Have You Ever Wondered?

This blog entry isn’t really starting out well, is it? Even the title is inconsistent. It’s paradoxical. It’s illogical. It sounds downright counter intuitive. And this is a maths inspired blog entry as well. Oh no. How am I going to get through this? Maths, the bastian of logic, the pillar of consistency, the base of my rational mind, and it seems to be going down the path of the nonsensical and the absurd. Please hold me hand will you? And let’s soldier on together.

Are imaginary numbers real? Do imaginary numbers exist? English teachers, philosophers, heck, almost every human being on this planet (including a certain Mr. Rene Descartes) will say no. It’s obvious. After all, if they are imaginary, how can they possible exist? But evidently they do.

But for those of you who are feeling a bit lost, let’s start by defining our beloved imaginary number, i as :

More notably, i is the square root of minus one. And all of us who have been paying attention during our high school maths classes should know that you can’t have a square root of a negative number. No way. Squares always end up positive, so you can’t have square roots of negatives - they just don’t exist.

Of course, mathematicians, being the stubborn folk that they are, decided that they didn’t want to follow this simple logical argument. Hence the birth of our beloved imaginary friend i.

Funny thing is, it turns out that this number i not only exists, but it is also particularly useful. When combined with real numbers (called complex numbers), they can be used to express things like vectors and phasors (no, not the star trek phasor), which in turn is used in fields such as fluid mechanics, modern control systems, signal analysis (urgh!) and electronics. In fact, and this sounds cooler (This is cool? Timo you’re a big fat nerd) than it really is, in things like control systems and fluid mechanics, you can actually ‘transform’ real number problems into the ‘imaginary’ plane (let’s call it the imaginary world, just to make is sound sexy), play around with it there and solve the problem there with our imaginary tools, while sitting in our imaginary convertible with an imaginary voluptous female friend wearing a lovely, albeit imaginary, red dress (ok I am going overboard here); and then we can transform the solved problem back into the real plane. It’s like magic. In fact, without this ability to transform stuff into the imaginary plane, most airplanes now would crash. We wouldn’t have been able to send shuttles into space. Our DVD players won’t work. Hard drives won’t work. Hand phones won’t work. Lots of stuff won’t work. All because of imaginary numbers.

But if you are still not convinced about the existence of imaginary numbers, hear this one out : The 240V (or 120 volts is you come from the devil land, or Japan!) mains electrical supply that you are using at the moment to power your pc, is expressed as a combination of a real number (240 volts), and an imaginary number (god knows how many volts), and even if the real voltage drops from 240 volts to zero volts, you still can get a shock (and kill youself!) from the imaginary voltage! Figure that one out!

Now, if you are waiting for me to explain how imaginary numbers, work, or how they exist, you are going to be disappointed, because I haven’t the slightest clue (anti-climatic, I know, but you try making a topic like this dramatic!). I am nearly freaking out as it is.

But before I leave, I would like to show you one of the best formulas in the mathematical world, and one of my personal favourite, Euler’s Identity  :

where e is Euler’s number (which is approximately 2.718….), i is our imaginary buddy, and pi is obviously pi. If you don’t know what pi is, you may as well stop reading right now. In fact stop reading 15 minutes ago.

Euler’s identity is particularly famous because it is very elegant, i.e., it is nice and simple. It also combines the most ‘important’ 5 constants of mathematics (0,1,pi,e and i), and it also combines all the major mathematical processes -multiplication, addition, equality, and exponentiation (power of … ).

Also, on a short side trip here, e and pi are also ‘irrational’ and ‘trancendental’. Irrational numbers are numbers that we can never ever fully determine, for example, for the case of pi = 3.14159…., the dot-dot-dot goes on forever. We can never ever fully calculate the value of pi. Trancendental numbers are numbers that are not solutions to non-zero polynomials, which basically mean that we can never find a polynomial equation (most common equations are polynomials) where the answer is pi, or e. So what does it mean for you and me? Nothing at all. Just thought it was interesting that the two most common numbers in nature, pi and e (e is found is many natural situations, most notably the growth of living populations), cannot be determined exactly, and can never be solutions to our equations. Makes you believe in God doesn’t it?

But to end, just as some take home work for you readers out there, if we take a look at Euler’s Identity again, how can it be that when you take an irrational and trancendental number (e), and exponentiate it by the product of another different irrational and trancendental number (pi) and an imaginary number (i), and add it to the first number of the universe (the number 1), it equals to zero, which is in fact not a number by itself, but an ABSENCE of a number. Work that one out!

On the landing of the first floor of a pretty looking shopping mall, in a pretty looking suburb in a developed country of the reader’s choice, an attractive lady who has just passed her 30th birthday is standing wearing a pair of trackpants, a slim fitting grey t-shirt and a newish pair of jogging shoes. Her hands are ice cold, and on the grooves of her palm there is an obvious amount of moisture building up. Her face is pale, and beads of sweat are forming on her brow, and on her upper lip. Her heart is racing, and not only is it pumping much faster than usual, it is pumping a larger volume of blood per stroke than usual. Her pupils are dilated, and her blood sugar levels are rising rapidly. Adrenaline is flooding into her bloodstream. To the outside observer however, these effects are hardly noticable when compared to her rapid rate of breathing ("Is she alright?" one shoppers asks. Another wandering shopper wonders whether she is having an asthma attack, or worse, a heart attack). She is standing completely still, and all her muscles, particularly the ones on her legs, and totally rigid. Ironically, she feels totally light headed.

Standing opposite her is a man, in his mid 40s, with a bushy moustache and a receding hairline. He is wearing a white coat with a name tag. On it, on a small font, the words "Dr. Stoichkov, Senior Psychiatrist, Centre for Mental Care". His face is calm, and he speaks slowly and softly,

"Just one more step Sally, you can do it. We have been through this many times before - there is nothing to worry about. We are perfectly safe. Nothing will happen to you. This is all in your head. You can do it, just one step at a time".

Sally however, does not move an inch. She is even paler than before, and she is starting to shiver. Passing shoppers are now taking longer glances towards her direction.

Just a few steps away, a young couple is leaning on a steel railing which divides the shopping area from the floor below. They are content to simply just stand there while looking at their fellow shoppers walking around on the ground floor. Their romantic bliss however is broken by someone tapping the steel railing next to them.

"You see Sally? Nothing at all to be afraid of!" says Dr. Stoichkov, while tapping the railing again. 

Sally does not budge. From the outside, it seems that she is frozen on the spot. On the inside however, her mind is faster and farther than anyone can imagine. She thinks of all the things that could go wrong - the steel could buckle, someone could push her off the railing, she could lose her balance, the floor may collapse, any number of things could go wrong. How can anyone be sure that they are safe there?

Then she makes a effortful U-turn. She tries to convince herself that if it wasn’t safe, other people wouldn’t dare stand there as well. See, even that mother is allowing her child to lean on the railing she thinks. What harm could it be to take one step forward? After all, there are so many malls around the world, each with a large number of people leaning on the railings on any particular day, and yet she hasn’t heard of anyone falling to their death. Surely if someone did, it would be in the news. With that in mind, with a gargantuan effort, Sally lifts up her left leg and slowly glides it forward.

And then it hits her : The newspapers could be lying! Yes! They are all lying, conspiring against her, to drop down her guard. They WANT her to believe that its safe, they WANT her to take that small step, and when she least expects it, someone will push her down - down and down all the way to her death.

"Why is this happening to me? Why?? ….No… no, I cannot do this Doctor, I just cannot…" Sally whimpers, while she turns on her heels and runs away as fast as she can, tears runnning down her cheek. Nearby shoppers look with a mixed reaction of bemusement and anxiety, as they see this normal looking person suddenly acting very abnormal. What they do not know, of course, is that Sally has been diagnosed with Acrophobia - the fear of heights.

Sally is only one of a great many people who suffer various types of phobias, some strange (eg. fear of antique furniture), and some not so strange (eg. fear of spiders). But all these people have one thing in common - every single one of them wishes they didn’t have a phobia. It is completely debilitating, and it ruins a person’s life. They will do almost anything to get rid of their phobias - including paying expensive psychiatric doctors to help them.

A few hundred meters from the scene of Sally hasty departure from the shopping mall, there is a large number of people who are queueing up to pay someone to experience the same symptoms as Sally had just experienced, albeit on a smaller scale. They are queueing to ride a roller coaster. They are paying money to go higher, faster, and to be in a more dangerous situation that Sally was in, and yet these people all have smiles on their faces. Why?

For the adrenaline ‘rush’ ?

But that doesn’t make sense either - adrenaline is not a ‘happy’ hormone, its not like pheromones that make us love, or testosterone that makes us want to fornicate - the body produces adrenaline for use when we are in dangerous situations (if we are bring chased by a tiger, for example), so the body doesn’t encourage us to get ourselves into dangerous situations by making an adrenaline rush an attractive experience.

It just doesn’t make sense - why do people like to get scared?

Personally, I hate being afraid (that’s because I am smart) and I am not afraid to say it (that’s because I am not egotistical and I am in touch with my feminine side heh heh) - give me bumper cars over roller coasters and ‘the sound of music’ over ‘the omen’ anytime. 

Now, let’s imagine you are a chinese man. A chinese man 5000 years ago. Somehow or rather, you find yourself wandering in a jungle, for whatever reason. Perhaps you are hunting wild boar or something. No matter. You are alone, and are only armed with a pointy shoot of bamboo. You are walking ever so softly, trying not to ruffle any leaves, trying not to make a single sound. But as you walk, things take a turn for the worse, you spot one of those stripey animals with big long teeth sitting on a big tree branch overhead. Later on in human history these stripey animals will be named Panthera Tigris, or more commonly known as - a tiger.

You’ve heard stories of these stripey guys before, but have never seen one. You take a moment to notice how beautiful these stripey guys are, but its one moment too long as the tiger leaps off the branch and attacks you. You turn and run, running as fast as your little legs will take you, jumping over fallen tree branches, you try to avoid overhanging tree vines, while trying to throw anything hard and sharp that you find on the ground towards the hungry looking beast that is after you. He (its clearly a HE) doesn’t flinch. And its also clear that he runs much faster than you can. You have no choice but to turn and fight.

With your heart racing, you turn around, close your eyes, and jab your bamboo stick forward with all your might. The tiger jumps over your body, you fall over, and the next thing you know, the tiger breathes his last gasp just over yours. You manage to survive, by some miracle. Getting up, and looking at the carcass of the beast that nearly just killed you, what’s the first thing that comes to mind?

a) Mother f***er that was close!

b) Take that you slime drinking tick on a camel’s rear!

c) Hmm, I wonder if this can solve my erection problems?

Now, some years later, after your bad experience with the tiger, you decide to hunt food in somewhere slightly safer - the big blue watery thingy - The ocean! You stand there, with your lucky bamboo stick, trying to poke fish so that you can feed your family tonight (yes, you are now married and have a daughter and two sons!). You stand under the sweltering heat, and after 4 hours without any luck catching fish, you decide to take a short swim to cool off.

La la la la la, you start to hum while perfecting the back stroke. Then, out of the corner of your eye, you notice something sparkle in the sun. Its translucent, moves around with the waves, and it looks like slightly solidified water. It’s got entrails, which look like strands of thick, transparent hair hanging from the bottom of it. You decide to say hello, after all, this guy doesn’t look dangerous at all (compared to the stripey guys). Of course, the minute you touch this creature (a jellyfish for those who haven’t already guessed), you feel a terrible pain throughout your body. You swim for the shore, and notice large chunks of your skin are now red raw, and the pain is unlike any you have felt before. As you look around the ocean, you now notice there are many of them, floating around innocently. What’s the first thing that comes to your mind?

a) I am going to get you, you sons of bitches!

b) Screw that, I am going home to some loving from my wife!

c) Hmm, I wonder if those guys will taste nice in a steamboat, perhaps will some lovely tom yam paste and some added vegetables?

Many years later, after countless jellyfish stings, and many other misfortunes will hunting for food, you have finally made enough to retire comfortably. You buy a house in the countryside, next to some limestone caves, where the sun doesn’t  beat down too harshly and you are covered from the cold chill of winter. On a lovely sunny spring day you decide to go for a walk with your lovely wife, and your pet dog Rover. You are walking, enjoying the company, watching the local birdlife, when you notice this particular bird, who hard at work collecting bits of twigs and leaves, and then vomits out a particular paste of saliva and bodily fluids which help bind these twigs and leaves to the side of the cave, and thus forms a lovely nest for the little hatchings. Your first thought is :

a) Oh how lovely, the birds are creating, bit by bit, through hard work and perseverence, their dream home

b) How I wish I was a bird, a bird who can fly, fly far far away! To see the world, to be totally freeee!

c) Hmm, I bet that bird’s nest would make a wonderful soup.

By now you guys should know what I am getting at. Delicacies. Things like tiger’s penis soup, bird’s nest soup, how did these delicacies start?

Personally, I am a eat to live rather than a live to eat person, so, even on the best of times I find it hard to understand how people can eat some things. But some delicacies just are beyond human comprehension. I have already indirectly named 3 chinese delicacies above. But there are other delicacies, from all around the globe. These include things like monkey brains (yes! monkey brains!), cobra blood, snails (yummmmyyy), frog’s legs, duck’s feet, shark’s fin, durian (a piece of spherical ‘fruit’ which grows sharp pointy thingies all over it and smells like old rotten wet thrash), and I haven’t even started on some of the more popular internal organs yet.

But before I continue, you’ll have to hear this one - The black pudding. Sound innocent enough, after all, its called a pudding (in the background you can hear Homer Simpson’s voice saying "MMmmmm pooooohh diinnngggg"). It’s a Scottish delicacy, and I believe they usually eat it for breakfast. To make black pudding, they take the blood of an animal, say a pig, put it in a big pot until its full, and then cook it with animal fat until it solifies. YUM YUM YUM.

Oh and the Japanese, cooking that Fugu pufferfish thingamajig. They know its poisonous, it kills people every year, yet they still eat it. And its not like the poison in the fish is painless either, its probably the worst way to die - the poison paralyses you will keeping you conscious - until you die of suffocation due to the fact that your lungs paralyse as well. Talk about a Happy Meal!

Look, I can go on forever. For every culture, for every region, there is at least one type of food that all other cultures will look upon with trepidation. To each his own I say, you eat whatever you want, and I’ll stick with chicken nuggets. I have no problems with that.

The only thing that really gets me is, who is the first person, the first person with the stroke of utter genius/delusion, who looks upon, say, a shark, and says, Hmm, I’ll bet that triangle thing sticking up the water would taste lovely in a soup…