Abacus

Abacus

October 2011, Issue 1

Quarterly

2

Inside this Issue...

From the Editor:

Welcome to Abacus pg 3

Looking Back on…

Blaise Pascal pg 6

Econ Recon

The Recession Pg 9

Just for Fun

Puzzles, comics, and jokes

pg 11

The Reluctant Mathematician

Dyscalculia pg 4

Shaping the Universe pg 10

Imaginary Numbers in the Real

World pg 7

Math on the Brain

Seeing Red pg 5

3

Mathematics, like so much else, is ever changing and growing. Sometimes I feel like we have been finding ways to make mathematics easier, but then you realize that many realms of math are becoming ever more complex. And always - and this is why I love math - you can always find a new way to look at a problem and a new way to think your way to the answer. Welcome to the first issue of Abacus, a quarterly magazine looking at mathematics - what we find fun, interesting, and thought-provoking. And here’s what has been ‘provoking my thoughts’ lately - the Abacus. Having named this magazine Abacus, I got to thinking, ‘how exactly did this toy-like contraption work? It was the first calculator pretty much, right?”’ So I took to the library. That’s right, I photocopied pages from an encyclopedia - no googling at all. And it turns out I had no idea how an abacus worked. An abacus, popular with the Babylonians of 5000 years ago, is a rectangular tool, usually made out of wood or plastic. It has at least 9 vertical bars and moveable beads on each; 1 horizontal bar crosses all the bars unevenly. The vertical bars each represent a decimal place (ones, tens, hundreds, etc). Using this simple looking tool you can add, subtract, multiply, and divide. Basic calculator, right? Not exactly. To use an abacus you need to have a good handle on the math. It’s an aid, but you still need to know your stuff. Whereas today’s calculators just need you to read the numbers in order to add, subtract, multiply, and divide. Adding and subtracting an abacus moves in the opposite direction as our conventional right to left written method. So let’s try this out - how about 342+765:

And there you have it. A nice new way to look at something you’ve been doing for years. Or, more accurately, an old way to look at it.

About the Author: Susie Taylor is a science communicator and lover of all things math. She works for Let’s Talk Science, supporting science outreach, and has a degrees in Biology and Mathematics and Science Communication. Susie’s non-math hobbies include sewing, jogging, and finding new great music to listen to.

Thoughts from the Editor The Abacus By Susie Taylor

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I think I should just jump right in and say it: I am not a “math-y” person. I play Sudoku puzzles with letters of the alphabet instead of with numbers; I rejoice when the debit machines at restaurants have an option that let’s you choose tip amounts by percent, so I don’t have to calculate what a reasonable tip might be; When I send people messages I will often write “Meet me at four o’clock”, because I find it easier than typing out the numbers. While I often joke that I have a mental block against numbers in general, I have to admit that I did manage to slog through university calculus and algebra courses relatively unscathed. It did make me wonder, however, whether others who claim to be “un-math-y” like myself might actually have a sort of brain-block against numbers. This wondering brings me to a conversation I had once with a coworker. She told me that she had always found the topic of mathematics interesting, and had even read math-related books, yet her history with numbers was rocky at best. In high school math class, tests would be handed back to her with notes that the steps had all been executed correctly, but that the numbers did not make sense. Later, she had difficulties at work when entering numbers into a computer – results would be incorrect even when she was carefully following instructions. Teachers and employers suggested that it must have simply been carelessness that led to these wrong number values, until finally a co-worker suggested that she might not be reading the numbers correctly. “Math Dyslexia” as described by my co-worker, is a form of dyscalculia – a math learning disability. It is much more common that you might think. In fact, it is estimated that 3 – 6% of the population may be affected by some form of dyscalculia. As with other learning disabilities such as dyslexia (difficulty reading), dyscalculia can come in many forms: some people, like my co-worker, transpose numbers (for example, 56 becomes 65); others have difficulty manipulating numbers or remembering basic math facts. Everyday tasks, such as counting change or estimating measurements, become much more challenging with specific forms of dyscalculia. The problem lies in the parietal lobe – the part of the brain responsible for body awareness and integrating information from our various senses. For this reason, people with dyscalculia often have other difficulties, including dyslexia, or difficulty differentiating left from right. Fortunately, there are tools to help dyscalculics: teaching

aids and software designed to assist children and adults with math learning disabilities do exist. The difficulty lies in identifying those who do need assistance. Unlike dyslexia and dysgraphia (difficulty writing), dyscalculia is not always as obvious. Math has earned a reputation as a difficult subject, and so it has been accepted and almost expected that students will struggle with numbers. For this reason, math learning disabilities are often over-looked, such as in the case of my co-worker. She has managed to work around her math dyslexia with memorizations and by learning to break down larger numbers into smaller, more manageable parts; however, she admits, numbers still sometimes manage to trip her up. I guess, though this gives me little excuse. If my coworker can overcome her ‘mental block’ for math, so can I.

About the Reluctant Mathematician: 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.

Reflections of a Reluctant Mathematician Dyscalculia By Nina Nesseth

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Does math make you see red? Literally? Metaphorically maybe, but not literally. While a lot of people find math difficult or stressful, a small portion of the population actually see red when they look at numbers. Not because numbers make them angry, but because seeing a number triggers a chain reaction in their brain that also makes them see a colour. About 4% of the population have brains that have extra wiring connecting different sensory pathways that are not usually linked. The condition is called synaesthesia, and comes from the Greek words meaning "senses" and "together", literally meaning that you experience more than one sense at the same time, but in response to just one stimulant. There are dozens of combinations of synaesthesia that can link any number of senses, and making people see colours when they hear sounds, taste spoken words, or even imbue numbers and letters with personalities. The type of synaesthesia we're interested in here are Grapheme-Colour Synaesthesia (seeing numbers and letters as having colour) and Numerical- Synaesthesia (seeing numbers and letters as having a personality). It seems that synaesthesia can be congenital or occur through conditioning (remember those coloured number and letter fridge magnets?) But in either case, studies have shown that most synesthetes often have the same colour associations. This is especially true for letters; 'A' is usually red, 'O' is black or white. This cross-sensory thinking is strongly associated with creativity, and synaesthesia is eight times more common in artistic professions than the rest of the population. But number synaesthesia also has a big effect on math and numerical understanding. While synesthetes usually perform extremely well in tests of memorization, it seems that just like the rest of us, a majority can find math extremely difficult. Algebra especially can become confusing when the coloured letters and numbers are mixed together. For people who give their numbers a personality, some equations simply do not make sense. One synesthete felt that a preppy pink 2 could not be a negative number; others cannot multiply numbers that don't "get along". Even in basic math things can get tricky. When adding a yellow number with a blue number, they should equal a green number. But what if they don't? Synesthetes don't choose the colours of their numbers – they see them spontaneously – so numerical equations don't necessarily agree with colour combinations. But plenty of synesthetes find that these same colours and personalities give them an edge at math, and use their unique perceptions to excel in the scientific world. Teachers go by the rule that learning requires you to take in information in more than one way – usually reading it, hearing it, and doing it. Synesthetes have the advantage

that they are learning in at least two ways all the time. They see a number and a colour, and that can be a great tool for math. The late Richard Feynman, a Novel Prize-winning physicist, had synaesthesia. He was also aware that his perception of equations was pretty unique. "When I see equations," says Feynman, "I see the letters in colours – I don't know why. As I'm talking, I see vague pictures of Bessel functions from Jahnke and Emde's book, with light-tan j's, slightly violet-bluish n's, and dark brown x's flying around. And I wonder what the hell it must look like to the students."

Want to learn more? http://mindhacks.com/2005/03/21/test-your-synaesthesia/ http://synesthete.org/ http://www.imprint.co.uk/pdf/R_H-follow-up.pdf

About the Author: Jalyn Neysmith 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.

Math on the Brain Seeing Red By Jalyn Naysmith

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Blaise Pascal (1623-1662)

Nationality: French Reputation: the “greatest might-have-been in the history of mathematics” -C.Boyer Put his stamp on: Pascal’s Triangle; Pascal’s Law; Pascal’s Wager (that it is logically prudent to believe in God); also the Pascal Math Contest from Waterloo, ON Style: Prosaic. Seems to not have liked using mathematical symbols at all as his treatises were written out in words.

Hot Topics: conics, probability, proof by induction (theology) Odd Invention: the Pascaline (humble, no?), a futuristic calculating machine thought by some to be the first computer. Was perhaps spurred on to this invention by his tax-man father. Fav. Pen Pals: Fermat and Chevalier de Méré Looks: 6/10 - great whimsical smile, but why bother with comb-over bangs? Also, was “sickly” from age 18 through his death by undetermined malady at 39 Contribution: 6/10 - flashes of brilliance amid a largely un-mathematical career Romance: 1/10 - all talk; lots about ‘heart’ in his theological writings - and no proof Chutspa: 2/10 - attended meetings with the big boys from age 14, wrote a paper at 16, but didn’t do much to ruffle any feathers About the Author: Jessi Linn Davies is a drama teacher and math tutor in Toronto, Ontario. Loves: proof by induction, number theory, and long division. Nerd Crush: Leibniz. Not a fan of: graphing, pencil-nib erasers, or Schrodinger’s cat.

Looking back on… Blaise Pascal By Jessi Linn Davies

When you delve into the popular history of mathematics, a lot of wild characters pop up. There is a clear stereotype of the genius savant, locked in his windowless room, hair wild, papers flying… As a math and drama major in university, I was fascinated by the vastly differing personalities I encountered in the two different faculty buildings between which I split my time. There will be plenty of time to discuss those brilliantly idiosyncratic folk I met throughout my undergrad, but instead, let’s start with one of the best-known mathematicians from our textbooks.

“Since we cannot know all that there is to be known about any-thing, we ought to know a little about everything.” - Blaise Pascal (This may be the reasoning behind his on-again-off-again relationship with mathematics.)

Great Story: Blaise had given up mathematics to pursue his theological work, but one evening he had a toothache during which he began to think about math. His toothache disappeared in what he was certain was God’s blessing upon his thought endeavours. He returned to math for a blazing week before giving it up again.

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Complex Numbers Imaginary numbers in the real world By Brokk Toggerson

We all know that two times two, is four.

And that 22 is 4.

Conversely, we say that the square-root of

four, √4, is two.

Have you ever wondered about √(−4)?

Clearly, it is not 2, as 2 × 2 = 4 not −4.

Similarly, the square-root of negative four is

not negative two as

(−2) × (−2) = 4 (this, by the way, is why

the √4 actually has two answers, +2 and

−2).

Now, if you are confused by this, don’t

worry. You are not alone. The entire

mathematical world was confused by the √

(−1) until the 18th century.

Eventually, mathematicians came up with a

solution. They defined the symbol

i (j to some engineers) to represent √ (−1).

One of the properties of all square roots is

that √AB = √A√B. So, finally we have an

answer to our original question:

√(-4) = √(−1)√4

= 2i.

This answer may feel unsatisfying. We

simply made up a symbol to represent the

problem. So what is the value of i? The

number i is completely different from the

familiar, “real numbers”: ..., −1, −2, 0, 1,

2, .... In fact, i is so different, that the great

mathematician and philosopher René

Descartes even called it “imaginary.”

For the moment, let’s just accept that i = √

(−1) and explore the implications.

One of the first things we learned in math

class was how to add real numbers.

Therefore, we should start by learning out

how to add imaginary numbers. Addi-

tion of two imaginary numbers is pretty

straight forward, for example

i + i = 2i.

However, if we try to add a real number

and an imaginary number, 6 + 2i, we run

into a problem. As I said before, i is a

number completely different from any of

the real numbers we are familiar with. So 6

+ 2i is just that, 6 + 2i. There is noth-

ing more we can do. Such combinations of

real and imaginary numbers are called

“complex”. The 6 is called the “real part”

and 2i the “imaginary part.” If we want to

add two complex numbers, say (6 + 2i) +

(4 + 8i) we just add the real components

and the imaginary components separately:

(6+4) +(2i +8i) = 10+11i.

When we ignore i = √(−1), we can happily

do addition, division, trigonometry and

so forth, and we always get one of those

real numbers of the type we grew up with.

When we include i, it seems like we have

taken a sudden sharp left-turn from our

usual thinking about mathematics. Luckily,

this “left-turn” feeling is a useful way

to visualize complex numbers! We can think

of a complex number, 2 + 2i√3 for

example, as existing in a plane, such as the

one shown in figure 1. The “real part”

of the number, 2, is represented by its

position along the horizontal line (its

longitude, to borrow from geography). The

“imaginary part”, 2i√3 is given by its

position on the vertical line (its latitude). There is also another way we could

describe the position of 2 + 2i√3. A way

where we

could specify both how far it is from the

center of the plane and also its angle

with respect to some other line; for

example, the line describing the positive

real

numbers. Using the Pythagorean Theorem,

a2 + b2 = c2, we can calculate the

distance from the center of the plane to 2 +

2i√3:

√[22 + (2√3)2]

= √16

= 4

Measuring the angle of the pink arrow in

figure 2 with respect to the positive real

numbers we get 60 . We have now shown

that an equally valid way to describe

2 + 2i√3 is to say that the distance from

the center of the plane, which we will

call r, is 4, and that the angle with respect

to the positive real numbers, which we

will call θ, is 60 .

We can further show the equality of 2 +

2i√3 and r = 4, θ = 60 by using a bit

of trigonometry.

The real part, 2, is equal to 4 cos(60◦ ), and

the imaginary part, 2i√3 = 4i sin(60◦ ). A

little bit of Taylor series magic from calculus

class can show us that cos(θ) + i sin(θ) can

be rewritten in terms of that seemingly

Figure 1. Representing a complex number graphically in

so-called “rectangular

coordinates”

Figure 2: Representing a complex number graphically

in so-called “polar coordinates”

8

random number e = 2.71828.. ., eiθ. Using these Taylor series, we can

represent 2 + 2i√3 as 4ei60 .

Think for a minute about cos(θ) + isin(θ) = eiθ,

Euler’s Theorem. In particular, if

θ = 180◦ , then

ei180 = cos(180 ) + isin(180 ) = −1.

Now if you replace 180 by the equivalent π

radians , then you get one of my favorite formulas,

eiπ = −1. In this simple formula, e, π, and i are all

connected in interesting way.

What happens if we raise e to a complex number

using the real variables x and y: x + iy? If you recall

some properties of exponents, ex+iy = ex eiy . Think of a line such as the

one shown in figure 3 where x is fixed to 2. Now, every point on this line

is 2 + iy where y goes from negative infinity to infinity. For the

points on this line, e2+iy = e2 eiy . Think back to how we represented

2 + 2i√3 as 4ei60 . Four was the distance from the center and 60 was the

angle made with respect to the positive real numbers. Now returning back

to e2 eiy , e2 is just some real number and represents some fixed distance

from the center of the plane.

The variable y occupies the same position as the angle, so each y along

the line corresponds to a different angle around a circle! Since the number

π characterizes a circle, we see the connection between e, i, and π. The

number e turns a line in the complex plane (i) into a circle (π).

Potentially more interesting than the connection between e, i, and π is the

fact there is a loss of information when we go from the vertical line 2 + yi

to the circle of radius e2. Each point on the circle does not correspond to a

unique point on the line. To illustrate, consider the point 2 + i(π/2): This

will correspond to a point on our circle π/2 radians (or 90 ) from the

positive real numbers. The point 2 + 5iπ/2 would also correspond to the

same point (since 5π/2 = π/2 + 2π or 450 = 90 + 360 ). We have simply

gone once more around the circle, ending up back in the same position. If

you think about it, each point on the circle corresponds to an infinite

number of points on the line. This multiplicity corresponds to an

unknown number of complete loops around the circle. Such

ambiguity is a common feature of functions of complex

numbers.

Now, this may all seem like philosophical rambling without real

world application. What possible application could imaginary

numbers have in the real world? Turns out, quite a bit.

The most common application is in describing systems that

oscillate, such as water waves on the ocean or the electrical

current in your home. The solutions to such problems often

involve imaginary numbers. A more explicit example is the

Schrodinger equation that can be used to describe how

electrons arrange themselves in atoms. By using complex

numbers we have been able to better understand a lot about

our world. Turns out that imagination really can go a long way.

Footnotes:

If you are familiar with Taylor series, the proof is pretty straight forward. Recall that the

Taylor series for ex is given by:

Mathematicians and physicists prefer to express angles in terms of “radians” rather than degrees. Basically radians are the fractional distance around the circle times 2π. Thus, one entire loop is 2π radians. Similarly 180 is one-half a circle so radians,

and

45 = Radians are nice, as they do not have a unit.

Figure 3: The transformation from a line into a circle. Note, how the red and the

orange points on the line map to the same point on the circle.

About the Author: Brokk Toggerson is working on the (hopefully!) final year of his Ph.D. in experimental particle physics from University of California, Irvine. His research involves searching for signs of an extension to the Standard Model of particle physics called supersymmetry at the ATLAS experiment on CERN's Large Hadron Collider. In the limited time he has that is not spent on finishing his Ph.D. he can be found dancing, or trying to find anything cheap in Geneva, Switzerland.

9

Over the past few years the state of the economy has consistently been one of the top headlines in the news. First, there was the 2008 collapse of the housing market and banking system in the US, which led to a recession experienced by many countries worldwide. In an attempt to stem this crisis, trillions of dollars were spent across the globe in the form of bailout funds and economic stimulus packages. And for some time this strategy seemed to work. But now with Greece teetering on the edge of economic collapse, many other European nations close to the brink, and the growing debt problem in America, many have come to the conclusion that another recession is inevitable. But what exactly is a recession? And how do these international problems affect the Canadian economy? Simply put, a recession occurs when the economy shrinks for two successive quarters. The expansion and contraction of the economy is measured in real gross domestic product, or real GDP. If for two successive quarters the GDP was calculated to be lower than the previous quarter, Canada would be in a recession. This is where the math comes in. The GDP of Canada is the value of all final goods and services produced in Canada during a given time period (it is usually measured quarterly and annually). Statistics Canada calculates GDP each quarter using the expenditure method. This involves summing personal expenditures on goods and services (C), business investment (I), government expenditures on good and services (G), and net exports of goods and services (or the value of exports minus the value of imports) (X-M), and deducting the value of the statistical discrepancy (S). For the second quarter of 2011 GDP was calculated like this with the values in millions of Canadian dollars at current prices, (data from Statistics Canada): GDP = C + I + G + (X-M) - S = 972,980 + 327,568 + 439,464 + (515,792 – 553,492) – 1,296 = 1,701,016 So, in the second quarter of 2011 the Canadian GDP was approximately $1.701 trillion. And in the first quarter of 2011 the Canadian GDP was $1.694 trillion. This is an upward trend, and so Canada couldn’t possibly be on the brink of another recession, right? Wrong. Unfortunately. when performing these calculations, inflation must also be taken into account. Inflation means that a dollar today has a different purchasing power than it did in the past. In Canada we currently use 2002 as our base year for the value of the dollar in these calculations. Once the GDP is converted to 2002 dollars the first quarter GDP becomes $1.350 trillion, and for the second quarter GDP is $1.348 trillion. The percentage change now becomes:

[($1.348 trillion – $1.350 trillion) / $1.350 trillion] x 100% = -0.1% And so we can see that Canada’s GDP has in fact shrunk from the first quarter of 2011 to the second quarter. This means that spending on final goods and services produced in Canada has gone down. Our businesses are selling less than they were previously. If this trend continues throughout the third quarter and the value of real GDP drops again Canada will have entered another recession. Looking at the state of other countries, this seems likely to occur. Exports to other nations makes up 30% of the Canadian GDP, with the United States being our largest trading partner. With their own economy buckling under the weight of government debt and citizens being encouraged to buy American to help bolster their own GDP, Canadian exports to the US will likely continue to fall. On top of this, the uncertainty over the economic fate of Greece, and the Eurozone as a whole, will lead to decreased spending around the world. Canada is well prepared to weather the coming storm,

and may even show slight growth this quarter, avoiding a

recession for the time being. But unless the world takes a

drastic turn another recession in the near future seems

likely, the numbers just don’t add up any other way.

About the author: Joshua 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 Bachelor’s 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.

Econ Recon

The Recession By Josh Osika

10

The universe. Think about it for a minute. If you were to step back from the universe and look at it from an outside per-spective, what would it look like? That is a question that many people have tried to answer. NASA has sent up balloons and satellites, such as The Wilkinson Microwave Anisotropy Probe (WMAP) and the Cosmic Background Explorer (COBE) to answer this, and other, questions. But, of course, it's impossible to get outside the universe to see what it looks like. So scien-tists have used the data from these missions, along with some intense mathematica fiddling, to determine some possibilities for the overall shape of the universe. Geometry behaves differently depending on how the uni-verse is shaped. In our modern geometry, all the angles of a triangle add up to 180 degrees. No matter how you draw a triangle, all of its angles must add up to 180. It is one of the core tenets of math in flat space. In other universes, however, this may not be the case. At small levels, the differences in how geometry behaves is es-sentially the same from universe to universe. Keep in mind, even the whole galaxy is considered small when compared to the entire universe. One possibility is that the universe is spherical. This is called a “positively curved” space. If we were in a posi-tively curved universe you could start flying your space ship in any direction and eventually end up back where you started. The strange thing about this type of uni-verse, geometrically, is that if you were to draw a trian-gle on the universe, the angles will add up to more than 180 degrees. But, like mentioned earlier, it also depends on scale. If you draw a tiny triangle on a piece of paper in front of you then add up the angles, you will see that they are only imperceptibly larger than 180. But if you draw a very large triangle that takes up most of the uni-verse, the angles will add up to much more than 180 de-grees. It's even possible to draw a triangle where the angles add up to almost 360 degrees! Another interesting possibility for the curvature of the universe is that it is negatively curved. In this case, the universe would be shaped somewhat like a saddle. Just like how in the positively curved universe the angles in a triangle add up to more than 180 degrees, the angles in a negatively curved universe add up to less than 180 de-grees. Once again, it depends on scale. The larger you draw your triangle, the more noticeable the difference is between these “curved” triangles and “flat” triangles.

A flat universe has no curvature. The angles of triangles in this type of universe, no matter how large or small they were drawn, will always add up to 180 degrees. Af-ter spending many years analyzing data and going through very complex mathematical computer modeling, cosmologists have concluded that this is probably the type of universe we live in. The universe has some slight bumpiness due to areas of large gravity, but overall it is flat. It may not be as interesting as living in a universe with some sort of curvature, but at least it makes geometry a little bit easier.

About the Author: Linda Henneberg recently received her Bachelor's Degree in Physics and Astronomy, and a Graduate Diploma in Science Communication. Despite lov-ing math, she’s terrible at it, which is only more incentive

to learn more about it. Linda was also recently doing an internship at CERN, the European Center for Nuclear Re-search. It was a great opportunity, and she got to learn so much more about particle physics, one of her favorite things. In her spare time, Linda likes to go swing danc-ing or read a good book.

Shaping the Universe By Linda Henneberg

11

The following problem can be solved in a couple different ways; some much more tedious than the other. See if you can figure out the answer whichever way you can.

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? (answer

coming in the next

issue of Abacus)

Puzzle submitted by

Nathaniel Tanti.

Red line is light ray; black lines

are mirror surface.

An infinite number of mathemati-

cians walk into a bar.

The first one orders a beer

The second one orders half a beer

The third orders a quarter of a beer

http://xkcd.com/435/

Just for Fun...

Q: What do you get if you divide the

circumference of a jack-o-lantern by

its diameter?

A special thanks to contributors to this issue:

Lauren Alvarado, Linda Henneberg, Brittney Sandul, and Nathaniel Tanti