougiveupontheperson,youmustmoveontothenextcandidateimmediatelywithoutever
th
comingback.Sohereisourstrategy:datekpeoplewithoutcommitment;startingfromthe(k+1) person,keep
datingandcommittotheonewhoisbetterthananyofthefirstkpeople.Weaimtofindthekwhichprovidesus
withthegreatestpossibilityoffindingthebestspousewhoappearsatpositionnfor1≤n≤N.
Sincethebestpersonmightbeinanyposition,thechancethatthebestpersonisatpositionnis1/N.We
alreadydecidedthatwewillnotchooseanyonefromthefirstkpeople,thustheprobabilityofanyoneamongthe
th
firstkpeopletobechosenis0.Startingfromthe(k+1) candidate,wemightstartconsideringtoaccept.Theonly
th
possiblesituationforthen candidatetobechosenisthatallthepreviousoptionsareworsethann.Inparticular,
thebestoptioninthe(n-1)previouscandidatesmusthaveoccurredinthefirstkcandidates(whenwewerenot
committingaccordingtothestrategy).Sothereare(n-1)positionsthatthebestpersoninthefirst(n-1)candidates
th
canbe,andtherearek"goodchoices",i.e.situationswhenthen candidateischosen.Thustheprobabilityfor
th
th
then persontobechosenisk/(n-1).Similarly,theprobabilityofthe(k+2) candidatebeingchosenisk/(k+1),
theprobabilityofk+3 tobechosenisk/(k+2)andsoonsoforth.Theprobabilityofthebestspousebeingatnis
1/Nasalways.Thenwecancalculatethetotalprobability.
!
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P(k)=( )*0+…+( )*0+( )*( )+ ( )*( )+ ( )*( )+…+ ( )*( )= ( )*( +
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+⋯+
), for
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k=1,2,…,N-1
FromGraph1.1,wefindthattheprobabilityincreasestoamaximumvalueandthenkeepsdecreasing.We
wanttofindkforwhichP(k)ishighest,i.e.wewanttofindthemaximumvalue.However,P(k)isnotafunction
fromℝtoℝ,sowecannottakeaderivative.Luckily,itturnsoutthatP(k)canbeapproximatedbythefunction
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!
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,-( ).Letxbek/N,wegetP(x)=xln( )=-xln(x),thusthederivativeP’(x)=-x( )-ln(x)=-1-ln(x).SettingP’(x)=0,we
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0
0
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solvetheequationandgetx= ≈0.368.Thismeanswhenkincreasestoapproximately0.37N,wereachthe
1
maximumpossibilitythatwefindour“best”spouse.Thisimpliesthatmaybeweshouldobservethefirst37%ofall
datingcandidatesandafterthatchoosetheonewhoisbetterthaneveryonewe’vedatedbefore.Apractical
advicecouldbethatdon’tbetooupsetifyoucannotendup
withyourpuppylove.Theloveandmemoryyouownareall
real,andthatiswhatmakesyouyou.Eventuallyyouwillfind
yourtruelove.However,beinghypercriticalmightleadyouto
theotherextreme.Althoughtheapproximationisprettyprecise
whenN≥15,wemightnothavesomanychoicesinourlife.
Thuswhenyou’vegainedenoughexperienceandarefully
Graph1.1PlotofP(k)againstkforN=15
prepared,don’tbehesitantandchoosethesuitableperson.
Thereisnowayforustoknowthetotalnumberofpeoplewearegoingtomeetinourlife,soeverything
aboveisjustforreference.“Atthebeginningofeveryproof,youusuallywrite,assumethereexistsx.Maybexis
fate.”HopeeveryonecanfindyourMr./Mrs.Right!
References:
Teague,Dan.HowtoFindaSpouse:AProbleminDiscreteMathematicsWithanAssistFromCalculus.n.d.
<http://courses.ncssm.edu/math/Talks/PDFS/Spouse.pdf>.
Vernon,Mark.Plato'sPodcasts:Theancients'guidetomodernliving.OneworldPublications,2013.
LessThanOne.ByArturoPerezJr.&SamanthaJayne.Dir.ArturoPerezJr.Prod.ZakFaust.2014.<http://www.lessthanone.com/>.
Recursion: An Application of Induction in Computer Programming
As one of my favorite professor, Harry Mairson, once said, “Whenever you are solving a problem
about programming, 90% of the time you are using the very neuron of recursion.” Nowadays, every
programmer needs to master the skill of recursion because it is one of the most frequently used and highly
effective programming methods. But, what is recursion? The general explanation of recursion, as the
beginning sentence in Wikipedia, is “the process of repeating items in a self-similar way”. Specifically in
computer programming, recursion refers to the action of a function calling another instance of itself. We
can prove that every recursive function has its desired functionality by an induction hypothesis, hence we
say that recursion is an application of induction.
One typical example of recursive function is the function to determine the n-th Fibonacci number,
denoted as FIB(n), which is defined as:
FUNCTION FIB(n):
IF n = 0 THEN return 0
IF n = 1 THEN return 1
-- base cases
ELSE return (FIB(n-1) + FIB(n-2))
-- inductive case
We know this function works properly because, by the definition of Fibonacci number, F0 = 0 and F1 = 1,
and Fn = Fn-1 + Fn-2 for all n > 1. When we calculate a specific Fibonacci number, say F10, we pass the
input 10 to the function. Since 10 ≠ 0 and 10 ≠ 1, FIB(10) will call another two instances of this function,
FIB(9) and FIB(8), and return the addition of the result of those two function instances. Similarly, FIB(9)
will call FIB(8) and FIB(7), and FIB(8) will call FIB(7) and FIB(6) and so on. This will process all the
way down to the base case, where FIB(2) will call FIB(1) and FIB(0). At this moment, FIB(1) will return
1 since 1 = 1, and FIB(0) will return 0 since 0 = 0. Then everything will add up all the way up to FIB(8) =
F8 = 21, and FIB(9) = F9 = 34. Then we can get FIB(10) = 21+ 34 = 55, which is the tenth Fibonacci
number, F10.
Another example of recursion which might not be that familiar to everyone is the function of
merge sort. A merge sort function will take an input string and return the sorted string. This function is
defined as:
FUNCTION MS(s):
IF |s| = 1 THEN return s
ELSE return MERGE(MS(sleft), MS(sright))
The input, s, is an arbitrary string over the set A. Every element in set A is comparable. That is to say, for
any two different elements in the set, there is always a relation, either larger or smaller, between these two
elements. For example, every letter is either larger or smaller than another letter based on the alphabetic
order. If the input is a string of length 1, it does not need to be sorted, and thus the function return exactly
the input string. Else, if the input is longer than 1, the function will call two instances of itself, whose
input is the left half of the input and the right half of the input, respectively. The MERGE function is
designed to merge the two sorted strings to one single string which is also sorted. Since the MERGE
function does not include induction, we will not talk about it with much detail.
This MS function is promised to return a sorted version of its input string, and we can prove this
by induction. The theorem is that MS(s) will return a sorted permutation of the input string s. The base
case is that when the input has length of 1. Since there is only 1 character in the string, it is sorted and we
can directly return the string. The inductive hypothesis is that, suppose MS(sleft) and MS(sright) returns
two sorted strings, where sleft is the left half of string s and sright is the right half of string s. Since the
MERGE function is designed to merge two sorted string to one single sorted string, the function MS(s)
will return a sorted permutation of the input string s.
One advantage of recursion over other loop structure is that there is no counter in recursion. The
existence of base case and inductive case replace the functionality of a counter, which is used to decide
whether to leave the loop structure or not. This advantage saves spaces in computer programming, which
makes recursion popular and effective.
Reference:
1. https://en.wikipedia.org/wiki/Recursion
2. https://en.wikipedia.org/wiki/Fibonacci_number
Why ripples are waves in concentric circles?
We all have the experience in throwing stones into a pond, to see who can throw further, or make a bigger noise.
However, there is one interesting thing that fascinates me most: no matter what shape the stone has, it always
generates circular waves around. Now imagine you throw a big square stone into water, do you expect square ripples?
Pretty sure the answer is no. Then we see how it works:
At first, the pond is in equilibrium and gravity serves as a restoring force. While the stone causes a disturbance to the
surface of water, this forms transverse waves that are the ripples you see and longitudinal waves that are the sound
you hear. We could explain this by the law of conservation of energy that the surface of water absorbs the kinetic
energy of stones and forms the ripples we are curious about. Hyperphysics illustrates that elasticity and a source of
energy are the preconditions for periodic motions and when the elastic object is an extended body, then the periodic
motion takes the form of travelling waves where elasticity means the capacity to return precisely to the original
configuration after being distorted. Here, water is the medium transferring the energy in which its property is always
the same, because water belongs to the same pond. So the waves travel in an equal speed in all directions, which
determines the ripples to be circular.
Figure1: equilibrium before the
stone is thrown.
Figure2: disturbance after the stone
touches the surface of water and
causes a void.
Figure3: the ripple moves outwards and
water overshoots above the equilibrium
level, and another ripple forms.
Think more about how ripples are formed: As the stones sinks, it forms a void that allows water to move upward to
fill in. Because of the kinetic energy of stone, the disturbed water rushes out which exceeds the equilibrium water
level. The due to gravity, the water rushes down and leaves a space again. The up-and-down repeats several times
until the pond comes to the equilibrium, as we know that this process will not keeps forever because part of energy
dissipates into moving outward to form your vision of several ripples.
Then, can we say that the shape of the object that creates the waves makes no difference to the shape of waves? In
fact, the initial image of disturbance depends on the shape of source. We take a big stone as an example:
Figure4: the stone is elliptical
and the hole is elliptical too.
Figure5: water moves inward
and the stone goes down
leaving the surface.
Figure6: isotropic nature,
cohesive forces, surface
Tension, etc. gives the force
to push water up.
Figure7: it goes down
down again, this horizontal interaction contributes
to the final circular shape.
For small stones, they act as point source, which their shape has little effect on the shape of waves. For big stones,
the shape of stone generates the initial shape of the ripple, but works less than the up-and-down energy in
determining shapes of following ripples. As waves travel outward, the density and composition of water becomes
prominent in providing an equal speed for water that makes the circular shape, because the shape of stone makes
little difference compared to the size of water pond.
References: 1. http://hyperphysics.phy-astr.gsu.edu/hbase/sound/tralon.html
2.http://hyperphysics.phy-astr.gsu.edu/hbase/sound/wavplt.html
Introduction of the Binary Number System
Nowadays, personal computers are becoming more and more popular in daily life. According to the data of
Internet World Stats, in North America the penetration of personal computers usage is 87.9%,
which means personal computers have significant effect on us humans. Computers can be divided
into two types---computer systems and the physical computer itself. The first generation
computer, ENIAC from 1946 was made up of vacuum tube and did not have computer system. In
1955, scientists incorporated a simple computer system into the second generation computer and
that is the origin of the modern computer system. They used the binary number system to operate
the computer system. The binary number system was invented by Gottfried Wilhelm Leibniz during the 18th century.
This essay mainly introduces how to translate the binary number system and the decimal system
When we face a binary number, for example 10110.01, how do we translate it into Decimal System? In general,
scientists note a binary number as follow:
The subscript “n” of the “an ” means the digits. For example, 10110.101 can be written as (10110.01)2 . The
subscript “2” means binary. Then we can use the single number on each digit of the binary number to multiply the (n1)th power of 2. We can translate each binary number to decimal by the following method:
For example, (10110.01)2 can be computed to be: (1*24 )+(0*23 )+(1*22 )+(1*21 )+(0* 20 )+(0*2-1 )+(1*2 - 2 )=
16+0+4+2+0+0.5+0.25=(22.75)10 . The subscript “10” means decimal. This is how to translate Binary system.
Next we discuss how to translate a decimal integer into binary. As above, we can write the decimal system integer
as (an-1 an-2 …a0 )10 . For example, 91 can be write as (91)10 . Then we divide this number by 2 and record the remainder.
Then keep using 2 to divide the quotient until the quotient equal to 0, the number behind “—“ is the quotient and the
number after“—“ is the remainder”
①91/2=45—1
④11/2= 5—1
②45/2=22—1
⑤5/2=2—1
③22/2=11—0
⑥2/2=1—0
⑦1/2=0—1
The remainders, written in reverse order, give the binary number and write it up: “1011011”. Thus, (1011011) 2
is the translation of (91)10 in the binary system, i.e. (1011011)2 =(91)10.
For the decimal number with the decimal point, for example (0.875)10 , we can multiply it by 2 and record the
integer digit part of the result. Then we can use the decimal digit part to multiply 2 and record the result’s integer digit
part until the result is equal to integer 1:
①0.875*2=1.75---1
②0.75*2=1.5---1
③0.5*2=1---1
Then we use the positive sequence of all integer parts and write it up: “0.111” .Thus. (0.111)2 is the translation
of (0.875)10 in the binary system.
After the introduction, you may have some question s about why binary number system are better suited for
computers. The reason is simple. Because currently we do not have the technology to create the “switches” that can
hold more than two possible states. The “0” and “1” can simply represent the “on” and “off” states in the electric
circuits. So binary number is better suited for computers. I hope everyone who has read this essay has the first sight
of the binary system and could learn Math better. Math is fantastic.
Reference:
1. "Binary to Decimal and How to Convert Binary to Decimal." Basic Electronics Tutorials. Ed. Wayne Storr. 25 July 2013. Web. 10 Apr. 2016.
2. Garlick, Mark A. Astronomy: A Visual Guide. Richmond Hill, Ont.: Firefly, 2004. Print.
3. "Why Computers Use Binary." Nookkincom RSS, Nookking.com. 15 May 2010. Web. 10 Apr. 2016.
The p-adic numbers
We generally take our basic number system for granted; we understand it so intuitively that we don’t
often think about how we define the system at all. The relevant property of the familiar real numbers is that we
define the size of a number by its absolute value, or distance from zero, and the closeness of two numbers (what
we call a metric) by the absolute value of their difference. We construct the real numbers by filling in all the gaps
between rationals numbers as described by the absolute value. For a non-mathematical explanation of p-adics, see
https://www.reddit.com/r/explainlikeimfive/comments/1700nr/eli5_padic_numbers/
Our goal here is to define a different p-adic metric for measuring distance, but this leads to constructing
an entirely new set of numbers (technically there is a different set for each prime). First, let’s take the natural
numbers, and express each of these in base p, where p is a prime. For this essay, we will use p=5. Consider 1303,
which is made up of a series of digits between 0 and 4, which we will call 5-adic digits. We can still add and multiply
these numbers the way we would any base 5 integer (1303 + 12 = 1320). A more detailed description of this
process is given in http://www.math.lsa.umich.edu/~lagarias/575chomework/padic-operations.pdf
Here is where normally we would add in negatives and decimals and get back the usual base 5 numbers.
Instead, we define a 5-adic integer as any number that can be expressed as a finite or infinite number of 5-adic
digits (which here describe positive powers of 5), still keeping the rules of addition and multiplication.
𝑖
Mathematically this can be expressed as ∑∞
𝑖=0 𝑎𝑖 𝑝 where ai is a digit from 0 to 4 and p in this case is 5.
Consider the number …44444444. Let’s try adding 1 to it. You end up with a result of …0000, showing that
-1 is …4444444 in the 5-adic integers. In fact, you can show that all natural numbers have a terminating 5-adic
representation and all other rationals except those with a factor of five in the denominator have a repeating
representation (for example …222223 in 5-adic is ½ in the rationals). That still leaves the non-repeating integers,
which happens to include the square root of -1 (or rather of …4444, which begins …32212) in the 5-adic integers
for example! By allowing these infinite digits, we have added “new” numbers in the p-adic integers not in the
regular rationals. For a more formal treatment, see https://www.win.tue.nl/~aeb/2WF02/p-adic.pdf
When we take a look at how we defined these 5-adic integers in terms of a series of powers of 5, we
realize that we can express any 5-adic integer as a product of 5r and another 5-adic integer, where 5r is the highest
power of 5 that divides the original 5-adic integer. For example,
… 343400000 = 56 · … 3434
We define p-adic numbers as the numbers you obtain when you divide a p-adic integer by another p-adic
integer. Using the technique shown in the last paragraph, they can be expressed in terms of a p-adic integer like
∞
𝑖
𝑟
𝑖
so: ∑∞
𝑖=𝑟 𝑎𝑖 𝑝 = 𝑝 · ∑𝑖=0 𝑎𝑖 𝑝 . A clearer derivation is here http://mathforum.org/library/drmath/view/65286.html
After all that work with confusing and non-intuitive expressions of numbers, we now have a very natural
basis for setting up p-adic absolute values and a p-adic metric. The p-adic absolute value for a p-adic number is
𝑖
defined to be ǀxǀp = p-r where 𝑥 = 𝑝𝑟 · ∑∞
𝑖=0 𝑎𝑖 𝑝 . In other words, the larger the power of p that divides the number,
the smaller that number is defined to be (that is why in p-adics it makes sense to have infinite left digits but finite
right decimals). The p-adic metric is similarly defined as the p-adic absolute value of the difference of two p-adic
numbers ǀx-yǀp.
The question is, why can’t we use the p-adic metric on things that aren’t the p-adic numbers, like the
rationals or the reals? It turns out that the way the real numbers are defined, they include numbers that are not in
the p-adic numbers and vice versa. On the other hand, the rationals are in fact a strict subset of the p-adic
numbers, and there are gaps that we want to fill in. The p-adic numbers are what you get when you fill in all the
gaps according to the p-adic metric.
For more math see http://www.asiapacific-mathnews.com/03/0304/0001_0006.pdf
For dense abstract math see http://www.maths.gla.ac.uk/~ajb/dvi-ps/padicnotes.pdf
It actually starts with things we did in class.
Fantastic Random Walk
“A drunk man will find his way home, but a drunk bird may get lost forever”, proposed by a
Japanese-born American mathematician, Shizuo Kakutani, is an interesting statement. When people
think the reason behind this statement, they may think the existence of some unknown superiority
of humans over animals. The true reason, however, is not something about biology but mathematics.
Random walk, a mathematical topic, accounts for that interesting statement. Random walk is
the process by which randomly-moving objects wander away from where they started. A drunk
man’s and a drunk bird’s movements are consistent with random walk, because they lose their
consciousness after getting drunk, and just randomly move away. When studying a new
mathematical topic, we usually start from 1-dimensional system into higher-dimensions. Therefore,
I will introduce random walks from 1-dimension and then generalize to 3-dimensions in this essay,
in order to prove that a drunk man is “smarter” than a drunk bird.
Suppose there is a number line, which is a typical 1-dimensional system. The black dot on the
origin, can be considered
as a drunk man, but he
can only move left or right, as the road is too narrow to move in other direction. Since he is drunk
and without any consciousness, the probability of each step, either being left or right is equal, calling
his first step as S1, the second step S2 and so on. Each of his step will result in a displacement of
either +1 or -1 in the number line. Now, suppose he starts wandering from his home “0”, takes N
steps and makes a total displacement D. Due to the N steps, we will have S1, S2, S3,…, Sn. The
displacement “D”, then equals S1+ S2+ S3+ ,…,+ Sn. Sn will either move right (+1) or left (-1). When
N tends to infinity, D will tend to zero, because the number of +1’s and -1’s will be equal. Therefore,
a drunk man will finally return to his home “0”, as his displacement will be zero as he keeps moving.
When it comes a 2-dimensionl system, the result will be also similar, which is the real situation
a drunk will face in the real life. Since, a person can not only move left and right but also forward
and backward. Therefore, we can consider that there is both an x-axis and an y-axis under the drunk’s
feet. Then, we can consider the drunk man’s x-coordinate as a random walk and y-coordinate as
another independent random walk. In 1-dimensional system, we have already proved in a number
line that the drunk man will return to origin “0” when he keeps moving, so both his x-coordinate
and y-coordinate will become “0” in some time point. If they reach the “0” simultaneous, the drunk
man will find his home.
The drunk bird, however, is not moving in the 2-dimensionnal system but 3-dimensional system,
which means the x-coordinate, y-coordinate and z-coordinate have to be “0” simultaneous, the
possibility of which is very small. Our findings on number line and coordinate plane can be proved
by Polya’s Recurrence Theorem. Let p(d) be the probability that a random walk on a d-dimensional
lattice returns to the origin, Pólya proved that p(1)=p(2)=1. If d＞2, then it is transient, which means
p(d)＜1. As for a drunk bird, the numerical value of p(3) proved by modern mathematician is
0.340537. Therefore, a drunk bird may never return to its home.
References:
1.MIT Class Note:http://www.mit.edu/~kardar/teaching/projects/chemotaxis(AndreaSchmidt)/random.htm
2. Pólya's Random Walk Constants:http://mathworld.wolfram.com/PolyasRandomWalkConstants.html
Currency of Three
The common currency denominations always used are the one bill, the two bill, the five bill and the ten bill.
In general, when a country has to determine the coin denomination, the highest denomination is always an integral
multiple of the lower denominations.
Generally speaking, currency denomination is designed according to the principle of mathematical
combination. For example, I’d like to divide the integers from 1 to 10 into two categories: “important” and
“unimportant”. For the numbers 1, 2, 5 and 10, they are “important” numbers because we can get the remaining
numbers with minimal additions and subtractions, such as 1 + 2 = 3, 2 + 2 = 4, 1+5=6, 2+5=7, 10-2=8, 10-1=9. The
number 4 is “unimportant” because if any one of the four "important" numbers is replaced by it, it will appear that
some numbers must add or subtract more than twice in order to get the desired numbers. For example, for the
combination of 1, 2, 4, 10, we get the remaining numbers with minimal additions and subtractions as: 1 + 2 = 3,
1+4=5, 2+4=6, 1+2+4=7, 10-2=8, 10-1=9. Similar to the number 4, it takes 1+2+2 to make the number 5, so it
makes sense to have the five bill cover all numbers with the last digit number 5. For the “unimportant” numbers 6
and 8, they are multiples of the number 2. It is unnecessary to print extra six and the eight bills. For the “unimportant”
numbers 7 and 9, it is inconvenient for daily use in public because of difficulty memorizing calculations of their
multiplications, especially for those who lack math training.
Moreover, from an economic perspective, when using the one bill, the two bill and the five bill, we could
use at most three pieces to create another number from 1 to 10. 1=1 (1 piece), 2=2 (1 piece), 3=1+2 (2 pieces),
4=2+2 (2 pieces), 5=5 (1 piece), 6=1+5 (2 piece), 7=2+5 (2 pieces), 8=1+2+5 (3 pieces), 9=2+2+5 (3 pieces).
Suppose we add a three bill into such combinations of numbers from 1 to 9. We get: 1=1 (1 piece), 2=2 (1 piece),
3=3 (1 piece), 4=2+2 (2 pieces), 5=5 (1 piece), 6=1+5 (2 piece), 7=2+5 (2 pieces), 8=3+5 (2 pieces), 9=2+2+5(3
pieces). You see only the numbers 3 and 8 become easier in exchange for printing an extra three bill in which is
costly and time-consuming. Additionally, it is silly to have the three bill when it is easy to make it by adding the one
bill and the two bill. So, in practical application, the one bill, the two bill, the five bill and the ten bill are enough for
combining the value from 1 to 9 and for easy change. For larger denominations, such as twenty bill, fifty bill and
hundred bill, their combinations of large amount of money are also more convenient for us to trade.
But this does not mean that three bill does not exist in human history. For example, the three note was
issued in Russian and Chinese history. As a collector’s item, this note is worth $9000 today. So why does Russia
and China appear to have the three-value denomination?
In fact, the use of notes denominated in ‘three’ by Russia has its origins in the currency that circulated in
that country in the first half of the nineteenth century. During the reign of Nicholas
I Russia introduced a 1½-ruble coin in an effort to integrate the currency of Russia
and its Polish possessions. Such denominations were well entrenched and a 3-ruble
note was introduced to match the value of the lowest valued 1½-ruble coin. Hence
forth, notes denominated in ‘three’ have been issued constantly in Russia and the
USSR. 1 At that time, China took the policy of comprehensive study from the
Soviet Union. On March 1, 1955, China also introduced the three-value
denomination and the Chinese call such denomination “Su three bills.” When the
relationship between Soviet Union and China worsened during the Cold War, the
Soviet Union utilized India to continue printing “Su three bills” and secretly
supplied to China through Xinjiang and the northeast border in order to disrupt the
financial order of the country. To counter the Soviets’ plot, in 1964, China
decided to forbid using the "Su three bill," and changed all "Su three bill" in
the bank into other denominations within one month or be destroyed. This is
how the three bill, "Su three coins," just showed up a short time in Chinese
history of currency and the rarest collections of denomination. Despite its
minimal use over the last 350 years, notes denominated in ‘three’ now appear
more likely to accumulate in albums of thematic collectors than to make an
appearance in new banknote issues.
1
http://www.pjsymes.com.au/articles/three.htm The Applications of Mathematics in the Arts
Mathematic is usually stereotyped as a regulated and rational approach of science. Through the
application of abstraction and logical deduction, mathematics is concerned with abundant
calculations, fact-based statistics, and pPrecise measurement. On the other hand, art is a cultural
phenomenon, which is always labeled with emotional, creative and visionary tags. By creating
beyond the real life, artists pour their emotions and feelings into their work. Although it seems that
the basic characteristics of math and the arts are opposite, the bond between them is actually deep
and strong.
Painting is an important branch of art. The application of mathematics enabled the most important
revolution in art, which is the Renaissance. To show this, let’s look at two paintings ; one from the
Middle Ages (left) and one from the Renaissance (right).
Figure1: The Bayeux Tapestry(The death of Harold), no name, 1077
Figure2: The School of Athens, Raffaello Sanzio da
Urbino(Raphael),1509–1511
It is obvious that Figure 1 does not convey a feeling of space, and the human figures look like “paper
man” in two dimensions. However, Figure 2 is a figurative painting, which conveys a strong sense of
distance and stereoscopic vision. The reason behind this change is the entrance of mathematics into
painting. Realistic painting became the mainstream during the Renaissance, and applying geometric
methods to painting enabled artists to implement this goal. Later, the appearance of scenography
facilitated the development of realistic painting during Renaissance.
Music, another important field of art, is also deeply connected to maths as well. In ancient Greece,
the great mathematician Pythagoras found that the sound made by plucking at the strings of an
instrument is deeply connected to the length of the string. Later, he also found that the length of the
strings for harmonic sound are proportional. Based on his observation about math, he invented the
Pythagoras Scale. This Greek musical scale dominated the western musical world for a long time.
Besides, the arrangement of the black keys on the piano is also connected to a mathematical sequence
—the Fibonacci Sequence. If we take one octaves, we could see that there are 13 keys for total. And
these keys are consisted of 8 white keys and 5 black keys. What is more, the 5 black keys are divided
in to two groups, with 2 in the first group and 3 in the second group. By extracting and organizing
these datas, we found that 2,3,5,8,13 are actually the first 5 numbers in the Fibonacci Sequence.
From all the examples, we see that mathematics does appear in arts, and its appearance is not a
coincidence. Applying mathematical methods enables artists to establish new ways of artistic
creation, and enrich the filed of art. Math and the arts are inseparable. I hope next time when you are
wandering in the Louvre museum and standing in front of the Mona Lisa, you will think of
something about the math behind her smile.
Reference:
No name. The Bayeux Tapestry(The death of Harold).1077. Bayeux Museum, Bayeux, France. The death of
Harold, Web. 10 April 2016. http://www.bayeux-tapestry.org.uk/deathofharold.htm
Raffaello Sanzio da Urbino(Raphael). The School of Athens. 1509-1511. Apostolic Palace, Vatican City. Khan
Academy, Web. 10 April 2016. https://www.khanacademy.org/test-prep/ap-art-history/early-europeand-colonial-americas/renaissance-art-europe-ap/a/raphael-school-of-athens
MATHEMATICS: THE SOUL OF ECONOMICS
THE OBAMA SPENDING PLAN
Economics is a social science that studies how individuals, firms, governments and nations make choices on allocating scare resources to
satisfy their unlimited wants. Before taking any economics classes, people would wonder what method we would use to study such
complicated subjects. What do economists know? How do they know it? More importantly, how do they know they are correct The
answer is Mathematic. Anyone would be surprised with the overriding importance of the use of Math in Economics. Economics can be
divided into Microeconomics which studies the specific choices made by consumers and producers and Macroeconomics which is the
study of the economy as a whole. In either branches of economics, mathematics is a powerful tool for economic analysis.
In macroeconomics, GDP ( Grossed Domestic Product ), which can be considered the total income of the economy, shows how healthy
the economy is. GDP is calculate as the sum of consumption, investment, net export and government purchase. The government’s use of
fiscal policies can greatly affect GDP and the economy. Therefore, economists, who study macroeconomics try to measure the effect of a
fiscal policy on GDP and the economy before imposing it on the economy.
In 2009, when president Barack Obama took office, the economy was suffering from the Great Recession. His advisor a sizable stimulus
package which would cost the federal government $800 billion. The package included some tax cuts and higher transfer payments, but
much of it was made up of increase in government spending. There were intense debates between economists at that time about the merit
of the plan. Many argued that bid increase in government spending will result in bigger budget deficit. Another concern was that
spending on infrastructure would take time while tax cuts could have more immediate effect. Despite skepticism and concern about the
effect of the package, Congress went ahead with President Obama’s proposed stimulus plans with a $787 billion bill on February 17,
2009. Obama’s administration economists when analyzing the merit of the plan calculated that the government purchases multiplier is
1.57 whereas the tax multiplier is only 0.99. These number is not made up number, therefore, there are methods to calculate and
understand these numbers.
As discussed above, GDP consists of four elements. Increase in government spending initially results in increase in GPD by the same
amount. However, as GDP is also the total income of the economy, increase in GDP will cause further increase in consumption. Thus,
government purchases multiplier is greater than 1.
Y= C + I + G + NX
ΔY=ΔC + ΔI + ΔG + ΔNX = MPC*ΔY+ΔG ( MPC: Marginal propensity to spend, MPC<1 )
(1-MPC) * ΔY=ΔG
ΔY=1/(1-MPC)*ΔG
On the other hand, tax cuts will have immediate effect on consumption as people now can have more money to spend. In order to
calculate the tax multiplier, we have:
Y= C + I + G
ΔY= ΔC + ΔI + ΔG + ΔNX = ΔC
ΔY= MPC * ( ΔY – ΔT )
(1-MPC) * ΔY = -MPC * ΔT
ΔY = -MPC/( 1- MPC ) * ΔT
As MPC <1 government purchase multiplier is always greater than tax multiplier. Therefore, increased government spending would be
better than reduced tax. In reality, this statement is also true as government spending can not only create more jobs but also help firms to
generate revenue. Moreover, when the government reduces tax, the amount saved from not paying tax might be saved by individual
rather than using on consumption.
In fact, GDP of the United States has been raising incessantly since the bill was passed. The effect of the bill was even greater than what
was calculated by Obama’s administration economist. With simple math knowledge, the economist was able to measure the outcome of a
fiscal policy rather than just make a conjecture. Math is used everywhere in economics, from building economic models to testing the
validity of economic theories. Without math, it would be hard to imagine what economics looks like.
References
Macroeconomics, N. Greogory Mankiw ( P. 292 – P. 294 )
The Earliest Known Matrices [2]
Jiu Zhang Suan Shu, or The Nine Chapters on the Mathematical Art, an ancient Chinese mathematics book,
is composed by several generations of scholars from the tenth to the second century BC. Liu Hui (225–295), one of
the greatest mathematicians of ancient China, edited and published it in the year 263. The contents of the book are
covered in nine chapters: Chapter 1, Fang Tian (Rectangular field), Chapter 2, Su Mi (Millet and rice), Chapter 3,
Cui Fen (Proportional distribution), Chapter 4 Shao Guang (Lesser breadth), Chapter 5, Shang Gong (Measuring
works), Chapter 6, Jun Shu (Equitable transportation), Chapter 7, Ying Bu Zu (Surplus and deficit), Chapter 8, Fang
Cheng (Rectangular arrays), Chapter 9, Gou Gu (Base and altitude) [1].
Chapter 8 of the book, Fang Cheng, involves solving real-life problems such as calculating yields of grain,
numbers of domestic animals, and prices of different products by solving linear equations [1]. Fang Cheng is
investigated in terms of modern linear or matrix algebra and it is apparent that it involves methods associated with
Leibniz (1646-1716), Seki Takaka- zu (1642-1708), and Gauss (1777-1855), but long predating those people [3].
The technique given in Chapter 8 for solving these problems is elimination, which is almost identical to the so-called
Gaussian elimination in the Western world [1].
In the book, Liu Hui first states a problem, and then illustrates the algorithm for solving the problem. Take
Problem I as an example, “there are three grades of grain: top, medium and low. Three sheaves of top-grade, two
sheaves of medium-grade and one sheaf of low-grade are 39 Dous (a unit of dry measurement for grain in ancient
China). Two sheaves of top-grade, three sheaves of medium-grade and one sheaf of low-grade are 34 Dous. One
sheaf of top-grade, two sheaves of medium-grade and three sheaves of low-grade are 26 Dous. How many Dous
does one sheaf of top-grade, medium-grade and low-grade grain yield respectively?” [1]
Immediately afterwards, Liu Hui explains the algorithm for solving it, “putting three sheaves of top-grade
grain, two sheaves of medium- grade grain, and one sheaf of low-grade grain and the total 39 Dous in a column on
the right, then putting the other two columns in the middle and on the left. Multiplying the middle column by topgrade grain of the right column, then eliminating top-grade grain from the middle column by repeated subtracting
the right column. Similarly, multiplying the right column and also doing the subtraction. Then, multiplying the left
column by medium-grade grain of the middle column, and carrying out the repeated subtraction. Now the remaining
two numbers in the left column decides the yield of low-grade grain: the upper one is the denominator, the lower
one is the numerator. Now, to obtain the yield of medium-grade grain from the middle column, the denominator is
the top number, and the numerator is the bottom number minus the middle number times the yield of low- grade
grain. To calculate the yield of top-grade grain by the right column, the denominator is the top number, and the
numerator is the bottom number minus the second number times the yield of medium-grade grain and the third
number times the yield of low-grade grain.” [1]
We can visualize the matrices Liu Hui describes with Figure 2 and Figure 3.
Figure 1: Algorithm descriptions, Chapter 8 of Jiu Zhang Suan Shu [1] Figure 2 and Figure 3: Matrices followed Liu Hui’s descriptions [4]
It is easy to see that the above procedure is almost identical to Gaussian Elimination [1]. Gaussian
elimination and general solutions of systems of linear equations are ultimately of oriental origin and were known in
China some seventeen centuries before Gauss [3]. The difference is the way the numbers are arranged in the arrays.
If we rotate the arrays anti-clockwise 90 degrees, we obtain the corresponding matrices of the Gaussian elimination.
The reason for this difference is that, as in ancient China, people wrote from top to bottom, and then from right to
left, while in the West, people write from left to right and then from top to bottom [1]. Therefore, from the algorithm
described in Chapter 8 of the book, we conclude that the Gaussian elimination was discovered at least over 2200
years ago in ancient China [1], while Gauss invented what Legendre named the method of least squares in around
1809 [4]. Recently, more and more western scholars credit this simple yet elegant elimination algorithm to ancient
Chinese mathematicians [1].
Reference
[1] Jiu Zhang Suan Shu and the Gauss Algorithm for Linear Equations, by Ya-Xiang Yuan, 2012 9-14
[2]Chinese Mathematics, http://www.crystalinks.com/chinamath.html
[3] Journal of the American Oriental Society, David Prager Brranner, 2011 October-December
[4] Mathematicians of Gaussian Elimination, Joseph F. Grcar, 2011 June/July