The Joy of X
The Joy of X – by Steven Strogatz
We carry a common myth that most of the Math that is being
taught in the schools and colleges is useless. But that’s not true. Math was formulated to satisfy the needs of
previous centuries and is now used to handle the modern technologies.The way
math was formulated and is being used is meticulously answered in this book by Steven in an interesting way
that truly connects practical life.
Why sum of the n odd natural numbers is n2?
Why is the sum of n natural numbers n(n+1)/2? 1+2 +3 + 4 is?
Put white marbles on the floor in the given order. 1, then 2, then 3 and then 4. Now make a rectangle by adding black marbles. Now it becomes a rectangle of four rows and five columns. (4 x 5). There are 20 marbles. So how many are white? Half of 20 = 10. Area of triangle is half of the area of rectangle. So n(n+1)/2.
The enemy of my enemy: Negativity multiplied by negativity turns it into positivity.
Scholars have used
these ideas to analyze the run-up to world war-I. It shows the shifting
alliances from Great Britain, France, Russia, Italy, Germany, Austria and
Hungary between 1872 and 1907. The bold segment between the countries shows
friendly relationship. The dotted
segment indicates Hostile relationship.
The bold line represents friendly relation whereas the
dotted line indicates enmity
Some part of what we observe is due to nothing more than the
primitive logic of “The enemy of my enemy” and this part is captured perfectly
by the multiplication of negative numbers.
Commutative law
application:
A)
10 % Tax to be applied after a discount of 20%
B)
20% discount to be applied on a value that is
increased by 10% tax.
Although
it is the same, quite a few find it different. You will end getting 12%
decrease on the given marked price. All it says is a x b = b x a
The
commutative law does not always work with our mindsets. Let’s say that a person
wants to get into IIM-B very desperately. Not getting there can lead her to
take a silly step of committing suicide. The person gets rejected by IIM-B but
gets a call from IIM-C. The girl cancels the idea of committing suicide. She
can go to IIM-C and commit suicide, but not the other way round.
Early
in the development of quantum mechanics, Werner Heisenberg and Paul Dirac had
discovered that nature follows a curious kind of logic in which p x q ≠ q
x p where p and q represent the momentum and position of quantum particle.
Without that break-down of commutative law, there would be no Heisenberg uncertainty
principle, atoms would collapse and nothing would exist. That’s why you better
mind your p’s and q’s.
0.12122122212222…..By design, the blocks of 2 get progressively longer as we move to the right. There is no way to express this decimal fraction. It’s irrational!
Grant
wiggins, an education consultant, has been posing a problem to students and
faculties for years. More than 50% people got it wrong.
Try
and make an equation for a relation, “One yard equals three feet”.
If
you think the answer is 1 y = 3f, welcome to the club of 50%. Seems to be a
straightforward translation but as soon as you try a few numbers, you’ll see
that this formula does not work. Say the hallway is 10 yards long. We all know
it is 30 feet. Now plug y = 10 and f =30 the formula does not work.
The
correct equation is f = 3y. Here 3 really means “3 feet per yard”. When you
multiply it by Y in yards, the units of yards cancel out and you’re left with
units of feet, as you should be.
What multiplying by i (imaginary number) looks like?
Multiplying by ‘I’ produces a rotation counterclockwise by a quarter turn. It takes an arrow of length 2 pointing east and changes it into a new arrow of the same length but now pointing north. Electrical engineers love complex numbers for exactly this reason. Having such a compact way to represent 90o rotations is very useful when working with Alternating currents and magnetic fields, because these often involve oscillations or waves that are quarter cycles. In aerospace engineering they eased the first calculations of the lift on an aero plane wing. Civil and mechanical engineers use them routinely to analyze the vibrations of footbridge, skyscrapers and cars driving on bumpy roads.
The
90o rotation property also sheds light on what i2 = -1
really means. If we multiply positive number by i2, the
corresponding arrow rotates 180o, flipping the arrow from east to
west.
In
the early part of ninth century, Muhammad bin Musa al-Khwarizmi, a
mathematician working in Baghdad, wrote a seminal text book in which he
highlighted the usefulness of restoring a quantity being subtracted by being
added to the other side of equation. Say x + 2 = 4. We subtract 2 on both sides
to get x +2 -2 = 4 -2. So x = 2. He called this process as Al-jabr (Arabic word
for ‘restoring’) which later morphed into ‘Algebra”. Then long after his death,
he hit the etymological jackpot again. His own name, Al- khwarizmi, lives on
today in word, “Algorithm”.
X2
+ 10 x = 39
Al-Khwarizmi
put the puzzle this way X2 + 10x = 39
The
missing corner was an irresistible step. That was filled in to turn the figure
into a perfect square.
A
mathematician needs Functions for the same reason that a builder needs hammers
and drills. Tools transform things. So does functions.
If we plot y = 4 –x2 on graph we get
The bowed shape of the curve is due to the action of mathematical pliers. The function that transforms x into x2 behaves a lot like the common tool for bending and pulling things. And what role does the 4 play in the equation? It acts like a nail for hanging a picture on a wall. It lifts the bent wire arch by 4 units. Since it raises all the points by the same amount it’s known as a constant function. The tool x2 bends the piece of X-axis and the ‘4’ lifts it and holds it. These building blocks 4 and -x2 can be regarded as component parts of a more complicated function, 4- x2, just as wires, batteries and transistors are components of a radio. If you look this way, you’ll start noticing functions everywhere.
Power
functions of the form xn in which the variable x is raised to a
fixed power n. changing n yields handy tools. For parabola n = 2 and for
constant n = 0. Whereas n =1 gives a function that works like a ramp, a steady
incline or a decline. It’s called linear function as it produces a line on the
graph. What it n = -2? (1/x2). This function is good to describe how
waves and forces attenuate as they spread out in three dimensions the way sound
softens as it moves away from the source.
In
x2, as x gets larger and larger an exponential function of x
eventually grows faster than any power function no matter how large the power.
Exponential growth in almost unimaginably rapid. That’s why it’s so hard to fold a piece of
paper in half more than seven or eight times. Each folding approximately
doubles the thickness of the wad, causing it to grow exponentially. Meanwhile
the Wad’s length shrinks in half every time and thus decreases exponentially
fast. For standard sheet of the note book paper, the wad becomes thicker than
it is long and so it can’t be folded again. For a sheet to be considered
legitimately folded n times, the resulting wad is required to have 2n
layers in a straight line, and this can’t happen if the wad is thicker than it
is long. In 2002 Britney Gallivan, then a junior in high school solved. She
made a formula that predicted the maximum number of times, n, that paper of a
given thickness T and length L could be folded in one direction.
L
=
Whispering
galleries are remarkable acoustic spaces found beneath certain domes, vaults or
curved ceilings. There is famous whispering gallery in New York, the Grand
central station. It’s a fun place to take a date. You can communicate by
whispering when you are forty feet apart and separated by a bustling
passageway. You’ll hear each other clearly but the passersby won’t hear a word
you’re saying.
Ellipse
displays a similar flair for focusing, though in a much simpler form. If we
fashion a reflector in the shape of an ellipse, two particular points inside it
(marked as F1 and F2) will act as foci in the following sense: All the rays
emanating from a light source at one of those points will be reflected to the
other.
Or
if pool is to be paled on an elliptical table, with the pocket on one of the
focus, there is an interesting trick that can be used in playing. To set up the
trick shot that is guaranteed to scratch every time, place the cue ball at the
other focus. No matter where you strike the ball and no matter where it caroms
off the cushion, it will always go to the pocket.
As Parabola is a set of points equidistant from given point and a given line (directrix) not containing that point the convergence of the light rays becomes obvious.
The
reason for calling F the focus becomes clear if we think of the parabola as a
curved mirror. It turns out that if you shine a beam of light straight at a
parabolic mirror all the reflected rays will intersect simultaneously at point
F producing an intensely focused spot of light.
Mathematicians
and the conspiracy theorists have this much in common: We’re suspicious of
coincidences – especially the convenient ones. There are no accidents.
Why
Parabola and ellipse only have such a fantastic powers of focusing?
They
are both cross-sections of the surface of Cone. So cone is the mother of
Circle, ellipse, parabola and Hyperbola together called conic section. In
algebra together they arise as the graphs of second degree equations
Ax2
+ Bxy + Cy2 + DX+ EY + F =0
The
constants A,B,C,D,E and F determine whether the graph is a circle, ellipse,
parabola or a hyperbola. So it’s no accident that planets move in elliptical
orbits with the sun at one focus; or that comets sail through the solar system
on elliptic, parabolic or hyperbolic trajectories; or that a child’s ball
tossed to parent follows a parabolic arc. They’re all manifestations of the
conic conspiracy.
Conversion of circular motion into sine waves is a pervasive though often unnoticed is a part of our daily experience. It creates hum of the florescent lights overhead in our offices, a reminder that somewhere in the power grid, generators are spinning sixty cycles per second, converting the rotary motion into alternating current, the electrical sine waves on which the modern life depends. When you speak and I hear, both our bodies are using sine waves- yours in the vibrations of your vocal cords to produce the sounds and mine in the swaying of hair cells in my ears to receive them.
Change we can believe in
Imagine
a mechanically controlled bank of anti-craft guns automatically firing at an
incoming fighter plane. Calculus, tells the guns where to aim. Calculus is the
mathematics of change. It describes everything from the epidemics to the Zigs
and Zags of well-thrown curveball. Roughly, the derivative tells you how fast
something is changing’; the integral tells you how much it’s accumulating. They
were born in separate times and places. Integrals in Greece around 250 BC and
Derivatives in England and Germany in the md-1600’s.Yet they’ve turned out to
be blood-relatives.
Change
is the most sluggish at the extremes, precisely because the derivative is zero
there. Things stand still momentarily. This zero derivative property of crests
and troughs underlies some of the most practical applications of calculus. It
allows us to use derivatives to figure out where a function reaches its maximum
or minimum, an issue that arises whenever we are looking for the best or the
cheapest or the fastest way to do something.
In
integration the symbol ‘ò’
simply means along ‘S’ that stands for sum. In astronomy, the gravitational
pull of the sun on the Earth is described by an Integral. It represents a
collective effect of all the miniscule forces generated by each solar atom at
the varying distances from the earth. In oncology, the growing mass of solid
tumor can be modeled by an integral. Why Sums like this require Integration and
not the school methods of adding? The first reason is that neither Sun nor
Earth are points. Both are gigantic balls made up of stupendous number of
atoms. Every atom in the sun exerts gravitational tug on every atom on the
earth. Of course, since atoms are tiny, their mutual attractions are almost
infinitesimally small. But there are infinitely many of them, in aggregate they
can still amount to something. Somehow we have to add them all up.
A
few numbers are such celebrities that they go by single-letter stage names,
something not even Amitabh Bachhan or Sachin Tendulkar can match. The most
famous is p, the
number that represents the ratio of the circumference to the diameter of a
circle. Closely behind is ‘i’ the imaginary number so radical that it changed
what it meant to be a number. Who’s the next celebrity?
Now
say hello to ‘e’ nicknamed for its breakout role in exponential growth, e is now
the Zelig of advanced Mathematics. From the insights it offers about chain
reactions and population booms, e has a thing or two to say about how many
people you should debate before settling down.
What’s
e?
It
is a numerical value 2.71828….
To make it simpler, understand this.
Imagine that you’ve deposited `1000 in a savings account that offers 100% interest rate compounded annually. A year later your amount should be `2000. What if you ask the bank to pay 50% semiannually instead of 100% annually? You’ll get better than before. It will be
1000
+ 500 =1500 and then 1500 + 750 = 2250. What if you further negotiate and ask
the bank to break further and further? Like 25% per quarter then, then
monthly…soon till hours, mins, secs or even Nano-seconds. Would you make a
small fortune of sorts?
To
make the numbers come out nicely, here’s the result for a year divided into
hundred equal periods, after each of which you’d be paid 1percent interest (The
100% annual rate divided evenly into 100 installments) your money would grow by
a factor of 1.01raised to100th power and that comes out to be about 2.70481….In
other words the `1000
would be come `20704.81…
But
what if the interest is compounded infinitely often?
This
is called as continuous compounding and your total after one year would be `2718.28…
How does their love ebb and flow over time? That’s where calculus comes in. By writing equations that summarize how Romeo and Juliet respond to each other’s affections and then solving those equations with calculus, we can predict the course of their affair. The resulting forecast for this couple is, tragically, a never ending cycle of love and hate. At least they manage to achieve simultaneous love a quarter of the time.
Romeo’s behavior can be modeled by the differential equation
Þ
which
describes how his love (represented by R) changes in the next instant
(represented by dt). According to this equation, the amount of change (dR) is
just a multiple (a) of Juliet’s current love (J) for him. This reflects what we
already know—that Romeo’s love goes up when Juliet loves him—but it assumes
something much stronger. It says that Romeo’s love increases in direct linear
proportion to how much Juliet loves him. This assumption of linearity is not
emotionally realistic, but it makes the equations much easier to solve.
Juliet’s behavior, by contrast, can be modeled by the equation
Þ
The
negative sign in front of the constant b reflects her tendency to cool off when
Romeo is hot for her
To
give a sense of what Maxwell accomplished and, more generally, what vector
calculus is about, let’s begin with the word “vector.” It comes from the Latin
root vehere, “to carry,” which also gives us words like “vehicle” and “conveyor
belt.” To an epidemiologist, a vector is the carrier of a pathogen, like the
mosquito that conveys malaria to your bloodstream. To a mathematician, a vector
(at least in its simplest form) is a step that carries you from one place to
another
We
know Log 10 = 1, Log 100 = 2, Log 1000 = 3. The interesting part here is that
the numbers inside the logarithms are growing multiplicatively, increasing
tenfold each time from 10 to 100 to 1000 their logarithms grew additively
increasing from 1 to 2 to 3. Our brain performs a similar trick when we listen
to music. The frequencies of the notes in a scale –do, re, mi, fa, sol, la, ti,
do – sound to us like they’re rising in equal steps. But objectively their
vibrational frequencies are rising by equal multiples. We perceive pitch logarithmically.
The
approach builds on ideas from linear algebra, the study of vectors and
matrices. Whether you want to detect patterns in large data sets or perform
gigantic computations involving millions of variables, linear algebra has the
tools you need. Along with underpinning Google’s PageRank algorithm, it has helped
scientists classify human faces, analyze the voting patterns of Supreme Court
justices, and win the million-dollar Netflix Prize (awarded to the person or
team who could improve by more than 10 percent Netflix’s system for
recommending movies to its customers)
For a case study of linear algebra in action, let’s look at how PageRank works. And to bring out its essence with a minimum of fuss, let’s imagine a toy Web that has just three pages, all connected like this:
The arrows indicate that page X contains a link to page Y, but Y does not return the favor. Instead, Y links to Z. Meanwhile X and Z link to each other in a frenzy of digital backscratching. The approach taken by Larry Page and Sergey Brin, the grad students who cofounded Google, was to let webpages rank themselves by voting with their feet—or, rather, with their links. In the example above, pages X and Y both link to Z, which makes Z the only page with two incoming links. So it’s the most popular page in the universe. That should count for something. However, if those links come from pages of dubious quality, that should count against them. Popularity means nothing on its own. What matters is having links from good pages. Google’s algorithm assigns a fractional score between 0 and 1 to each page. That score is called its PageRank; it measures how important that page is relative to the others by computing the proportion of time that a hypothetical Web surfer would spend there. Whenever there is more than one outgoing link to choose from, the surfer selects one at random, with equal probability. Under this interpretation, pages are regarded as more important if they’re visited more frequently (by this idealized surfer, not by actual Web traffic)
And
because the Page Ranks are defined as proportions, they have to add up to 1
when summed over the whole network. This conservation law suggests another,
perhaps more palpable, way to visualize PageRank. Picture it as a fluid, a
watery substance that flows through the network, draining away from bad pages and
pooling at good ones. The algorithm seeks to determine how this fluid
distributes itself across the network in the long run. The answer emerges from
a clever iterative process. The algorithm starts with a guess, then updates all
the Page Ranks by apportioning the fluid in equal shares to the outgoing links,
and it keeps doing that in a series of rounds until everything settles down and
all the pages get their rightful shares.
The value of the page rank of Z after update changes. The scores are updated to better reflect each page’s true importance. The rule is that each page takes its PageRank from the last round and parcels it out equally to all the pages it links to. Thus, after one round, the updated value of X would still equal 1/3, because that’s how much PageRank it receives from Z, the only page that links to it. But Y’s score drops to a measly 1/6, since it gets only half of X’s PageRank from the previous round. The other half goes to Z, which makes Z the big winner at this stage, since along with the 1/6 it receives from X, it also gets the full 1/3 from Y, for a total of ½.
The wonderful book by Steven Strogatz has
given many interesting points, which are being assembled by me in
this summary. To do this I have used my discretion, personal examples to relate
it to me in a better way.
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