EXPLORING AND DISCOVERING WITH MATH

Prediction is very difficult, especially if it’s about the future

- Niels Bohr

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JANUARY 2012, ISSUE 2

Ringing in 2012

Interested in writing for Abacus? Or do you just want to chat? Contact us at susie.r.taylor@gmail.com

In this Issue...

From the Editor:

by Susie Taylor

page 3

The Reluctant MathmaticianNew Years Resolutions

by Nina Nesseth

page 7

It’s Universal

Charles Babbage and the Difference

Engineby Linda Henneberg

page 9

Math on he Brain

by Jalyn Neysmith

page 12

Econ Recon

For Love or Money

by Josh Osika

page 4

Puzzling Puzzles & Other

Distractions

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Welcome to 2012! Another year, another 365 days of math and change. So far, my 2012 has been quite relaxing. I have been cleaning and organizing. Organizing my time, my room, and my multitudes of stuff. And as such, I have been spending a good amount of time thinking about categorization. Today my roommate and I spent the better part of the afternoon categorizing our movie collection. It was a slow day. We ended up with 20 categories ranging from “takes place in a school” movies to “saving the world” movies to “made pre-1995” movies. Now we have approximately 160 movies (I know, itʼs insane), so thatʼs an average of 8 movies in each category. If we were to get into the probabilities associated with this, we could have come up with thousands of different category arrangements. But weʼll save that for another day.

From my quick look into categorization (thank you, Wikipedia), I learned that there are 3 theories of categorization: classical categorization, conceptual clustering, and prototype theory. Now my movie categorization was far from classical. The classical view of categorization states that each category needs to be mutually exclusive - and let me tell you, there are a few movies in our collection that take place in a high school AND are made pre-1995. So thatʼs out. Iʼd say our movie collection fits best into conceptual clustering, which allows for that bit of overlap in the categorizations. Now aside from movies, categorization has many uses. We categorize numbers all the time: real numbers, rational numbers, primes, perfect squares, etc. We categorize time: hours, months, days, and years. It just helps us to make better sense of things. Having entered into our new category of 2012, it is time to look ahead and think back. What concepts

might overlap this year from our past? And what aspects from last year do we want to keep exclusive to last yearʼs category?

Best of luck in 2012!

Susie Taylor

About the Author:

Susie Taylor is a science communicator and lover of all things math. She works for Letʼs Talk Science,

supporting science outreach, and has a degrees in Biology and Mathematics and Science Communication. Susieʼs non- math hobbies include sewing, jogging, and finding new great music to listen to.

Thoughts from the editor

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Econ Recon By Josh Osika

For Love or MoneyI have a feeling one of my memories of 2011 which will be with me the longest is attending the U2 concert in July at the Rogers Centre in Toronto. I had held onto tickets for this concert for more than a year; originally scheduled for the summer before, the show – and the entire North American leg of the tour – was postponed by a year after Bono injured his back during rehearsals. The wait was worth it when I was able to find a spot right against the stage and be a part of the highest grossing tour of all time. In 2011, U2 sold out all 44 of their shows and earned nearly $300 millionz, and the U2 360° Tour grossed $736 million during its two year run. These numbers are massive, but still do not represent the true earning potential of the band. One

of the most basic concepts of economics is that of supply and demand equilibrium. In a perfectly competitive market supply and demand for a product

will adjust until the two are in equilibrium – as the price of a product increases the quantity demanded by sellers will also increase while the quantity demanded by consumers will decrease. In the market for a U2 concert this process does not occur. U2 are the only providers of this good,

other acts may put on similar shows but no one would say that a U2 concert and a Bon Jovi concert are perfect substitutes for each other.

Thus, the band holds a monopoly in the market for their own concerts and can maximize profits through increasing prices or increasing supply. The point at which profits are maximized could be determined through the use of derivatives if the market supply function is known.

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There are a multitude factors which can contribute to this function, including many which are difficult to quantify such as talent, but when these are estimated calculus can be used to determine the maximum level of profit. However, without going through all of that it can still be said that U2 was nowhere near their maximum earning potential with this tour. With every show selling out and scalpers able to sell a large number of tickets at highly inflated prices it is obvious that demand for the concerts was not nearly

met – if supply and demand were in equilibrium than each concert would sell out but there would be no additional demand for tickets amongst scalpers. So why

didnʼt U2 choose to make more money if the potential to do so was there?

Firstly, they were limited by the size of the venues. Throughout the tour they performed only in stadiums which held tens of thousands of people, allowing for more revenue than if they had done shows in arenas. However, in most areas these stadiums are arguably the largest venues available and did not have capacity which would equal the demand. U2 could have also decided to charge a higher

price for their tickets. This would cause demand to decrease, and if the right price is found demand and supply would come into equilibrium. All of the stadiums would be full but there would be no excess of demand. This practice would likely result in a much higher ticket price, which could incite anger and resentment amongst many fans who would then buy less of the bands products in the future.

Many factors must be taken into consideration that the supply and demand functions can quickly become quite complicated. But from the math, tour managers are able to make inferences and decisions for planning a concert tour. Despite the mantra of many economists that the goal of any business is to maximize profits, I am very happy U2 decided not to.

Econ Recon By Josh Osika

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Econ Recon By Josh Osika

Happy New Year and I hope you all have many amazing and memorable experiences in 2012, and that they donʼt cost you too much money.

The title of this article is a lyric from “Desire” by U2, ten points for you if you figured it out.

multitude factors which can contribute to this function, including many which are difficult to quantify such as talent, but when these are estimated calculus can be used to determine the maximum level of profit.

About JoshJoshua Osika lives in Sudbury, ON and has worked at Science North as a bluecoat engaging visitors in learning about science since 2004. He attends Laurentian University, where he has earned a Bachelorʼs of Science degree with a combined specialization in biomedical biology and psychology, as well as a graduate diploma in science communication and is currently studying economics. When not studying or writing for Abacus, Josh can usually be found reading, watching a movie, or out somewhere in Northern Ontario searching for the diner with the best pie.

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Letʼs face it - you donʼt have to be a statistician to realize that most New Yearʼs resolutions remain unaccomplished. The reasons for failing to eat healthier or to quit smoking may be varying and often unpredictable, so here are a few suggestions for making your New Yearʼs resolutions a little more attainable.

Keep it Simple

While it might be satisfying to write out a long, detailed list of everything you want to accomplish, having an overly complicated list of resolutions can actually reduce your likelihood of success. This can be expressed in its most basic form as a joint probability:

P(A ∩ B) = P(A) * P(B)

Most resolutions are independent events – one does not affect the outcome of the other. For example, if a person has resolved to

visit a new country in 2012, this will not dictate that he will also be successful at his other resolution of flossing his teeth every day. Letʼs say thereʼs a 40% chance that this person will travel in 2012, P(A) = 0.4, and an 80% that heʼll floss regularly, P(B) = 0.8. The probability of both events occurring then becomes

P(A ∩ B) = (0.4) * (0.8)P(A ∩ B) = 0.32

This example only shows the likelihood of two resolutions being accomplished. The more resolutions that you add to your list, the less likely it becomes that all of them will be achieved.

Be Specific

Another way to make your resolutions more manageable is to tone down the ambiguity. A quick web search of “2012 resolutions” turns up a

mountain of blog posts of resolution lists, most of them announcing sweeping resolutions like “lose weight”, “be fit”, and “travel”. These are great starting points, but from there you have to ask yourself questions to whittle down your resolution: how much weight do you want to lose, and by when? In what ways do you plan to become more fit? Where specifically do you want to travel? Once you understand that what you really mean by “lose weight” is that you want to lose 5 lbs by April, or that by “travel” is that you want to fly to Spain on that two-week break you have in July, your list becomes a set of discrete goals for which you can start planning and taking steps.

Be Realistic

The most important things to understand when hashing out a list of

The Reluctant MathematicianBy Nina Nesseth

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New Years Resolutions

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resolutions are your limitations. You may want more than anything to run the Boston marathon this year, but if, say, you donʼt have ample time to commit to training, or you lack the funds to travel to Boston in the first place, then this goal just might not be feasible. It is up to you to recognize your own limitations, whether they present in the form of health and physical limitations, amount of time, funds, or motivations. Sometimes, these limitations pop up unexpectedly, but for the most part, you can predict your success based on your knowledge of yourself.

So, hereʼs to a new year, a new start, and new goals to achieve. Maybe for 2012, with the help of these guidelines and some well-planned resolutions, the odds will be in our favour.

About Nina: Nina Nesseth is a graduate of the Science Communication program at Laurentian University and a reporter for The Lambda. In her spare time she does Alphabet Sudoku and plays the ukulele.

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The Reluctant MathematicianBy Nina Nesseth

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Before the age of electric calculators and computers, tables of trigonometric and logarithmic functions (for use in astronomy, physics, and mathematics) were calculated by hand—a slow and error-prone process. In the 1800s, Charles Babbage sought to reduce or eliminate the errors in these tables by performing the calculations mechanically.

Logarithmic and trigonometric functions can be approximated by higher order polynomial functions. The Difference Engine actually calculated the values of polynomials which approximate other functions.

Charles Babbage was an English inventor, philosopher, mathematician, and mechanical engineer who lived from December 26, 1791 to October 18, 1871. Among many other accomplishments, he is credited as having invented the first programmable computer. As a child Babbage was often ill, so he had a lot of private tutors. His parents said that his “brain was not to be taxed too much,” and so Babbage felt that “this great idleness may have led to some of my childish reasonings.”

How the Difference Engine Works Babbageʼs Difference Engine uses the method of differences for calculating the values of various polynomial functions.

Mechanically, the Difference Engine works by turning a hand crank. The machine is roughly eight feet high, and weighs fifteen tons. It has thirty-one figure wheels in eight columns, which means it can store eight numbers of thirty-one digits each. One iteration happens for every four turns of the crank, and each time the crank is turned one of the following steps is performed:

1.Even numbered columns are added to odd numbered columns by an interior arm.

2.Carry propagation is performed by increasing the next highest wheel by

Difference Engine built in 1991 by the Science Museum of London to celebrate the 200th anniversary of Babbage's birth.

It’s Universal By Linda Henneberg

Charles Babbage and the Difference Engine

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It’s Universal By Linda Henneberg

one, and dropping the current wheel to its original digit.

3.Similar to step one, odd numbered columns are added to even numbered columns.

4.Carry propagation is performed again.

The Fate of the Difference EngineThe project was originally funded by

the British government, but they withdrew when Babbage kept asking for more money, while making no appreciable progress of building the Difference Engine. So Babbage started making plans for a Difference Engine #2, which would be more efficient and perform calculations faster, but he had the same funding complications as he did with Difference Engine #1.

Finally, Babbage started on plans for his next mechanical computer, the Analytical Engine. He died, however, before completing the plans for the Analytical Engine, and none of his machines were built in his lifetime.

In 1991 the London Science Museum constructed a working Difference Engine #2, built to the tolerances available in the 19th century, which successfully calculated functions out to thirty-one digits.

Other AccomplishmentsWhile in college in 1812, Babbage,

along with John Herschel, George Peacock, and others, founded the Analytical Society, whose mission was to promote the standardized use of Leibnizian calculus over Newtonian calculus. He also received

the Gold Medal of the Royal Astronomical Society in 1824 for his work on the Difference Engine. He helped found the Astronomical Society and the Statistical Society, and from 1828 to 1839 was the Lucasian Professor of Mathematics at Cambridge University, the same position held by Isaac Newton and Stephen Hawking..

Charles Babbage contributed to the field of cryptography by breaking Vigenereʼs autokey cipher, which until that time was known as the “undecipherable cipher.” In 1838 he invented both the cow-catcher, to remove obstacles from in front of trains, and the dynamometer, which measured various aspects of the trainʼs performance.

He was also the first to articulate what is now known as the Babbage principle, which is the principle of division of labor between workers that are highly skilled and highly paid, and workers that have fewer skills and are paid less. This idea was criticized by Karl Marx.

A Few Eccentricities...In 1857 Babbage counted all the

broken panes of glass in the windows of a factory and published a paper titled “Table of the Relative Frequency of the Causes of Breakage of Plate Glass Windows.” Of 464 broken window panes, fourteen were caused by “drunken men, women, and boys.”

He also contacted the Poet Laureate of Britain at the time, Alfred Lord Tennyson,

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to correct a statement in one of Tennysonʼs poems. In his poem titled “The Vision of Sin,” one verse reads, “Every moment dies a man/Every moment one is born.” Babbage wrote to Tennyson to say that since the birth rate exceeds the death rate, the verse should actually read “Every moment dies a man/Every moment 1 1/16 is born.” He followed this up by saying, “Strictly speaking, the actual figure is so long I cannot get it into a line, but I believe the figure 1 1/16 will be sufficiently accurate for poetry.”

About LindaI recently got a Bachelor's Degree in Physics and Astronomy, and a Graduate Diploma in Science Communication. Despite loving math, I'm terrible at it, which is only more incentive to

learn more about it. I was also recently doing an internship at CERN, the European Center for Nuclear Research. It was a great opportunity, and I got to learn so much more about particle physics, which is one of my favorite things. In my spare time I like to go swing dancing or read a good book.

It’s Universal By Linda Henneberg

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It's the new year - 2012 - and one of my resolutions is to not shop ever again. That's going to be a hard resolution to keep (so what else is new), but after being bombarded with sales for two months I feel I need a break from buying. I found myself in the United States this holiday season, so first there was Black Friday, then Christmas sales, then post-Christmas sales, then New Year sales! Yikes! And to top it off, most of these sales aren't even very good deals. For example, I was once in a grocery store and saw two bags of those delicious President's Choice chocolate chip cookies. You know the ones I'm talking about, the ones that taste like ambrosia. There were two sizes. You could buy a

350g pack for $1.99, or a 700g pack for $2.99. This particular time, the 350g packs were on sale at 2 for $3.50. It's still cheaper to buy one big pack than two small packs, but of course the small packs were sold out. Why do people fall for these tricks? Our brains see numbers in a whole new light when they're on a price tag instead of on a homework assignment. Numbers in prices have their own secret language that most of the time only our subconscious brains pick up on. Here's a short pricing dictionary:

9sThis is the classic sales gimmick. Ending a price with the number nine does two things. First, our brains remember a price before

we've finished reading all the number. It's called "left-digit effect". When we see $34.99 we remember it as thirty four dollars. The only 9 that's a no-no is ".09". That's not a deal, that's almost ten cents more expensive!Nines have also been grilled into our brains as sale numbers. $39 is way cheaper than $40, right?

8sWalmart ends most of its prices in the number eight to show that it's just that little bit cheaper than everyone else's nines.

7s and 4sThese numbers are especially tricky. Scientists have found that the sound of a number influences our preconceived notions of how big or small it is.

Math on the brain By Jalyn Neysmith The Most Wonderful Time of the Year... to shop

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Numbers with vowel sounds along with soft consonants like s, f, and z are perceived as smaller than numbers with hard sounds, like ten. So nine, seven, five, four, and one all have the effect of sounding cheaper. People also seem to think that prices with fours and sevens in them (not necessarily just the decimal points) are precise prices. Clearly the price has been thought out and is reasonable.

5sNine's less popular cousin, the five also triggers our brains to say, "good deal! On sale!" Five is also a common threshold number. When you are planning on buying an item that has a variable price like a car, you usually decide how much you're willing to pay in advance.

Fives and tens are often chosen by people to demarcate the boundaries of their price range. Moving from $3,400 to $3,500 doesn't seem like as big a step as moving from $3,500 to $3,600, because that 3,500 was a psychological threshold.

0sFancy brand items often end in a zero. A precise price ending in a zero indicates that the seller is confident that you're going to pay the price they set. Most items ending in a zero are brands that offer more than just the product. When you purchase the product, you also get let's say, social esteem. Prices with no decimal points are saying "this is my price, and you're going to buy me anyway".

So with your inside knowledge of numbers you'll be ready to navigate

the next round of sales. And since that's Valentine's Day, here's one more trick to watch out for. When you take your date out for a fancy dinner, look for the priciest item on the menu. Is there a ridiculous seventy dollar steak? That's there as an anchoring price. Once you've seen a seventy dollar steak, the forty dollar roast chicken seems practically reasonable. Bon appétit!

Learn more at:Science DailyCBS NewsNPR

Math on the brain By Jalyn Neysmith

About Jalyn

Jalyn is from Calgary and has degrees in science communication and archaeology. Her favourite math subject is definitely calculus!

She loves to travel, her favourite destinations so far being Israel and Jordan. When not planning her next trip, she can be found running, snowboarding, or playing the fiddle.

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Worth a Laugh?

Q: What does the B in Benoit B. Mandelbrot stand for?

A: Benoit B. Mandelbrot.

_______________________________

All the numbers went to a party and numbers being what they are, all the evens stayed around each other and all the odds did the same and neither group interacted with each other. Whilst two was chatting to four he noticed zero was on his own in the corner and suggested to four that because zero is sort of even he should be encouraged to mix with even numbers - four agreed. So off went two to invite zero into their little group. "Would you like to join our little group" enquired two, to which zero replied "I have nothing to add!"

Challenge your Brain!Instead of a puzzle, I’m going to share a cool website and video with you. My friend shared this for me and it is a great way to learn math and, well, many many other things. It’s called Khan Academy.

And here is the TedTalk video by the creator, Salmon Khan.

Solution from last issue:

Puzzling Puzzles & Other Distractions

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Puzzle Challenge SolutionsMirror Challenge (from Issue 1) Submitted by Nathaniel Tanti

Puzzle: You shine a light ray incident on a mirror in the shape of a V trough (see picture to left). Can you figure out how many times light will hit the surface of the mirror before coming out of the trough?

Solution:The most obvious, but difficult method is to use the law of reflection to figure out in what direction light will leave the surface at each point of incidence. You repeat this method until light escapes the cone, and the number of reflections is your answer. The other method is to use a result of symmetry, illustrated to the right.

So, instead of drawing a reflected light ray at each point of incidence, the V-trough is instead flipped about the plane of reflection and the light ray continues on a straight path. This is

repeated until the straight light ray no longer intersects the V-trough. As highlighted in blue in the diagram bellow, the number of reflections in this case is four and all I used was paint to draw the diagram!

Puzzling Puzzles & Other Distractions

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