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1.6.3.3: The very short history of the S and R - Mathematics


R is the statistical environment. The idea was to make independent realization of S language concept which would differ from S-Plus in some details (for example, in the way it works with local and global variables).

Practically, R is not an imitation of S-Plus but the new “branch” in the family of S software. In 1990s, R was developing slowly, but when users finally realized its truly amazing opportunities (like the system of R extensions—packages, or libraries) and started to migrate from other statistical systems, R started to grow exponentially. Now, there are thousands of R packages, and R is used almost everywhere! Without any exaggeration, R is now the most important software tool for data analysis.


Srinivasa Ramanujan

After demonstrating an intuitive grasp of mathematics at a young age, Srinivasa Ramanujanꂾgan to develop his own theories and in 1911, he published his first paper in India. Two years later Ramanujan began a correspondence with British mathematician G. H. Hardy that resulted in a five-year-long mentorship for Ramanujan at Cambridge, where he published numerous papers on his work and received a B.S. for research. His early work focused on infinite series and integrals, which extended into the remainder of his career.ꂯter contracting tuberculosis, Ramanujan returned to India, where he died in 1920 at 32 years of age.


Qin Dynasty Construction

Though the beginning of the Great Wall of China can be traced to the fifth century B.C., many of the fortifications included in the wall date from hundreds of years earlier, when China was divided into a number of individual kingdoms during the so-called Warring States Period.

Around 220 B.C., Qin Shi Huang, the first emperor of a unified China under the Qin Dynasty, ordered that earlier fortifications between states be removed and a number of existing walls along the northern border be joined into a single system that would extend for more than 10,000 li (a li is about one-third of a mile) and protect China against attacks from the north.

Construction of the “Wan Li Chang Cheng,” or 10,000-Li-Long Wall, was one of the most ambitious building projects ever undertaken by any civilization. The famous Chinese general Meng Tian initially directed the project, and was said to have used a massive army of soldiers, convicts and commoners as workers.

Made mostly of earth and stone, the wall stretched from the China Sea port of Shanhaiguan over 3,000 miles west into Gansu province. In some strategic areas, sections of the wall overlapped for maximum security (including the Badaling stretch, north of Beijing, that was later restored during the Ming Dynasty).

From a base of 15 to 50 feet, the Great Wall rose some 15-30 feet high and was topped by ramparts 12 feet or higher guard towers were distributed at intervals along it.

Did you know? When Emperor Qin Shi Huang ordered construction of the Great Wall around 221 B.C., the labor force that built the wall was made up largely of soldiers and convicts. It is said that as many as 400,000 people died during the wall&aposs construction many of these workers were buried within the wall itself.


A Quick Puzzle to Test Your Problem Solving

We’ve chosen a rule that some sequences of three numbers obey — and some do not. Your job is to guess what the rule is.

We’ll start by telling you that the sequence 2, 4, 8 obeys the rule:

Now it’s your turn. Enter a number sequence in the boxes below, and we’ll tell you whether it satisfies the rule or not. You can test as many sequences as you want.

Enter your first sequence here:

When you think you know the rule, describe it in words below and then submit your answer. press this button. Make sure you’re right you won’t get a second chance.

Guess wrong

The answer was extremely basic. The rule was simply: Each number must be larger than the one before it. 5, 10, 20 satisfies the rule, as does 1, 2, 3 and -17, 14.6, 845. Children in kindergarten can understand this rule.

But most people start off with the incorrect assumption that if we’re asking them to solve a problem, it must be a somewhat tricky problem. They come up with a theory for what the answer is, like: Each number is double the previous number. And then they make a classic psychological mistake.

They don’t want to hear the answer “no.” In fact, it may not occur to them to ask a question that may yield a no.

Remarkably, 80 percent of people who have played this game so far have guessed the answer without first hearing a single no. A mere 7 percent heard at least three nos — even though there is no penalty or cost for being told no, save the small disappointment that every human being feels when hearing “no.”

It’s a lot more pleasant to hear “yes.” That, in a nutshell, is why so many people struggle with this problem.

Confirmation Bias

This disappointment is a version of what psychologists and economists call confirmation bias. Not only are people more likely to believe information that fits their pre-existing beliefs, but they’re also more likely to go looking for such information. This experiment is a version of one that the English psychologist Peter Cathcart Wason used in a seminal 1960 paper on confirmation bias. (He used the even simpler 2, 4 and 6, rather than our 2, 4 and 8.)

Most of us can quickly come up with other forms of confirmation bias — and yet the examples we prefer tend to be, themselves, examples of confirmation bias. If you’re politically liberal, maybe you’re thinking of the way that many conservatives ignore strong evidence of global warming and its consequences and instead glom onto weaker contrary evidence. Liberals are less likely to recall the many incorrect predictions over the decades, often strident and often from the left, that population growth would create widespread food shortages. It hasn’t.

This puzzle exposes a particular kind of confirmation bias that bedevils companies, governments and people every day: the internal yes-man (and yes-woman) tendency. We’re much more likely to think about positive situations than negative ones, about why something might go right than wrong and about questions to which the answer is yes, not no.

Sometimes, the reluctance to think negatively has nothing to do with political views or with a conscious fear of being told no. Often, people never even think about asking questions that would produce a negative answer when trying to solve a problem — like this one. They instead restrict the universe of possible questions to those that might potentially yield a “yes.”

Government Policy

In this exercise, the overwhelming majority of readers gravitated toward confirming their theory rather than trying to disprove it. A version of this same problem compromised the Obama administration’s and Federal Reserve’s (mostly successful) response to the financial crisis. They were too eager to find “green shoots” of economic recovery that would suggest that the answer to the big question in their minds was, just as they hoped and believed: “Yes, the crisis response is aggressive enough, and it’s working.” More damaging was the approach that President George W. Bush’s administration, and others, took toward trying to determine whether Iraq had weapons of mass destruction a decade ago — and how the Iraqi people would react to an invasion. Vice President Dick Cheney predicted in 2003, “We will, in fact, be greeted as liberators.”

Corporate America

Corporate America is full of more examples. Executives of Detroit’s Big Three didn’t spend enough time brainstorming in the 1970s and 1980s about how their theory of the car market might be wrong. Wall Street and the Fed made the same mistake during the dot-com and housing bubbles. To pick an example close to home, newspapers didn’t spend enough time challenging the assumption that classified advertisements would remain plentiful for decades.

One of the best-selling business books in history — about negotiation strategy — is “Getting to Yes.” But the more important advice for us may instead be to go out of our way to get to no. When you want to test a theory, don’t just look for examples that prove it. When you’re considering a plan, think in detail about how it might go wrong.

Some businesses have made this approach a formal part of their decision-making: Imagine our strategy has failed what are the most likely reasons it did? As Jason Zweig has written in The Wall Street Journal, “Gary Klein, a psychologist at Applied Research Associates, of Albuquerque, N.M., recommends imagining that you have looked into a crystal ball and have seen that your investment has gone bust.”

When you seek to disprove your idea, you sometimes end up proving it — and other times you can save yourself from making a big mistake. But you need to start by being willing to hear no. And even if you think that you are right, you need to make sure you’re asking questions that might actually produce an answer of no. If you still need to work on this trait, don’t worry: You’re only human.


Choose your category

THE LOVE ECLIPSE: LOVE IS NOT A FEELING, IT IS A CHOICE

Leo Felix is a young man whose life became miserable at the age of five. He grows up as a gambler… full of anger, grief, vengeance and regrets─ his life is in.

THE LOVE ECLIPSE: LOVE IS NOT A FEELING, IT IS A CHOICE

Leo Felix is a young man whose life became miserable at the age of five. He grows up as a gambler… full of anger, grief, vengeance and regrets─ his life is in darkness. He’s forced to attend the music school where he bumps into a girl and fall for her on first sight. He is not alone… there is a tough rival after the girl too. He is determined to win the girl’s heart for he believes love can get him out of darkness─ it once did. As he gets closer to the girl, his first and only lover who disappeared years ago reappears. He has to make a decision. What decision will he make? Will the decision take him to light or lead him deep into darkness?

Formats: PDF, Epub, Kindle, TXT

My Bodyguard

The forbidden is always irresistible. Being married to a powerful businessman, feared all-over Portland, Mia Kingston has a reputation to uphold. Her life .

My Bodyguard

The forbidden is always irresistible. Being married to a powerful businessman, feared all-over Portland, Mia Kingston has a reputation to uphold. Her life is filled with glamour from the outside, but quite a void she doesn't discern until she falls deeply into her desire for true love and freedom when she gets involved with the man she isn't supposed to. But what if the price to pay is too high? Will Mia succeed to keep her love and freedom?

Formats: PDF, Epub, Kindle, TXT

When I grow up!

In this story, Joanna will tell her friends, what she wants to be when she grows up and encourage them to also think about that!

When I grow up!

In this story, Joanna will tell her friends, what she wants to be when she grows up and encourage them to also think about that!

Formats: PDF, Epub, Kindle

The Coldest Summer

When Kira Jones finally decides to take a six-week summer vacation, her best and only friend, Samantha, drags her into a trip out of California. What awaits i.

The Coldest Summer

When Kira Jones finally decides to take a six-week summer vacation, her best and only friend, Samantha, drags her into a trip out of California. What awaits in the way is something Kira has never fathomed at all, as her life gets a serious turn. She meets a mysterious ranch owner whom her friend already has eyes for, and Kira finds herself drawn near him in a very strange way. What will win in the end between the power of love and friendship?

Formats: PDF, Epub, Kindle, TXT

The Secret Cave

A compelling time travelling romance novel. Shirley meets, falls in love and marries Jeffrey. Shirley however, has a secret will Jeffrey discover it? He becom.

The Secret Cave

A compelling time travelling romance novel. Shirley meets, falls in love and marries Jeffrey. Shirley however, has a secret will Jeffrey discover it? He becomes very suspicious of her strange behaviour and comings and goings. Thinking she is having an affair he follows her to The Secret Cave. This is an emotional and exciting Science fiction tale full of twists and turns.


A Brief History of Pi ( π )

Pi ( π ) has been known for almost 4000 years—but even if we calculated the number of seconds in those 4000 years and calculated π to that number of places, we would still only be approximating its actual value. Here’s a brief history of finding π .

The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave a value of pi = 3. One Babylonian tablet (ca. 1900–1680 BC) indicates a value of 3.125 for π , which is a closer approximation.

The Rhind Papyrus (ca.1650 BC) gives us insight into the mathematics of ancient Egypt. The Egyptians calculated the area of a circle by a formula that gave the approximate value of 3.1605 for π .

The first calculation of π was done by Archimedes of Syracuse (287–212 BC), one of the greatest mathematicians of the ancient world. Archimedes approximated the area of a circle by using the Pythagorean Theorem to find the areas of two regular polygons: the polygon inscribed within the circle and the polygon within which the circle was circumscribed. Since the actual area of the circle lies between the areas of the inscribed and circumscribed polygons, the areas of the polygons gave upper and lower bounds for the area of the circle. Archimedes knew that he had not found the value of π but only an approximation within those limits. In this way, Archimedes showed that π is between 3 1/7 and 3 10/71.

A similar approach was used by Zu Chongzhi (429–501), a brilliant Chinese mathematician and astronomer. Zu Chongzhi would not have been familiar with Archimedes’ method—but because his book has been lost, little is known of his work. He calculated the value of the ratio of the circumference of a circle to its diameter to be 355/113. To compute this accuracy for π , he must have started with an inscribed regular 24,576-gon and performed lengthy calculations involving hundreds of square roots carried out to 9 decimal places.

Mathematicians began using the Greek letter π in the 1700s. Introduced by William Jones in 1706, use of the symbol was popularized by Leonhard Euler, who adopted it in 1737.

An eighteenth-century French mathematician named Georges Buffon devised a way to calculate π based on probability. You can try it yourself at the Exploratorium's Pi Toss exhibit.

Shown: Thomas Degeorge (1786–1854), The Death of Archimedes (detail), 1815. Collection of the Musée d’Art Roger-Quilliot Museum [MARQ], City of Clermont-Ferrand, France.


Types of Learning Disabilities

Learning disabilities are due to genetic and/or neurobiological factors that alter brain functioning in a manner which affects one or more cognitive processes related to learning. These processing problems can interfere with learning basic skills such as reading, writing and/or math. They can also interfere with higher level skills such as organization, time planning, abstract reasoning, long or short term memory and attention. It is important to realize that learning disabilities can affect an individual’s life beyond academics and can impact relationships with family, friends and in the workplace.

Since difficulties with reading, writing and/or math are recognizable problems during the school years, the signs and symptoms of learning disabilities are most often diagnosed during that time. However, some individuals do not receive an evaluation until they are in post-secondary education or adults in the workforce. Other individuals with learning disabilities may never receive an evaluation and go through life, never knowing why they have difficulties with academics and why they may be having problems in their jobs or in relationships with family and friends.

Learning disabilities should not be confused with learning problems which are primarily the result of visual, hearing, or motor handicaps of intellectual disability of emotional disturbance or of environmental, cultural or economic disadvantages.

Generally speaking, people with learning disabilities are of average or above average intelligence. There often appears to be a gap between the individual’s potential and actual achievement. This is why learning disabilities are referred to as “hidden disabilities”: the person looks perfectly “normal” and seems to be a very bright and intelligent person, yet may be unable to demonstrate the skill level expected from someone of a similar age.

A learning disability cannot be cured or fixed it is a lifelong challenge. However, with appropriate support and intervention, people with learning disabilities can achieve success in school, at work, in relationships, and in the community.

In Federal law, under the Individuals with Disabilities Education Act (IDEA), the term is “specific learning disability,” one of 13 categories of disability under that law.

“Learning Disabilities” is an “umbrella” term describing a number of other, more specific learning disabilities, such as dyslexia and dysgraphia. Find the signs and symptoms of each, plus strategies to help below.


Understanding a Monte Carlo Simulation

When faced with significant uncertainty in the process of making a forecast or estimation, rather than just replacing the uncertain variable with a single average number, the Monte Carlo Simulation might prove to be a better solution by using multiple values.

Since business and finance are plagued by random variables, Monte Carlo simulations have a vast array of potential applications in these fields. They are used to estimate the probability of cost overruns in large projects and the likelihood that an asset price will move in a certain way.

Telecoms use them to assess network performance in different scenarios, helping them to optimize the network. Analysts use them to assess the risk that an entity will default, and to analyze derivatives such as options.

Insurers and oil well drillers also use them. Monte Carlo simulations have countless applications outside of business and finance, such as in meteorology, astronomy, and particle physics.


Online Courses

Our short online courses take place in a virtual learning environment. Most courses are 10 weeks in duration and they all run asynchronously &ndash they have no live-time meetings - so you do not have to be online at any specific time to take the course. You can access the course whenever it is convenient for you, from anywhere in the world.

Class sizes are kept small to maximise interaction between you and your classmates and tutor in the online forums. Students are able to take part in in-depth discussions and receive personalised tutor guidance and feedback.

Sample units from online courses are available to view from the course demonstration site.

Course credit

Coursework is an integral part of all our short online courses. Everyone enrolled will be expected to do coursework, but only those who have registered for credit will be awarded CATS points for completing work at the required standard.

Please refer to our CATS Points System guidelines to find out more about CATS points and how to register for credit.

Credit earned from our short online courses is transferable towards our undergraduate award programme, the Certificate of Higher Education.


Pi in pop culture

But wait—the obsession with pi isn’t just limited to mathematicians and scientists. Pi has a special place in popular culture, thanks to its prevalence in mathematical formulae and its mysterious nature. Even completely non-cerebral shows, books, and movies can’t help but mention the popular constant.

For example, pi gets mentioned in a scene from Twilight, in which vampire-boy Robert Pattinson recites the square root of pi (and on-the-ball Kristin Stewart quickly shuts him down).

The Simpsons is also pretty into pi (and math references in general). In one scene, two young girls at a school for the gifted play patty-cake and say “Cross my heart and hope to die, here’s the digits that make pi, 3. 1415926535897932384…” In another scene, a sign at the Springfield graveyard says “Come for the funeral, stay for the π.”

Albert the Intern contemplates pi.

Yep, whether you like it or not, pi is everywhere. Here are a few more places it’s popped up:

  • The main character in the award-winning novel (and 2012 film) Life of Pi nicknames himself after the constant.
  • A circular room in the Palais de la Découverte science museum in Paris is called the pi room. The room has 707 digits of pi inscribed on its wall (though there is an error beginning at the 528th digit, thanks to William Shanks’ erroneous calculations).
  • In an episode of Star Trek: The Original Series, Spock commands an evil computer to compute pi to the last digit—which it cannot do, of course, because, as Spock explains, “the value of pi is a transcendental figure without resolution.” is advertised as a scent that "embodies the confidence of genius."
  • Both MIT and the Georgia Institute of Technology have cheers that include “3.14159.”
  • Several other movies reference pi, including the 1966 Alfred Hitchcock film Torn Curtain, the 1995 Sandra Bullock thriller The Net, 1998 indie thriller Pi.

Finally, pi is perhaps most rampant in pop culture on March 14—Pi Day! On Pi Day, nerds, geeks, and mildly interested geometry students alike come together and wear pi-themed clothing, read pi-themed books, and watch pi-themed movies, all while eating pi-themed pie.

Just think of how excited everyone will get two years from now, when Pi Day falls on 3/14/15.

Correction, March 14, 2013: An earlier version of this story mistakenly stated that Archimedes' estimate for pi was 3.1485. His actual estimate calculated pi to be between 3.1408 and 3.14285. (If you average these two figures, you get an in-between point of 3.141851.) We regret the error.

Article originally published March 13, 2010 updated March 13, 2013.

Sarah is a freelance writer and editor based in Los Angeles. She has a love/hate relationship with social media and a bad habit of describing technology as "sexy."


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