Different Dots!

I wonder if people really know what differentiation is? What does this monster of a “dee-wai-dee-axe” mean to everyone? To be frank, there are so many sides of differentiation that I can’t put my finger down on an exact meaning that fits the nature of differentiation. Differentiation is simply one of the most beautiful tools in mathematics that allows you to obtain wonderful relations.

Differentiation, to Richard Feynman, was a game; he allowed for the convention:

And then differentiation was reduced to nothing but a mere game of placing the dots:

Differentiation has never been easier! Dot placing has been the most interesting way to do differentiation:

And it’s all about playing with dots. Sometimes one dot, sometimes two! But it’s all about using dots efficiently. I’ve just illustrated an extended version of the product rule in differentiation. I very much like to call it the ‘dot’ rule.

How about the quotient rule? Well, that’s easy! Looky here:

And if I put it altogether, we have:

Mod Sum Does It All - Part I

I modified a mathematical trick that is a favourite amongst some, well, I like to call them Mathemagicians! And it goes like this:

First, pick out a three digit number, any one will do, that has its digits in decreasing order. Now remember this number, don’t tell me what you have with you.

Second, reverse the order of that number. Then subtract this new number from the original number you thought of. Having done this, remember also this new number that you’ve obtained via subtraction.

Finally, reverse this new number and add to itself.

Well, you have obtained a very, very, very special number! And because it’s so special, I know what it is already, because it’s simply 1089!

And of course this isn’t the trick, because now I’m going to up the difficulty level of this trick and impress you some more; I’m going to give you a range of numbers to choose from, and I’ll say this before hand: ‘the larger the number you choose, the harder it’ll be for me’. So here goes:

54, 1089, 3060, 42012, 80001, 103410, 500031

Now, just choose any one of the above numbers (don’t worry, they aren’t that special actually, I could give you one whole list of other numbers!). Have you chosen one yet? Well if you have, I’m going to up the difficulty one more time!

I want you to choose any number you want, any number with any number of digits, and I want you to multiply your chosen number by that new number you just chose. To make it really hard for my, I implore you to multiply it by say, a 3 digit number.

Now, if you’re trying this out, you’ve to tell me one thing, and I’ll tell you one thing in exchange:

If you tell me all the digits of your final number, save for one, I’ll tell you the last digit you left out.

I’m going to up the difficulty for myself again: you needn’t tell me the digits of your number in order and you can jumble them up. Still, I can tell you the last digit which you didn’t tell me.

Try it! And then you can always come ask me and challenge me, or just tag on the tagboard on the right. :)

An Equation for Bacon Sandwiches

Well, well, what do we have here:

This is none other than what Britons refer to as a bacon butty! To me it's just a bacon sandwich, but it is apparently so important, that the Britons have come up with an equation just to determine how much force is required to produce how much crunch for every bite of the sandwich! Believe it or not, the research fellows at Leeds University spent more than 1000 hours testing (with their mouths and guts I presume!) dozens of such sandwiches!

And their brainchild? Haha, it amounts to no more than a mere equation:

To quote, "0.4 newtons should be applied to crunch the sandwich, creating 0.5 decibels of noise. The formula uses these values: N = force in newtons; fb is the function of the bacon type; fc is the function of the condiment or filling effect; Ts is the serving temperature; tc is cooking time; ta is the time taken to insert the condiment or filling; cm is the cooking method and C represents the breaking strain in newtons of uncooked bacon."

Well, don't take my word for it! View the original article here at The New York Times: http://www.nytimes.com/2007/04/11/world/europe/11bacon.html?_r=2&ref=world&oref=slogin&oref=slogin

Clapping for Kepler! - Part II

I've been thinking of a way to explain Kepler's 2nd Law, and well, Newton provided two methods; in his Principle of Natural Philosophy, he outlined a geometrical method which involves highly accurate sketches.

Well too bad for you guys, I'm not about to go drawing things on my computer, because the angles to work out are simply too tedious! So kudos to Isaac Newton for working it out by hand.

But hey, he did come up with another easier method, which we all know now as Calculus - however, his work on his three laws of motion never once included the mechanics of calculus in his proofs.

So... how?

Like that lor.

No lah, of course not - let's just go through the main idea of using Calculus in proving Kepler's 2nd Law. Now, let us say we have an area A that is swept out by a planet moving around the Sun. If we can prove that the rate of change of area swept out (i.e. dA/dt) is zero, then we can then prove that the area swept out doesn't change with time!

Hurhur. Sounds easy.

Well, strictly speaking the area we should be considering should really be the area below, which shows the area swept out by Earth with the Sun at one focus:


But now we're considering the infinitesimal area that the planet sweeps out, and thus we are allowed to consider dA instead - that is, we consider the area swept out in a very small amount of time, such that the area swept out can be represented adequately by a triangle as such:


Well, I've removed the Earths for clarity's sake; and you may not yet be convinced as to why this small area can be a triangle - it is clearly a segment, is it not? Well, let me enlarge the diagram for you:


Are you now convinced that the segment is effectively reduced to a simple isoceles triangle when infinitesimally small areas are considered? It's as if the radius vector's length didn't change much! That's why we consider dA instead of A itself, and this allows us to work with nice approximations that hold. With this in mind, let us determine a formula for dA and I've enlarged the triangle OAB (yes, enlarged again!) for your viewing pleasure:


Notice that I've obliterated the Sun and the trajectory as well for clarity (Physicists tend to do that, haha!). Now,

Clapping for Kepler! - Part I

Have you heard of Kepler's three laws of planetary motion? Do you even know who Johannes Kepler is? Well, he was the great astronomer and mathematician who worked on information on Tycho Brahe (another interesting astronomer!), and realised that planets don't move in perfect circular paths around the Sun!

Well, you might say that's rather commonsense - but try saying it about 250 years ago, when no one knew of Newton's Laws of Motion, and when everyone thought that force was required to sustain motion! Clearly then, it would've been an astronomical task (no pun intended!) to derive and deduce the path taken by the celestial planets.

Let's dive in immediately, because I simply cannot stand dwelling on simple historical facts anymore; to get into the heat of action, let us have Kepler's 2nd Law in a nice statement:

Equal areas are swept out by the radius vector connecting the planet to the Sun in equal times.

Well, I'll explain that in another post, but harkening back to what I'm really interested in for this entry, let us assume for a second that there is no gravitational force or whatsoever in this Universe. Would Kepler's 2nd Law still hold? Well, very much! And I shall endeavour to prove this to you now.

First of all, what does a force do to the state of motion? Easy, it changes either the direction and/or the magnitude of the velocity of the body in motion. Next, where exactly and in what direction does gravity act if you're travelling around the Sun? Easy also: gravity acts also in a straight line towards the centre of the Sun and acts at your centre of mass. Basically this means that the gravitational force is a radial force, a force that acts towards the centre of the mass providing the gravitational field.

So, let us now assume that the Sun is infinitely massive with respect to the Earth or any other object (and it is rather much so!), which is justified by its very much larger mass - if so, then we can neglect the Sun's motion and say that it is effectively in rest in our mathematical treatment. Now, let us have a moving rocket - this rocket moves up, in a straight line, at constant velocity of 10 m/s. And we have three positions outlined below, A, B and C, all in equal intervals of 5 s. So let's refer to the picture below that I've so kindly (ahem!) drawn out for your ease of interpretation of the physical situation at hand:

Now, I've mentioned already this is a thought experiment to see what happens to Kepler's 2nd Law if there were no gravity, and this explains why the rocket has no force on it and continues moving in a straight line with constant speed.

So, the two triangles to compare now, are obviously triangles OAB and OBC; notice that both have the same height h, and because the speed of the rocket is the same, and equal time intervals have passed, we must agree that the rocket travels the same distance in the same amount of time, and therefore AB = BC.

What then is the formula giving the area of triangles? Well, if you've forgotten, it's:

Area of a triangle = 1/2(base)(height)

Since OAB and OBC have the same height h, and the same base (AB = BC), we conclude that they must have the same area! And this must then mean that Kepler's 2nd Law holds true even in the absence of gravity! That equal areas are swept out in equal times. :)

Mathematical Synthesis

I know all of us know what long division is, but do you know what synthetic division is? Well it's a very fast method of calculation taught to me by my Additional Mathematics teacher in Secondary Three, Mr Michael Doyle. However, there are some limitations and downsides to using this method. I shall illustrate using an example:

Let's say I'd like to simplify the following fraction using long division first:


Notice that I haven't done it step by step because I'm assuming you, the reader, knows enough of long division already. Well notice the important things: we have a polynomial in x, being divided by a divisor D(x), which results in a quotient Q(x) and a remainder R(x).

We now introduce what we call the remainder theorem, that is, for any polynomial, we can write it as such:


If the remainder R(x) = 0, then we say that the divisor D(x) is a factor of the polynomial. So clearly for this particular polynomial we have it written as:

Now, that being done, we naturally have to ask ourselves: is there an easier method? Well, there is! And this method is known as synthetic division. We shall use the previous polynomial to illustrate. This method requires us to write down the following first:

Notice that I've written down the coefficients of the polynomial - these coefficients will represent our polynomial of interest. Now, to the left of these coefficients is the number -3, which you should recognise as representing the divisor (x + 3). We put -3 to represent (x + 3) because earlier we said that:


Now clearly when x = -3, notice that the divisor term (x + 3) goes to zero, and we're only left with the remainder term (x - 6). So to divide by (x + 3), a really simple method is to substitute in the value that makes (x + 3) = 0. Now, there is a basic limitation: clearly then the remainder is obtained no longer as a polynomial, but as a specific number. So this explains why long division is preferred in some instances.

No matter the case, we move on the next step, as shown below:


And of course, the next step:


And the last step:


Our final presentation is therefore:

Voila! Do you see how synthetic division has worked out the same exact answer as long division? Try it yourself today!

On The Dimensional God

In trying to determine whether it is essential to have an intelligent Creator or not, I have written, rewritten and investigated many possible reasons, and none are so persuasive to the human intellect as to those which I shall mention and discuss here. To discuss at length the exact mathematics involved would be a rather lofty task, one which I dare not undertake in view of the numerous other more qualified experts in this world can verify, but nevertheless, I’ll persist in providing a more technical view because I think through it one can see the validity of my claims and judge for yourself whether it makes sense to have an intelligent creator or not.

I however, admit that all of these evidence merely point towards an engineer of precision who placed the universe into its intricate clockwork, and it is in no way sufficient to prove and convince anyone that it is the Christian God who single-handedly created this world. I leave it up to faith and conscientious work on the part of the individual to realize for himself that the God of the Christian Bible, is the God that created this world.

There are main intellectual barriers that simply dismiss that the universe itself could have ever come into being and existence solely due to chance alone:

1) The rejection of evolution based on statistics and thermodynamics.

2) The Big Bang model that necessitates a Creator.

3) The use of dimensional analogy to explain how the nature of a Creator allows for his role.

4) The precise intricacy of the universe that can only result from Creation.

Perhaps it is a little mundane for me to harp on the same piece of evidence over and over again, in which case I've done so with several friends over the course of this week itself. That being said, let me offer you my views of retrodimensional analysis, which makes use of dimensional analogy to show how Jesus himself can be fully God, and yet fully man at the same time. And that brings the emphasis of this entry to point number 3.

You got that right (wow, what a sudden change in tone!), Jesus was fully man and God, but how do we even come to understand such a fact? Surely it can be justified and rationalised? Of course it can, and we do so first by doing away with certain disputes.

The first dispute is that Jesus died, and God does not ever die, and therefore Jesus can't be God, for he wouldn't have died. Now, in saying this, you have just proven yourself wrong, because Jesus didn't die - He rose again on the third day (well, technically speaking, it wasn't a full three days actually) and therefore He is said to have overcome death itself. Surely that is a feat that only God himself can achieve?

The second dispute is, if Jesus really were God, then how was it possible that He was manifested in the image of man, and yet not lose his deity? That is, I'm asking, how is it possible that both man and God can both be two sides of the same coin.

Alright, I'm really making a big claim here, so if you know I'm wrong, please do correct me; for the scriptures we have read clearly say that "So God created man in his own image, in the image of God he created him; male and female he created them." - Genesis 1:27.

In His image, get that settled in your head first; and I would take that to mean that man is but an image of God, and therefore it is possible that in each and every one of us, we see the slightest hint of God.

So scripture says so, does science say otherwise? Well, this question has to be dealt with by geometry, a branch in mathematics, but to be more specific, hypergeometry. Let me explain further the basics of dimensional analogy before we go on; dimensional analogy is simply the study of the relationships between a dimension of order n and another dimension of order n-1. That is, the study of how higher and lower dimensions relate to one another.

Sounds rather abstract at this point doesn't it? Well let's give you an example: we all know all 3-dimensional objects are encased by 2-dimensional surfaces (for instance, a cube has six surfaces) and all 2-dimensional surfaces are encased by 1-dimensional lines (like how a square is enclosed by four lines). Of course, a single 1-dimensional point would be enclosed by empty space (zero dimensions). The intellectual difficulty in visualising four dimensions can then be partially reduced by making the logical leap that a four dimensional object is enclosed by 3-dimensional volumes.

Well, that doesn't make things all that better, but at least we've now one extra piece of information to use to analyse things of a higher dimension. So let's give you another piece of information to help you bolster your thoughts; 3 dimensional beings like us, naturally live in a 3 dimensional world (we have 3D space and time), while 2 dimensional things will exist in a 3 dimensional space (like how a piece of paper must exist in a 3D space, because stacking two 2D pieces of paper automatically generates the existence of depth, introducing 3D qualities and thus 2D objects, in order to exist independently of its own geometry, must be in a 3D world). So of course, a 1D object must be living in a 2D world, and in a 1D world, only things of zero dimensions can exist (that is to say, nothing really exists in a 1D world except for dots).

May I pause for a second to say that God was wise to create us in a 4D world? :) For life without time is simply a life without existence, for it is time that gives meaning to many things, and it is time that builds up many things in life.

Now, I shall give you one more piece of information for your understanding; when you have a three dimensional object and you cast light say, directly overhead, you'll obtain a 2D shadow. Of course, a 2D object like a sheet of paper, if you cast light along its edge, you'll obtain a 1D shadow in the form of a line. And if you have a 1D object like a string, and cast light head on, you'll obtain a point like shadow, that is, a shadow of effectively no dimensions.

So have you caught on to what I'm saying yet?

If you haven't made the logical jump, it's alright, let me illustrate. For instance, let us assume we have a 4D being - wouldn't this 4D being first of all live in a world of higher dimensions as we've just said? So this world would then be effectively 5D, and of course I'm assuming this higher being is a 4D being, but it could be of any dimensions higher anyway. Now, when a shadow of this 4D being is being cast, wouldn't it be a 3D shadow then, which we realise to be an object that can be visualised in real space and time.

Now let's make the bold hypothesis that this 4D being is God - then the 5D realm could be heaven (notice I'm being very conservative with my language here for this is just a mere hypothesis). Now that explains why we can't physic ally see God nor heaven, for they are of higher dimensions!

Now, you're now probably getting what I'm talking about - earlier, I was hinting at God being able to manifest himself as Jesus to us, visible to the naked eye, because a shadow of God would be something that can be materialised as an object in real space, or at least in our world. But to say Jesus is a shadow of God completely misses the picture, for a shadow is not the object that casted it, but a mere aspect of the object, and shadows can be distorted, and they aren't the complete replica of the object, for the sense of depth, length and breadth can be distorted using shadows.

I know I know, you're probably saying now 'but you just talked about shadows, so why not?' Well I'm saying now that the example I gave isn't really that adequate, for it doesn't tie in with the true nature of who Jesus really was. Let me now borrow an idea from Edwin Abbott, the author of Flatland (a truly exciting and mind blowing book written way back in the 1900s!).

Well, imagine you have a 2D world (well, a piece of paper!), in which 1D men and women live (oh those poor souls! I'm talking about stick men basically); imagine one day, there's this 3D person that comes by - let's say he's a cube, in quite the literal sense. So if this cube were to sit on the 2D world (ouch!), what would the 1D people see? Well, the 1D people can only perceive as much as the 2D world will allow, and what does the 2D world allow? The 2D world allows only one surface of the cube to be on the paper, meaning all the stick men will see, is a square.

That's it! Jesus, is simply God, and the above analogy helps us to understand that. Let me elaborate a little further - the square that the 1D people see, is not a square at all, but is simply something that the 1D people are forced to see because they cannot see further. So is it right to say that the 2D square is a shadow of the 3D cube? Well, no! And a rather resounding no at that!

Why? Because the square is part of the cube, and the cube is part of the square, but of course we do see that the square must eventually return to be part of the greater cube, for it is in the realm of higher dimensions that the cube can really be in all its glory of having 3 full dimensions.

So isn't that what Jesus is saying? He says: "You heard me say, 'I am going away and I am coming back to you.' If you loved me, you would be glad that I am going to the Father, for the Father is greater than I." - John 14:28

Clearly, Jesus is saying, if you love Him, then you must understand through death, he will return to where he truly belongs. If you read the original Greek, it doesn't mean Jesus is going back to his father, but that Jesus will be glorified in going back. (Tells you how important Greek is huh?)

So clearly, the square isn't separate from the cube, for the square is just what they see of the cube, and the square was never apart from the cube - the square retains the quality of the cube just as much as it appears as a square to the eyes of the stick men.

And just as this has made clear, that is how God shows Himself as Jesus. Now, I'm making a very bold claim in saying this, and it is by no means, scientifically justifiable - but I am making an assertion that an engineer once made in a VCF (Varsity Christian Fellowship) meeting, just that I'm now piling up more information, now that I've acquainted myself more with the topic of dimensional analysis.

So why must God be a 4D or higher dimensional being? Well, let me put it this way: if you believe in a Creator, what attributes would he have? For starters, he must exist outside of our dimensions, for only one who is of higher dimensions can effectively create something of lower dimensions. It completely makes sense. 2D beings, with no sense of height, could never visualise buildings, for buildings require the vision of height and depth. But they could most certainly feel it, because they would know that something that has height cannot be simply passed through. Similarly, 3D beings can't visualise time - it is simply something to be felt.

Only the Creator of higher dimensions could effect the Big Bang (now that is to be included in a later discussion), for the Big Bang was something that happen in spacetime fabric.

Now, how does God see into your heart? This can be explain by retrodimensional analysis as well. For instance, if you're a three dimensional being, you see all sides of a 2D object at once. If you're a 2D being, you see the entire 1D object (all sides doesn't make sense here). If you're a 1D being, well then, haha, there's nothing much left for you to see.

But the point is, a being of higher dimensions, sees all aspects of objects of lower dimensions. Now, the higher the order of your dimensional existence, the more you can see of and into objects of lower dimensions. That being said, I shall leave the rest to your intellectual wit. :p

This concludes all I have to say on the dimensional nature of God - but please note that God is more than just plain dimensions!