I would like to express my gratitude for your Margaret help with the computer. I'm getting it fixed tomorrow. Student B: page Student C: page SoftYvare" '. With a partner, discuss what you as a customer can do about these problems. C8J I For your next project, I need you to take a look at our Kirkcaldy contact centre. Our 04 customer satisfaction survey ISdisastrous see charts and report extract attached and this represents a major risk to our corporate image and future sales.
We have serious recruitment problems and a high staff turnover. These two problems are obviously linked. I'd like you to come up with some proposals but without significantly increasing costs, which are another problem. They spend about the same time with an adviser, although If possible we need to reduce this because at the moment we can't take enough calls.
Obviously, we can't afford to increase salaries. I'm counting on you to come up with some creative ideas; get back to me as soon as you can. What are employees' rights on mea I and rest breaks in the USA? Are they similar to those in your country? In her opinion, what is the biggest problem for advisers? What are the effects on the contact centre of the following?
S olving proble m s o Work in small groups. You are the consultants that Hamish Hamilton wrote to in Exercise2. Hold a meeting to discussthe agenda below. Solutions Brainstorm solutions to the problems prioritized as a and b above. Recommendations Define recommendations for short- and long-term policy. Exercise3 so that they have the sam e m eaning. T true or F false.
Personnel is a function of Human Resources. Please hang and I'll call you back. Could you speak , please? Crossout the one verb that does not collocate III Complete the email using the words in the box. Put them in the correct places. It still hasn't arrived. I 2 be 3 if you 4 ship the order immediately. Helpline operator: Before I can locate the problem, I first For your reference, I have 5 a copy of the need to escalate exactly where the fault is.
Customer: OK, no problem, we can do that. But if it's still not working properly, can you sort out the product, or at I confirmation you sent me. Do not 6 7 I would 9 me if you need any 8 to informatiof1. Helpline operator: Yes, that's possible, but I'm not Best regards authorized to do it. I would first have to give the problem J Garcia to my supervisor. Useone word to 'I'll into it, it out, and back to you fill each gap. The clues in bracketswill help you. D iscussion D W ith a partner, think of three exam ples of products which are packaged well or badly, and say why.
Think about protection, identification, transport, storage, display and security. Why does he think packaging is so important? P a ra phra sing D Reformulate these phrases from the article in your own words. In small groups, discuss how you will package it.
W hat are their interface with advantages and disadvantages? Think about shape, rage colour, photos, logos and text. SURVEYSshow that intense frustration and even injury caused by modern packaging is on the increase, especially amongst seniors. Seventy per cent of over 50s admit to injuring fingers, hands and shoulders as a result of 'wrap rage', a new term used 5 to describe the irritation and loss of self-control experienced when struggling to open wrapping.
In recent years manufacturers have been under increasing pressure to keep food items sterile, to provide child-proof packaging for dangerous or toxic household cleaning products, 10 to protect products during transport and to reduce theft from shops. At the same time, they are forced to keep costs to a minimum. As a result, packaging has become ever more resistant to fingers, nails and even teeth. In their 15 frustration with plastic packs, which defeat a" attempts to open them, even with scissors,consumers use all kinds of tools and knives.
At best, the product inside the packaging is at risk; 20 at worst, it is hardly surprising that 60, people a year are injured in Great Britain alone. Some of the most common triggers of wrap rage are processed cheese packages, tightly wrapped CDs,child-proof tops on medicine bottles, and milk and juice cartons. Ring-pull 25 cans are particularly problematic for older fingers and delicate skin. Even fighting to remove price tags from items bought as gifts can raise blood pressure, and unnecessary overpackaging is pure provocation to the ecologically-minded.
However it seems there's light at the end of the tunnel. The bottom line is that if they don't react, they risk losing sales if customers simply stop buying products with packaging that offers too much resistance. IT] Generate new ideas in focus groups and brainstorming meetings. It is 7. With a of g when empty, it can be used to ship plans, posters, blueprints, etc.
Describing products o Describethe dimensions of objects in your pockets or your bag. Your partner should try to guess what they are. Use collocations from Exercise 4 to describe what is being discussed. What are its three main features? Search for the keywords recording vocabulary and make a list of possible techniques. Rank the techniques on your list from the most to the least useful for you personally.
Glossary PAGE attention-grabbing 11 Match the expressions in Exercise 7 with their function in the presentation a-d. Use the vocabulary and expressions in the previous exercises to present the specifications and features of an electronic device of your choice.
IZwBusiness 2. Re fre sh y o ur 2 Who was the woman who is famous for her research on radioactivity? Relative clauses 4 Who was the. Usethe Articles Internet if you have a connection. Then challenge another group.
Grammar and practice page Noun combinations The main noun comes at the end. Any others describe it. N on-de fining re la tive cla use s o With a partner, take turns making sentencesand adding relative clausesabout famous people, products and organizations.
How much information can you add? B : S tevejobs, w ho w as from C alifornia, w as the co-founder of A pple. A : S tevejobs, w ho w as from C alifornia, w as the co-founder of A pple, w hich is fam ous for high-end electronic products like the iP hone. B : S tevejobs, w ho w as from C alifornia, and w hose parents adopted him at birth, w as the co- founder of A pple, w hich is fam ous for high-end electronic products like the iP hone.
Chooseyour top three. Some popular products take a long time to get to the marketplace. It worked just as Babbage had intended. Listen to a product review and write the words you hear which the customer omitted in the notes below. Then listen and compare your versions with the recordings. Search for the noun information. Is information countable o Hate the phone. Toosmall- can't open flip cover with one hand. No screen on outside to or uncountable?
What see caller identity. Reception - horrible. Very long key is its informal form? Anxiously awaiting day can upgrade and get rid of monstrosity. Which prepositions is it used with? What typical collocations and constructions is it Browse several online Had FoYle. Qbout tl-lr'e. S d too. EQSjzyxwvutsrqpon dictionaries and find e. Ver' aielr''''' c. GO;V13 tr'j Vle. As: turn to page Bs: turn to page 12l.
The zyxwvutsrqponmlkjihgfedcb Business 2. With a partner, draw up a checklist. Compare the presentation with your checklist. Then listen again to Version 2 and check your answers. I'm here this morning to the Pingman,.. My objective today is The goal of this meeting is Agenda Summary Feel to interrupt me. I'd like to up the presentationand move on to Last,but not , I have given you I'll take any questionsat the end.
Callfor action Close Theseare the why I am askingyou to If you haveany questions,I'll do my bestto answerthem. Then listen and compare your answers. After that, I'll be talking about - 1 How long should a the prototype, and the data we've collected from tests, focus groups and market studies. Finally, I'd good presentation like to present a business plan; this will demonstrate why we expect a return on investment that is be?
Is everybody happy with that agenda? Then practise reading the extract with ways to practise a correct intonation, stress and linking. II In small groups, prepare the introduction and conclusion of a presentation of one of these 6 How should you new products to a group of department store buyers. Hook: What is the most surprising, exciting or unusual aspect of your product? Now watch the video Agenda: How will you organize your presentation and what will happen after the talk?
Summary: What are the highlights of your talk? Call for action: What do you want your audience to do now? Assume the overwhelmingly body of the talk has been presented. The rest of the ping classare the buyers.
As a class,vote for the best tracking product presentation. Then discuss the questions below. ABS air-conditioned comfort alloy wheels automatic climate control head-turning good looks power on demand safe braking and cornering 3.
W hat about other products? M ode l o Read the product description and list the main benefits of using Open Office.
O pe nO ffic e. E asy to use It is available in many languages and works on all common computers. It stores all your data in an international open standard format and can also read and write files from other common office software packages.
It can be downloaded and used completely free of charge for any purpose, A new approach to office productivity software o OpenOffice. You can create dynamic documents, analyze data, design eye-catching presentations, produce dramatic illustrations and open up your databases. However, as you become used to OpenOffice. You can of course continue to use your old Microsoft Office files without any problems - and if you need to exchange files with people still using Microsoft Office, that's no problem either.
What's in the suite? You can easily integrate images and charts in documents, create everything from business letters to complete books and web content. CALC - a feature-packed spreadsheet. Use advanced spreadsheet functions and decision-making tools to perform sophisticated data analysis.
Use built-in charting tools to generate impressive 2D and 3D charts. Your presentations will truly stand out with special effects, animation and high-impact drawing tools. DRAW- produce everything from simple diagrams to dynamic 3D illustrations and special effects. Find out more - try it today! Get OpenOffice. Go to www. Alternatively, stand out use a product of your choice.
Focus on the benefits to consumers, adding any details you feel are appropriate. SOi1Atio1't B usiness zyxwvutsrqponm 2. How does each of them try recommendations to differentiate itself from its competitors? Scan reading o Read the documents and answer the questions. Big pizzas, big value! Re: Marketing meeting tomorrow Billie, Mick, I've attached the latest figures and customer-feedback summary, which seem to confirm what we discussed last time. Restaurant sales are holding up but, as expected, our takeout and delivery revenues are down again this month.
If we want to defend our market share against Pizza Hut and the other international majors, and attract new franchisees, we desperately need to relaunch our product. So, here's the agenda for the meeting: 1 a new, more exciting range of pizzas 3 a new or updated logo and color scheme 2 new promotional ideas 4 a new box for takeout and delivery Looking forward to hearing your ideas on all these points tomorrow. What two decisions are made?
Which ones does the home. Jack like? W hy not? Big Ja ck's P izza wishe s to stre ngthe n its bra nd a nd im prove its pa cka ging. The zyxwvutsrqponmlkjihgfedcbaZ Business 2. With a partner, explain and justify your choices. I haveno idea how my career I havea clear idea of my will develop. I expectto work for one company I expectto work for several all my life.
Money,statusand a comfortable Job satisfaction,variety and being lifestyle are my priorities. In part one, Charlie Schumann, a popular careers coach, talks about two things you zyxwvutsrqpon Search for the keywords p e rs o n a lity p ro file te s t shouldn't do, and two things you have to do.
Before you listen, try to predict what those and do at least one things might be. Then listen and check your predictions. Money, variety,fame, autonomy, beauty, recognition, team spirit, job security, helping people. D iscussion D In small groups, discussyour reactions to these statements from the talk. Glossary PAGE S um m a riz ing earth-shattering o Summarize each of the eight remaining tips in one inertia sentence. In small groups, compare your sentences with other jump in at the deep end people and choose the best summary for each tip.
Make a list of all the things in your life that you have really enjoyed. It could be work or play, an event, a period of time in your life, etc. Pick one and start digging 5 into the reasons why. Get beyond what you love doing, and break it down into the underlying characteristics.
Think of it as identifying your passion's building blocks. Write them down 10 alone or with friends, in one session or on a small pad of paper you carry with you. Above all, be creative. You never know what crazy idea is going to spark the Big One. Once you have identified some things you think you might be interested in, identify people who are knowledgeable in those 15 areas and contact them.
Explain that you are exploring your options and ask if you can pick their brains. You'll get some fantastic insights if you make this a habit, not to mention making some great contacts along the way. Rememberthere's a shallow end too, 20 so you can still enjoy splashing in the water.
Look for baby steps you can take that will bring your passion into your life and keep you moving towards your long-term goal. Make a list. Maybe they're real financial obstacles, or perhaps the need for 25 more training. Maybe they are internal. What's stopping you? Simple inertia? To browse Academia. Log in with Facebook Log in with Google. Remember me on this computer.
Enter the email address you signed up with and we'll email you a reset link. Need an account? Click here to sign up. Download Free PDF. Consumer Behavior, Global Edition. Haxan Kingin. How long is forever? How powerful is God? It has been banished, outlawed, and shunned. During the Inquisition, the renegade monk Giordano Bruno was burned alive at the stake for suggesting that God, in His infinite power, created innumerable worlds.
Zeno of Elea c. These conundrums anticipated ideas at the heart of calculus and are still being debated today. Bertrand Russell called them immeasur- ably subtle and profound. His arguments have come down to us through Plato and Aris- totle, who summarized them mainly to demolish them. In their tell- ing, Zeno was trying to prove that change is impossible. Our senses tell us otherwise, but our senses deceive us.
Change, according to Zeno, is an illusion. The first of them, the Paradox of the Dichotomy, is similar to the Riddle of the Wall but vastly more frustrating. And before you can do that, you need to take a quarter of a step, and so on. Who would have thought that taking a step required completing infinitely many subtasks? Worse still, there is no first task to complete. If you thought you had a lot to do before breakfast, imagine having to finish an infinite number of tasks just to get to the kitchen.
Another paradox, called Achilles and the Tortoise, maintains that a swift runner Achilles can never catch up to a slow runner a tortoise if the slow runner has been given a head start in a race.
For by the time Achilles reaches the spot where the tortoise started, the tortoise will have moved a little bit farther down the track. And by the time Achilles reaches that new location, the tortoise will have crept slightly farther ahead. In these first two paradoxes, Zeno seemed to be arguing against space and time being fundamentally continuous, meaning that they can be divided endlessly.
His clever rhetorical strategy some say he invented it was proof by contradiction, known to lawyers and logi- cians as reductio ad absurdum, reduction to an absurdity. In both paradoxes, Zeno assumed the continuity of space and time and then deduced a contradiction from that assumption; therefore, the as- sumption of continuity must be false.
Calculus is founded on that very assumption and so has a lot at stake in this fight. It rebuts Zeno by showing where his reasoning went wrong. During that time the tortoise will have moved 1 meter farther ahead. It takes Achilles another 0. As the math shows, he can do them all in a finite amount of time. This line of reasoning qualifies as a calculus argument. We just summed an infinite series and calculated a limit, as we did earlier when we discussed why 0. We could use algebra instead.
To do so, we first need to figure out where each runner is on the track at an arbitrary time t seconds af- ter the race begins. Since Achilles runs at a speed of 10 meters per second and since distance equals rate times time, his distance down the track is 10t. To solve this equation, subtract t from both sides. Then divide both sides by 9. So from the perspective of calculus, there really is no paradox about Achilles and the tortoise.
If space and time are continuous, everything works out nicely. Zeno Goes Digital In a third paradox, the Paradox of the Arrow, Zeno argued against an alternative possibility — that space and time are fundamentally discrete, meaning that they are composed of tiny indivisible units, something like pixels of space and time.
The paradox goes like this. If space and time are discrete, an arrow in flight can never move, because at each instant a pixel of time the arrow is at some definite place a specific set of pixels in space. Hence, at any given instant, the arrow is not moving. It is also not moving between instants be- cause, by assumption, there is no time between instants. Therefore, at no time is the arrow ever moving.
Philosophers are still debating its status, but it seems to me that Zeno got it two-thirds right. In a world where space and time are discrete, an arrow in flight would behave as Zeno said. It would strangely materialize at one place after another as time clicks forward in discrete steps. And he was also right that our senses tell us that the real world is not like that, at least not as we ordinarily perceive it. But Zeno was wrong that motion would be impossible in such a world.
We all know this from our experience of watching mov- ies and videos on our digital devices. But because of our percep- tual limitations, it would look like a smooth trajectory. Sometimes our senses really do deceive us. Of course, if the chopping is too blocky, we can tell the differ- ence between the continuous and the discrete, and we often find it bothersome.
On the analog clock, the second hand sweeps around in a beautifully uniform mo- tion. It depicts time as flowing. Whereas on the digital clock, the second hand jerks forward in discrete steps, thwack, thwack, thwack. It depicts time as jumping. Infinity can build a bridge between these two very different conceptions of time. Imagine a digital clock that advances through trillions of little clicks per second instead of one loud thwack. We would no longer be able to tell the difference between that kind of digital clock and a true analog clock.
Likewise with movies and vid- eos; as long as the frames flash by fast enough, say at thirty frames a second, they give the impression of seamless flow. And if there were infinitely many frames per second, the flow truly would be seamless.
Consider how music is recorded and played back. This is a quintessential analog experience. Whereas when you listen to her on digital, every aspect of her music is minced into tiny, discrete steps and converted into strings of 0s and 1s. So in everyday life, the gulf between the discrete and the con- tinuous can often be bridged, at least to a good approximation.
For many practical purposes, the discrete can stand in for the continu- ous, as long as we slice things thinly enough. In the ideal world of calculus, we can go one better. With limits and infinity, the discrete and the continuous become one. Zeno Meets the Quantum The Infinity Principle asks us to pretend that everything can be sliced and diced endlessly.
Imagining pizzas that can be cut into arbitrarily thin pieces enabled us to find the area of a circle exactly. The question naturally arises: Do such infinitesimally small things exist in the real world?
Quantum mechanics has something to say about that. Its terminology, with its zoo of leptons, quarks, and neutrinos, sounds like something out of Lewis Carroll.
The behavior it describes is often weird as well. At the atomic scale, things can happen that would never occur in the macroscopic world. For instance, consider the Riddle of the Wall from a quantum perspective. This effect is known as quantum tun- neling.
It actually occurs. This means there is some small but nonzero probability that the electron will be detected on the far side of the barrier, as if it had tunneled through the wall.
With the help of calculus, we can calcu- late the rate at which such tunneling events occur, and experiments have confirmed the predictions. Tunneling is real. Alpha particles tunnel out of uranium nuclei at the predicted rate to produce the effect known as radioactivity. Tunneling also plays an important role in the nuclear-fusion processes that make the sun shine, so life on Earth depends partially on tunneling.
And it has many technologi- cal uses; scanning tunneling microscopy, which allows scientists to image and manipulate individual atoms, is based on the concept. We have no intuition for such events at the atomic scale, be- ing the gargantuan creatures composed of trillions upon trillions of atoms that we are.
Fortunately, calculus can take the place of intu- ition. By applying calculus and quantum mechanics, physicists have opened a theoretical window on the microworld. The fruits of their insights include lasers and transistors, the chips in our computers, and the LEDs in our flat-screen TVs. Maxwell made the same assumption in his theory of electricity and magnetism; so did Newton in his theory of gravity and Einstein in his theory of relativ- ity. All of calculus, and hence all of theoretical physics, hinges on this assumption of continuous space and time.
That assumption of continuity has been resoundingly successful so far. But there is reason to believe that at much, much smaller scales of the universe, far below the atomic scale, space and time may ultimately lose their continuous character. At such small scales, space and time might seethe and roil at random. They might fluctuate like bubbling foam.
Although there is no consensus about how to visualize space and time at these ultimate scales, there is universal agreement about how small those scales are likely to be. They are forced upon us by three fundamental constants of nature. One of them is the gravitational constant, G. It measures the strength of gravity in the universe. It is bound to occur in any future theory that supersedes them.
The third constant is the speed of light, c. It is the speed limit for the universe. No signal of any kind can travel faster than c. This speed must necessarily enter any theory of space and time because it ties the two of them together via the prin- ciple that distance equals rate times time, where c plays the role of the rate or speed. In , the father of quantum theory, a German physicist named Max Planck, realized that there was one and only one way to combine these fundamental constants to produce a scale of length.
That unique length, he concluded, was a natural yardstick for the universe. In his honor, it is now called the Planck length.
It is given by the algebraic combination! Space and time would no longer make sense below these scales. These numbers put a bound on how fine we could ever slice space or time. Take the largest pos- sible distance, the estimated diameter of the known universe, and divide it by the smallest possible distance, the Planck length.
That unfathomably extreme ratio of distances is a number with only sixty digits in it. I want to stress that — only sixty digits. Us- ing more digits than that — say a hundred digits, let alone infinitely many — would be colossal overkill, way more than we would ever need to describe any real distances out there in the material world.
And yet in calculus, we use infinitely many digits all the time. As early as middle school, students are asked to think about num- bers like 0. We call these real numbers, but there is nothing real about them. The requirement to specify a real number by an infinite number of digits after the decimal point is exactly what it means to be not real, at least as far as we understand reality through physics today. If real numbers are not real, why do mathematicians love them so much?
And why are schoolchildren forced to learn about them? Because calculus needs them. From the beginning, calculus has stubbornly insisted that everything — space and time, matter and energy, all objects that ever have been or will be — should be re- garded as continuous.
Accordingly, everything can and should be quantified by real numbers. In this idealized, imaginary world, we pretend that everything can be split finer and finer without end. The whole theory of calculus is built on that assumption. If all we ever used were decimals with only sixty digits of precision, the number line would be pockmarked and cratered.
There would be holes where pi, the square root of two, and any other numbers that need infinitely many digits after the decimal point should exist. If we want to think of the totality of all numbers as forming a continuous line, those numbers have to be real numbers. They may be an approximation of reality, but they work amazingly well. Reality is too hard to model any other way. With infinite decimals, as with the rest of calculus, infinity makes everything simpler.
His name was Archimedes. For one thing, there are a lot of funny stories about him. Several portray him as the original math geek. Both of these ideas have countless practical applications. It also underlies all of naval architecture, the theory of ship stability, and the design of oil-drilling platforms at sea. And you rely on his law of the lever, even if unknowingly, every time you use a nail clip- per or a crowbar. Archimedes might have been a formidable maker of war ma- chines, and he undoubtedly was a brilliant scientist and engineer, but what really puts him in the pantheon is what he did for math- ematics.
He paved the way for integral calculus. To say he was ahead of his time would be putting it mildly. Has anyone ever been more ahead of his time? Two strategies appear again and again in his work.
The first was his ardent use of the Infinity Principle. To probe the mysteries of circles, spheres, and other curved shapes, he always approximated them with rectilinear shapes made of lots of straight, flat pieces, fac- eted like jewels. By imagining more and more pieces and making them smaller and smaller, he pushed his approximations ever closer to the truth, approaching exactitude in the limit of infinitely many pieces. This strategy demanded that he be a wizard with sums and puzzles, since he ended up having to add many numbers or pieces back together to arrive at his conclusions.
Specifically, he mingled ge- ometry, the study of shapes, with mechanics, the study of motion and force. Sometimes he used geometry to illuminate mechanics; sometimes the flow went in the other direction, with mechanical arguments providing insight into pure form. It was by using both strategies with consummate skill that Archimedes was able to pen- etrate so deeply into the mystery of curves.
Squeezing Pi When I walk to my office or go out with my dog for an evening stroll, the pedometer on my iPhone keeps track of how far I walk. The distance traveled equals stride length times the number of steps taken. Archimedes used a similar idea to calculate the circumference of a circle and to estimate pi. Think of the circle as a track. It takes a lot of steps to walk all the way around.
The path would look something like this. Each step is represented by a tiny straight line. By multiplying the number of steps by the length of each one, we can estimate the length of the track.
And so the approximation is sure to underestimate the true length of the circular track. Archimedes did a series of calculations like this, starting with a path made up of six straight steps. He began with a hexagon because it was a convenient base camp from which to embark on the more arduous calculations ahead. The advantage of the hexagon was that he could easily calculate its pe- rimeter, the total length around the hexagon. Why six? Of course, six is a ridiculously small number of steps, and the resulting hexagon is obviously a very crude caricature of a circle, but Archimedes was just getting started.
Once he figured out what the hexagon was telling him, he shortened the steps and took twice as many of them. He did that by detouring to the midpoint of each arc, taking two baby steps instead of striding across the arc in one big step.
A man obsessed, he went from six steps to twelve, then twenty-four, forty-eight, and, ultimately, ninety-six steps, working out their ever-shrinking lengths to migraine-inducing precision.
That required him to calculate square roots, a nasty chore to do by hand. Forget about math for a minute. The squeeze technique that Archimedes used building on ear- lier work by the Greek mathematician Eudoxus is now known as the method of exhaustion because of the way it traps the unknown number pi between two known numbers. The bounds tighten with each doubling, thus exhausting the wiggle room for pi. Circles are the simplest curves in geometry.
Yet, surprisingly, measuring them — quantifying their properties with numbers — transcends geometry.
How to measure the length of a curved line or the area of a curved surface or the volume of a curved solid — these were the cutting-edge questions that consumed Archimedes and led him to take the first steps toward what we now call integral calculus. Pi was its first triumph.
He avoided doing all that because pi was not a number to him. It was a magnitude, not a number. We no longer make this distinction between magnitude and number, but it was important in ancient Greek mathematics.
It seems to have arisen from the tension between the discrete as represented by whole numbers and the continuous as represented by shapes. The historical details are murky, but it appears that sometime be- tween Pythagoras and Eudoxus, between the sixth and the fourth centuries bce, somebody proved that the diagonal of a square was incommensurable with its side, meaning that the ratio of those two lengths could not be expressed as the ratio of two whole numbers. In modern language, someone discovered the existence of irrational numbers.
The suspicion is that this discovery shocked and disap- pointed the Greeks, since it belied the Pythagorean credo. This de- flating letdown may explain why later Greek mathematicians always elevated geometry over arithmetic. They were inadequate as a foundation for mathematics. To describe continuous quantities and reason about them, the ancient Greek mathematicians realized they needed to invent some- thing more powerful than whole numbers.
So they developed a sys- tem based on shapes and their proportions. It relied on measures of geometrical objects: lengths of lines, areas of squares, volumes of cubes.
All of these they called magnitudes. They thought of them as distinct from numbers and superior to them. It was a strange, transcendent crea- ture, more exotic than any number. My children certainly were intrigued by it. They used to stare at a pie plate hanging in our kitchen that had the digits of pi running around the rim and spiral- ing in toward the center, shrinking in size as they swirled into the abyss.
For them, the fascination had to do with the random-looking sequence of digits, never repeating, never showing any pattern at all, going on forever, infinity on a platter. We will never know all the digits of pi. Nevertheless, those digits are out there, waiting to be discovered. Yet twenty-two trillion is nothing compared to the infinitude of digits that define the actual pi.
Think of how philosophically disturbing this is. I said that the digits of pi are out there, but where are they exactly? They exist in some Platonic realm, along with abstract concepts like truth and justice.
On the one hand, it represents order, as embodied by the shape of a circle, long held to be a symbol of perfection and eternity. On the other hand, pi is unruly, disheveled in appearance, its digits obeying no obvious rule, or at least none that we can perceive.
Pi is elusive and mysterious, forever beyond reach. Its mix of order and disorder is what makes it so bewitching. The Brain and Behaviour Initiative BBI enables cross-faculty, multidisciplinary, collaborative research in the cognitive and affective neurosciences and brings together expertise on phenotyping, genotyping, cognotyping, brain imaging and molecular signatures to address brain-behaviour issues.
New experimental techniques including brain imaging, genetic testing and neuropsychological assessment combined with new theoretical insights have opened up significant potential for the advancement of novel diagnostic tools and treatments for people with mental disorders.
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In she received the World Lung Health Award, awarded by the American Thoracic Society at a ceremony in San Diego, in recognition of work that has "the potential to eliminate gender, racial, ethnic, or economic health disparities worldwide". Currently regarded as a thought leader in Rheumatic Heart Disease, both on the continent and internationally. Communities in Manitoba Community Documents Find community resource documents to facilitate municipal administration, public works, recreation and wellness, environmental services, protective services, community development, land-use planning, community planning, and infrastructure development.
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