To infinity and its negative - a demonstration of the nature of space in the universe

This graph is analogical to movement in the universe. It is in fact the graph of the reciprocal of the hyperbolic tan function. What I wanted to demonstrate was the seemingly impossible nature of the graph switching from a value of infinity to negative infinity across the line x=0. I think that when we do discover the exact nature of space in the universe it will be dependent on a curved model in so that the three dimensional plane of space will fold back in on itself, metaphorically speaking, like a sphere so that anyone attempting to reach the edge of the universe will find it is in fact impossible.

A Mathematical Anecdote Analogous to Chemical Resonance

What this little conundrum suggests is that in infinitely switching between two states, the final result is halfway in between. This is the reasoning behind the structure of benzene and the ionic carboxylate group.

s = 1 - 1 + 1 - 1 + ...

(-1) * s = (-1) * (1 - 1 + 1 - 1 + ...)   | *-1

1 + (-1) * s = 1 + (-1) * (1 - 1 + 1 - 1 + ...) | +1

1 - s = 1 - 1 + 1 - 1 + ...

1 - s = s

1 = 2 * s

s = 1/2


Source 

Probability, chaos and the nature of creation

Probability and an elementary, almost axiomatically governing rule of the universe (thoughts given through the analogy of a complex puzzle)

I will attempt in this article to describe my views on how probability plays an intrinsic role in the governing of chaos and the universe as we know it.
Many subscribe to the belief that the universe is fundamentally predictable through mathematics. In fact Einstein once famously said that "God does not play dice with the universe". It turned out, in a sense, that Einstein was wrong. But he was wrong only in the sense that betting companies are wrong to hedge their bets with making money through gambling. The only thing that makes betting companies money is that a probability describing a single event never has a certain outcome, but is in fact a way of describing a myriad of many complex factors which are all mathematical and will hence yield the same result in a great enough sample. Remember that if anything can be conceived to possibly (or impossibly) happen it is possible. It is just that the more likely events occur a higher proportion of the time. Just as we must define limits for integration, without which the sum of any specific value is zero, probability is meaningless when applied to single values. Probabilities are proportions of a sample on a large scale. As the sample size tends to infinity, it is 100% certain that the probabilities will also tend to the actual proportions obtained from the infinite sample.
In fact, the classical mechanics of the universe are simply a large scale description of predictable patterns in probability on a very large scale: the laws which govern these mechanics provide the footholds for mathematics and are maths' only claim to validity. Force is equal to mass multiplied by acceleration because of a repertoire of different subatomic forces, energies, masses - among other things - that all interplay to produce this predictable pattern on the larger scale. At the smallest scale the universe is in fact still affected by what goes on around it. So, in fact one day we might in fact realise - if such a statement is true - that the quantum mechanics are in fact affected by the large scale mechanics of physics and visa versa.
This might be another "strange loop" which could further support Hofstadter's opinion/hypothesis that strange loops are very important. In fact, they might prove to be a stepping stone in understanding how patterns in the universe arise.
On to my main point - an essay to describe evolution in terms of an infinitely complex puzzle and to attempt to shed some of my self-validated insight on intelligence's origins and nature.
Think of an infinitely complex maze, further convoluted by the fact that the maze is affected in strange ways by the way you move through it, so going through the maze to a certain spot - you may only return to your original state through backtracking your steps and actions exactly to your starting point. This can be likened to travelling back in time. You cannot take shortcuts through the maze. Think of intelligence as one possible location in the maze. The achievement of reaching this location (creating intelligence) is only possibly attained through one single route, since there is only one possible route to any single place in the maze. Even if two different routes are exactly the same but for one footstep for example taking a step with the right foot rather than the left foot, this small effect will produce a butterfly effect, meaning that a difference between the two paths will arise which shall become more and more pronounced as you progress further and further away from the location at which you committed this deed.
In fact, the pathways can be thought of as progress through time and the different pathways, in this case, as evolutionary branches on the evolutionary tree of life. In fact, there are many other situations in which this analogy applies anywhere where probability and infinite possibilities play a key role.
Evolution follows the concept of probability. It was infinitely unlikely that "intelligence" as we know it should ever have been developed. In fact, intelligent humans (homo sapiens) have only existed for the equivalent of the blink of an eye in the timeline of the universe, even in the comparatively minute period of existence of life on earth. Had one idiosyncratic genetic mutation not occurred at exactly the right time and place in the past, intelligence might never have existed - or perhaps existed but in an altered form. In fact, we also have to think outside of the internal development of humans. For instance, what could have happened if the individual human or lifeform in which this mutation occurred was killed before it could reproduce. That lifeform would never have passed on the line of material necessary for humans to develop. Therefore the events in the past leading up to this were all important in the timeline in which intelligence was created. In fact, the external effect need not be as extreme as death, but only a small interaction occurring in a different way. Generally - as I alluded to earlier through mentioning the butterfly effect - the further back a change occurs in timelines, the more exponential its effects will become in the future. This is simply due to the nature of chaos.
Chaos is a thing we use to describe infinitely complex physical - or other - processes which we cannot possibly predict - since such an attempt would be paradoxical. This would be because in the act of trying to assess the future of a chaotic system, we ourselves would be changing its course, and would have to include the act of our own prediction in the context: the instrument affects the experiment. It is in fact impossible to ever conduct any "experiment" whether that be a thought experiment or a physical experiment, without in fact affecting the outcome itself. Such an experiment would need to infinitely recurrently apply corrections to the experiment to account for the influence of the experimenting tools themselves. One such thing is the theoretical model of the universe in a computer, which would have to include the computer itself and its effects. This is my view of what we mean by relativity in the universe, and why I believe constancy is the only possible perspective which could help us truly understand how the universe works.
If the impossibility of this supposed trial by error (natural selection in other words) method of obtaining a pathway to intelligence seems too unlikely, then consider this: intelligence did not in fact exist until the moment it was created. We only place such a high value on intelligence because it mystifies us: we do not fully understand the universe or intelligence, because we are using intelligence itself to understand it. In fact, it is questionable whether 'understanding' is something inherent in the universe, or simply another function of intelligence itself - something created from nothing. This also goes for mathematics, science and the content of this article, all of which are functions or consequences of intelligence.
It is almost certain that something infinitely more amazing than intelligence could have been created through natural selection, and that something infinitely more amazing than that could also have been created ad infinitum. There will always be a better possibility for perfection would imply that there are a finite number of outcomes. This can be explained through the nature of creating something from nothing: that something is not defined and so we assume it has "infinite value" - because 'something' is every possibility ever conceived or not conceived. That is the brilliance of creation and possibly why mathematics has evolved as a tool used to describe the universe:
I believe explanation of the power of mathematics can be found through starting with the simple statement 0=0. The rules of algebra dictate that we may apply any transformation to either side as long as we apply it to the other. In the mathematical sense, we might add 2 to either side to give 2=2. But this could also be written 1+1=2. I firmly subscribe to the belief that every action has an equal and opposite reaction is an underlying principle in the creation of the universe. If something is created, its equal and opposite must also be created. Through various transformations things might appear not to sum to 0 in the universe: there are a growing multitude of different physical phenomena that we must look at and consider in describing the universe. Yet, I believe that if all things are reduced to their simplest form - rearranging the equation - this will not result in a solvable equation with universal variables, but a simple statement that can be reduced to 0=0. Of course you would have to "solve simultaneous equations on a grand scale" to achieve this, but in doing so would better understand the patterns of the universe. I think that you must think of forces, energy, fields etc. to all be intrinsically linked and to be of the same sort of substance (i.e. things which exist in this universe the traditional sense) in order for this to be possible, however the universe as we know it is such a complicated soup of all these things that it becomes difficult, almost impossible, to make sense of it in this fashion. I also believe that relativity arises from everything having an equal and opposite as shown by the diagram below, which I shall leave for the reader to consider:

Imaginatively Real - Understanding Imaginary Numbers and why they're not really all that 'Imaginary'

Imaginary numbers are one of the mysterious parts of mathematics, which most people have heard about - but not many people truly understand.
The first thing that will spring to mind for most when we talk about imaginary numbers, is the mysterious symbol $i$.
Now, we ask ourselves, what does $i$ actually mean? Well, $i$ is simply the number which is equal to $-1$ when squared, so:
$$i^2 = -1$$
Or you will sometimes see:
$$i = \sqrt{-1}$$
Now that we "know" what $i$ is, we can use this number to solve a lot of previously unsolvable problems - a notable case being electronics. In fact, you may have heard of $i$ sometimes being written as $j$. This is because in physics, the letter $i$ is often used to denote current:

Engineers use a 'j' to indicate the square root of minus one since they tend to use 'i' as a current. Mathematicians use 'i' for this since they don't know a current from a hole in the ground! 
University of St Andrews

However, I suppose you might still be thinking this is all very well and good, $i$ is the square root of minus 1 - but what does it mean in practice? Show me some evidence!

In fact, there is a very good video on YouTube which explains how $i$ actually works in a very intuitive way. Here is a summary of what he is trying to say - you will soon see how exactly this fits into electronics.

$i$, just like any other symbol in our number system, is simply used to describe a quantity. $i$ is not all too dissimilar to negative numbers in that sense - it is equally as valid as the positive set of numbers and yet because we cannot "touch" or "interact" with $i$ very practically, we often lose sense of what it really is. Professor Arthur T. Benjamin of Harvey Mudd College sums this up in one of his lectures in the series 'The Joy of Mathematics'. Professor Benjamin challenges us to think about the ways in which the concept of simple negative numbers, which seem so obvious and necessary to us in a society where we use them everyday (profits and losses; increases and decreases etc.) might have seemed as alien as complex numbers like $i$ for example, to people of another age simply because they did not really use them in practice. How can you have a negative amount of rocks? Surely the only way to represent something in real life is using positive, tangible, whole objects? We now know this not to be true, since negative numbers are a necessity, but it is very much so that imaginary numbers are just the same - because most people do not really come into contact with them in a tangible sense, they lose sight of what imaginary numbers actually are.
Now, how can we understand imaginary numbers and how they fit into real life? Using something simple of course: circles!
Imagine an object travelling in one direction. We know from the laws of physics that this object might have a velocity in that direction. A velocity is a vector quantity and has magnitude, but more critically direction. Say, we wanted to reverse the velocity of this object, i.e. make it travel at an equal speed in an opposite direction. We need to keep the magnitude the same but rotate it through 180 degrees, or $2\pi$ radians - whatever floats your boat.

N.B. In fact, radians might be more sensible here - since they're more natural to use with circles (or else it would have been pretty superficial and unnecessary to come up with them!) - I might do a post about this later.

If we want to make an object travel in the opposite direction, once again from physics, we know that all we need to do is multiply its current velocity by $-1$. This makes sense in terms of the conservation of momentum for example, where we can see an explosion as valid when two particles with zero initial velocity then have velocities which are of opposite sign to one another. ($-1 + 1 = 0$ so the conservation of momentum applies)

So, we know that we can make an object travel in an opposite direction by multiplying its velocity by $-1$ - this gives its opposite. However, what if we wanted to make it travel in a perpendicular direction? Think about it - perpendicular means at 90 degrees (or $\pi$ radians) and since two turns by this number of degrees/radians would give you a full 180 degree ($2\pi$) turn, we simply need to multiply by the square-root of this half-turn. In other words the square-root of minus one.

But we need a name for this new value and guess what?
$$i$$
If you're still going to try to prove to me that $i$ "doesn't exist" now, then I'd like to hear how? $i$ seems to be essential - it must exist since the square root of minus one "doesn't" in the set of real numbers. In another world, maybe $i$ is the norm - in fact it could be if you looked at the problem from a perspective starting 90 degrees/ $\pi$ radians later!

Now we can draw up a new scale - with values of $i$ included. We can pretend each point is a state of motion of the particle. Those that involve 2 axes (real and imaginary parts) are called complex numbers.  But, you can also have vectors in complex space - vectors are just transformations relative to a point of origin. If there is no point of origin you can simply think of a vector as coming from the point (0,0) or $O$ as you may have seen it written in all those maths problems that now make a lot more sense. That's really all there is to it. So, the point $[1,2i]$ simply represents a magnitude in a fixed direction. It can also be written as $1 + 2i$ if you are representing the complex number relative to the origin.

Lets say we call that point $z$, so $z = 1 + 2i$ is the complex number (vector). Remember - you can add/subtract vectors (since they represent translations) so you can also do this with complex numbers. In short: if you represent the complex number as a set of coordinates, you've already defined the starting point (the origin), however if you define it as a vector equation - this will be assumed to have started from the origin, however you can also say it is relative to another point. Lets say this was $[4, 4i]$ - then the end point of the vector $z = 1 + 2i$ relative to this point would be $4 + 4i + 1 + 2i = 5 + 6i$ or $[5, 6i]$ - a new complex number - and remember that you simply group the separate real and imaginary components to add or subtract. You can also perform operations on them similar to those when dealing with polynomials, where when equating or performing mathematical operations with two or more polynomials, you simply equate coefficients of the same type e.g. $x^2$. As with vectors, you could also represent a complex number in the form $[\frac{x}{yi}]$ - a vector translation.
Moving on: guess what we can use to calculate each part (the magnitude and the direction)? The omnipresent trigonometry

Lets see - the direction and magnitude of a complex number is very similar to the direction of a vector in physics, except when talking about complex numbers the direction (or angle from the horizontal) is called the argument and the modulus (similar to magnitude in vectors). These two components can also be combined into one, as you will see, to represent the complex number in a different form: the trigonometric form. It all seems to link together.

So, lets take the vector "$z$" above again. If we want to represent it on a diagram (as a vector, we can simply draw it as above. However, if we want to simply look at the magnitude of that line (in practicality the speed at which the point is travelling) we can use Pythagoras' Theorem.

As can be seen on the diagram (simply click to zoom) I have now split the complex number up into its two original components - the real and the imaginary. Now, before I tell you how to use the Pythagorean Theorem to work out the modulus (magnitude) I need to explain an insight I had into signs of numbers to you).

Signs of Numbers

How I see it, the signs of numbers are not actually part of the numbers themselves - numbers such as 4,5,6 which we take as the "normal" numbers are simply numbers of the dimension (note this is not official - I simply use this description since I think it makes sense) $+r$, in other words real, positive. I see this $+r$ as the descriptor of the number - used to describe which dimension it belongs to. The exact opposite dimension is $-r$ or real, negative. These real dimensions, are not by any means the most important however - there are of course the imaginary dimensions, which are equally as valid. These dimensions are exactly perpendicular to the real dimensions as seen on the diagram, and is composed of both an imaginary, poistive and an imaginary, negative dimension. What I am trying to say is that when we describe numbers as "$2$" or "$-2$" we should really describe them as "$2r$" or "$-2r$", since we do this for the imaginary numbers by adding "$i$". Now I can continue with the explanation.

So, the diagram above should really have "$+1r$" in place of the "$1$" along the real axis. We know however, that Pythagoras' Theorem states that we must calculate the modulus of the two components added together to find the magnitude. Now, when you calculate the modulus of an expression, you take each component in turn, take away its sign (which in this case we will be thinking of as the dimension) and then finally square each part and take the square root of the overall expression. That is how you do it: the reasons behind this are in the Pythagorean Theorem itself. However, I suspect you want a proof of why $a^2 + b^2 = c^2$ in a triangle of hypotenuse $c$. Well, the geometric proof can be found here. (There are many other possible proofs - I think I found website listing upwards of 40!)
So, now that we know how to find the modulus, we merely square the 'coefficients' of the two components. We therefore obtain:
$$2^2 + 1^2 = M^2$$
$$M = \sqrt{2^2 + 1^2}$$
$$M = \sqrt{5}$$
In fact, the generally accepted notation for the "magnitude" or modulus of a complex number is simply... the modulus notation, so for our $z$ this would be:
$$|z| = \sqrt{5}$$
So, now that we have the length of the line (with no dimensions - this isn't actually in the real, positive form) we can calculate the argument - this is simply the angle which the complex number 'vector' makes with the real, positive axis.
To do this we use trigonometry. In fact, we can actually draw a similarity between the complex number vector we are working on and that of the many triangles which make the unit circle in a CAST diagram, often used in trigonometry - an interesting point to note.

Source

We can see that the complex number's modulus is very similar to the magnitude of a force/velocity vector. Ask yourself now - how would you express a vector in terms of its two components? Well, it should of course be the two separate vector components (here components means the components including the angle and the magnitude) added together. i.e. if you had a road and you wanted to get from its start to its end, you could simply follow a horizontal path then a vertical path, once you had completed the horizontal component. The total change in displacement would be the two displacements added together.
So, working out the angle, as you might've guessed, is just simple trigonometry. Since our components are real, positive ($+1r$)and imaginary, positive ($+2i$), it follows we can get a positive angle using the tan ratio (I will do an article on trigonometry and how sin, cos and tan are all linked perhaps, as well as how both they and the logarithmic functions can be estimated using the infinite Taylor series), since:
$$tan(\theta) = \frac{opposite}{adjacent}$$
So...
$$tan(arg(z)) = \frac{2}{1}$$
$$tan(arg(z)) = 2$$
$$arg(z) = tan^{-1}(2)$$
Once again using the magnitude of the numbers only and not their "dimensions".
You could of course work in radians or in degrees, but you must stick to your guns once you have picked one. Once again, I would advise radians since circles are involved - since it is more "natural" (since radians are in terms of $\pi$ and $\pi$ is at the heart of all circles).
So, what do we do now that we have used Pythagoras and Trigonometry to work out the modulus and argument? Well, we can now express the original vector in trigonometric form. Instead of each component having a magnitude and a dimension, each component will now have a magnitude and an angle which can be helpful if you need both the real and imaginary components in the same "form".
Think about how you might do this for a normal physics vector - you have the hypotenuse and the argument. You simply need to obtain an expression for the imaginary component using the $cos$ ratio and the real component using the $sin$ ratio.
$$sin(\theta) = \frac{opposite}{hypotenuse}$$
$$opposite (imaginary) = hypotenuse[sin(\theta)]$$
$$opposite (imaginary) = 5[sin(tan^{-1}(2))]$$

$$cos(\theta) = \frac{adjacent}{hypotenuse}$$
$$adjacent (real) = hypotenuse[cos(\theta)]$$
$$adjacent (real) = 5[cos(tan^{-1}(2))]$$

There we go, and to get from the origin to the end of the vector we just add the two components (doesn't matter about the order:
$$z = 5[cos(tan^{-1}(2))] + 5[sin(tan^{-1}(2))]$$
$$z = 5[[sin(tan^{-1}(2)) + cos(tan^{-1}(2))]$$
The above is the trigonometric form.

Now that you understand how to work with complex numbers a bit better: here's a little bit of insight into how they're vital to electronics:
I obtained the following images from this website: you should go and check it out for a more expanded explanation.

Look at the moving diagram - its a model of how an Alternating Current Works - a current will not suddenly change from I to -I and back again, it will oscillate, with the sine curve modelling its motion in one plane. This plane can be regarded as our real plane. This is shown in the last diagram - where physics equations are substituted in to the complex and real parts to get an overall expression for the phase or the angle $\theta$.
It just goes to show how circles, trigonometry and circular motion are all inherently linked (article?).
However, simply doing this ignores a major part of the picture: the horizontal plane. Unfortunately our real numbers have all been used up on the axis modelling the vertical motion. But we can still include it using imaginary numbers. Do you see where this is going?
Imaginary numbers, as you have seen are linked to almost every important topic in mathematics: circular motion, vectors, diagrams, trigonometry, calculus (the change of the trigonometric curve over time in the AC), etc. etc. But one question: is it imaginary numbers that links all of these, or all all of these simply inherently linked themselves? I agree with the latter and I hope reading this Blog will convince you to agree with me.

Here's a nice video to finish off with by Sixty Symbols of Nottingham University.
If nothing else, Philip Moriarty's confusion should reassure you.

Important note: I write this Blog on my own and undoubtedly I will make mistakes. Please do not hesitate to correct me in the comments if I am wrong or slightly misled about something, since this Blog is an essay in learning for myself as much as its readers. Discussions are just as good. I really appreciate your involvement. Any contribution is part of the Blog itself.

We'll be counting... numbers

You may have learnt the formula in school:

$\sum\limits_{r=1}^n r = \frac{(n+1)(n)}{2}$

$n$ here is the last value in the series to be added up. $a$ is the starting value. $r$, the value after the summation sign is the formula for any term - and here it is simply on its own, meaning if you are on the 3rd term of the series, r will be equal to 3. In other words, the series is simply a number line from a to n.
However, this formula in itself is not very intuitive, what makes that formula work? Well, you may also have learnt "the" proof, which involves pairing up the numbers in reverse order and adding them up to give the sum of double the series, then dividing by 2. However, as you will soon discover, this is definitely not the only way of thinking about this problem. There are various other ways which may help you to get a much more in-tune understanding of what makes this formula tick.
N.B. This is of course the general formula for the sum of numbers going up in 1's, I will attempt to expand this formula to be valid for any arithmetic progression later.

For the purposes of this exercise, I will be using the set of numbers 1 to 10, who's sum (work it out on a calculator if you do not trust me) happens to be 55.

Preamble

Interesting note: the word preamble comes from the prefix' pre', meaning 'before' and the addition amble, coming from the Latin verb 'ambulare' meaning 'to walk'. It is meant to signify a stroll before a run, or a gentle introduction.

Any operation you perform on a number can be regarded as a transformation, much like you would do to a graph. You could, say, take the number 1. You could then ADD 2 to it. Here, we can think of the reference point as being the number 1 and the operation to be to add 2 to it. On a number line, this would simply be jumping two whole number spaces ahead. Very simple. However, you could achieve the same operation, by multiplying by 3. This can be regarded as an addition to the reference point 0 of 3 lots of 1. It can also be regarded as a sort of stretch as you would stretch out a piece of rubber to multiply its length by 3. Subtraction is simply the opposite of division. You take away a set of whole numbers to get your new value, with your original value acting as the reference point. Division is similar. Lets say you wanted to divide the number 1 by 3 this time. This can be regarded as either splitting the number up into 3, or in terms of a multiplication again, with x being the division's answer:

$3x = 1$

So, it is the number you add to zero three times to get one. What I am saying is that you can think of multiplication and division as addition or subtraction.

Method 1 - even numbers

Suppose we start with the numbers 1 to 10. We need to add them up. You could either use a calculator, or you could do as the mathematician Carl Gauss supposedly did when his teacher tried to occupy his students by telling them to add up the numbers 1 to 100: see the patterns in the numbers.

The numbers 1 to 10 can be laid out as follows:

 1  2 3 4 5

10 9 8 7 6

Let us take $x$ to equal the number last term ($x$th term in a sequence) and let us take $n$ as the actual value of the last term. For the purposes of jumps of 1, starting on 1 these values happen to be equal for a sequence. However, if the value of the "jump" changes or we start on a value other than 1 then they will not be. For now, we can consider them to be equal.

$\sum\limits_{r=1}^n r = \frac{(n+1)(n)}{2}$

is therefore better written (in more general cases):

$\sum\limits_{r=1}^x r = \frac{(n+1)(x)}{2}$

N.B. This is simply the notation I have chosen to use. Please do not assume that this is the convention.

As can be seen from the layout of the numbers 1 to 10, each row contains half the numbers: the lower row contains the bigger ones, the upper row contains the smaller ones. If we add up the columns, we find every column will add up to 11. There are five columns and five times 11 is 55 - our answer.

The sum of each column is in fact equal to $(n+1)$ with n being the final number. There are $\frac{x}{2}$ columns, where $x$ is the number of numbers to be added, and so the sum of the numbers from 1 to n can be generalised (for now for even values of n only, with a jump of 1 between each number):

$\sum\limits_{r=1}^n r = \frac{(n+1)(n)}{2}$

But say our starting number was not 1, but another number which we will call $a$, how do we now use this formula?

Well, its pretty simple if you think about it. If you want to add up the numbers 3 to 5, then you can simply add up the numbers 1 to 5 and take away the values in the unwanted range: 1 and 2. These values are simply in the range 1 to one below the starting number. The maximum value of the range -the range is here 1 to 2 - will always be one less than the starting value of the range you actually want. (In a more general sense, it is the x=0th term of ANY arithmetic series, where a is x=1th term - arithmetic series are simply formulae to calculate the nth term of a series which have the form $nth term = a + nr$).

So...

$\sum\limits_{r=a}^n r = \frac{(n+1)(n)}{2} - \frac{(a)(a-1)}{2}$

Here $\frac{(a)(a-1)}{2}$ simply gives you the sum of all the values up to one less than the value a.

Note: we can only actually use $\frac{(a)(a-1)}{2}$ as a formula, since once again, the value of a will be equal to its number in the overall sequence on the first term of the sequence we want to add up.

Method 2 - odd numbers

The above operation is very good for sums of numbers up to even numbers - but what happens when the value of n is odd? A good question. Say we had to add up all the numbers from 1 to 9 instead? What happens then?

Well, actually - all you need to do is think about the problem in a different way. We can obviously not make pairs from 9 numbers, since that would require 9 to be divisible by 2, which it isn't.

We can however, approach the problem by trying to convert the set of 9 numbers into a set of 10 numbers without affecting the overall sum: the way we do this is by including 0 as one of our numbers.

0 1 2 3 4

9 8 7 6 5

There... all those numbers added up would equal $\sum\limits_{1}^9$. Here, every column added up would equal 9. There are in fact 5 columns which is equal to the value of $\frac{(n+1)}{2}$. Hang on a minute... so the value of the sum of a series of odd numbers is...

$\sum\limits_{r=1}^n r = \frac{(n+1)(n)}{2}$

Reminder: for these sequences, n=x.

They are both the same! Except they are derived in slightly different ways. However - this now seems to be a general formula.

However, we can also use another method: calculate the value of the numbers 1 to 10 instead and simply take away the number 10.

$\sum\limits_{r=1}^n r = \frac{(n+1)(n+2)}{2} - (n+1)$

$\sum\limits_{r=1}^n r = \frac{(n+1)(n+2) - 2(n+1)}{2}$

$\sum\limits_{r=1}^n r = \frac{(n+1)(n+2-2)}{2}$

$\sum\limits_{r=1}^n r = \frac{(n+1)(n)}{2}$

Again, here I am using (n+2) as x since the sequence is equivalent to the number line and n=x.

Again - the same formula!

Method 3 - averages

Now, we have not yet thought about this very nifty trick! It should make sense that if we have a set of numbers which are each separated by one unit (e.g. 1 to 10 or 1 to 9) then we can calculate the average of these numbers by finding their sum and dividing by the number of terms we have.

$Sum = \frac{(n+a)(x)}{2}$

$Average = \frac{(n+a)(x)}{2x}$
$Average = \frac{(n+a)}{2}$

Here, as you may have noticed, we are using the value $a$ instead of 1, to make the thing more general, as well as $x$, since $x$ may not be equal to $n$ in some cases.

It works equally well using the above proofs, look at this diagram:

a (a+1) (a+2)
n (n-1) (n-2)

This is a similar construct to that which I used to prove that the sum of all the pairs in series which start at 1 were equal to $(n+1)$. You can now see that more generally:

$Sum = \frac{(n+a)(x)}{2}$

Where $x = n$ if the numbers go up in 1's. This general formula will work for all differences (usually denoted using the letter $d$ in arithmetic sequences).

Now - this general formula for the average makes sense! Instead of adding up ALL the numbers, we simply take the biggest number - $n$ in this case, and the smallest number - $a$. Adding the biggest and the smallest number and dividing by 2 gives you the number exactly in between both - the average in the case that the numbers which separate the biggest and smallest numbers are all equally spaced.

Now, we can simply multiply the average by the number of terms, $x$, to get the sum of the numbers from $a$ to $n$.

$Sum = \frac{(n+a)(x)}{2}$

Although this proof seems like it loops back on itself, since we proved that the average of a number by deriving from a formula for the sum, it is useful, since you can prove that:

$Average = \frac{(n+a)}{2}$

simply by applying a bit of common sense. In effect this proof can worth both ways, depending on whether you are sure the average of a number follows the formula given above, or that the sum of numbers follows the formula.

Another note on finding the sum of series

There is another clever way to find the sum of a series which does not start with the number 1. That is to imagine that it is a series that starts with 1, and then simply trim off the start of the series up to 1 before your starting value.

For example, we already have the formula:

Visualising what we have learned using geometry

Now, this is a very useful thing to do if you want an intuitive understanding, since we as humans operate in a world filled with geometry and physical objects. You will see in a later post that  the same thing helps to really explain imaginary numbers and i.

So, we've got this formula:

$\sum\limits_{r=1}^n r = \frac{(n+a)(x)}{2}$

But how do we visualise this? Using a rectangle of course!

We can visualise, for example, the sum of the numbers 3 to 10 as a series of dots.

ooo

oooo

ooooo

oooooo

ooooooo

oooooooo

ooooooooo

oooooooooo

If we've got two of these:

xxx

xxxx

xxxxx

xxxxxx

xxxxxxx

xxxxxxxx

xxxxxxxxx

xxxxxxxxxx

Now we add them together.

oooxxxxxxxxxx

ooooxxxxxxxxx

oooooxxxxxxxx

ooooooxxxxxxx

oooooooxxxxxx

ooooooooxxxxx

oooooooooxxxx

ooooooooooxxx

We have made a rectangle, of height $x$ and width $(n + a)$ I'm sure you agree: the area of which is simply

$x(n+a)$

But we've got double the amount of dots, so the area is simply...

$Sum = \frac{x(n + a)}{2}$

Once again... the same formula!

Another cool way of adding numbers together

Now that we know the formula 

$Sum = \frac{x(n + a)}{2}$

to be true for any series, here's a cool way to work out the sum of all the odd or even numbers between a set of two values.

Lets say you want to add up all the even numbers between 1 and 20. We can think about this problem in a few ways, one of which being that we can simply work out the sum of the numbers 1 to 10 and then double the sum. This will, in effect, double each term in the series, giving us all the even numbers between 1 and 20. You can see this by looking at the upper and lower limits: the lowest number you can get is 2, this is the first even number between 1 and 20. The highest number is 20, this is the last number you get between 1 and 20.

$EVEN \sum\limits_{r=1}^{20} r =( \frac{(10+1)(10)}{2})(2)$
$EVEN \sum\limits_{r=1}^{20} r =110$

Now we have calculated that value, why not simply calculate the odd values? Since aren't they just the negative space of the even values?

We can do this by taking the value above away from the sum of the values to 20.

$ODD\sum\limits_{r=1}^{20} r =( \frac{(20+1)(20)}{2}) - ( \frac{(10+1)(10)}{2})(2)$
$ODD\sum\limits_{r=1}^{20} r =100$

You can see here that the value of the odd and even numbers added together are simply $\frac{n}{2}$ apart, when the highest value in the series is even - with the sum of the evens being higher. If the highest value is odd, then they are $\frac{n+1}{2}$ apart - with the sum of the odds being higher.

Now that you have read this article: go and look at this question for an interesting discussion.

Credit goes to Better Explained for some of the ideas. This website is fantastic!

The Self Referencing Proof or (God's) Pure Essence Tainted Upon Entering this World

Watching this video by VSauce recently, reminded me of a question that I had once asked myself: do we all see colours in the same way? When Michael mentioned Tommy Edison in the video, I was absolutely dumbfounded, since a mere month ago I had visited his channel and gone through the same thought process mentioned in the video. If colours are in fact merely relative illusions which require you to actually be somebody to experience, then is everything else different for everyone else, and is our perception of reality blinding us to others' perceptions? Who is to say that the things we touch, feel, see and generally experience through our senses are experienced in a similar way by anybody else? In fact, if everything you think about in life depends on your 'view' on it (assuming your 'view' has been shaped by the experiences you have gone through since you were born) then must the physical senses also by that conclusion be subjective - were they shaped by what happened to you while developing from one lowly cell into a fully functional organism?
Anyway, watching the video highlighted something I had never really considered in depth before: the theory of mind. In the video, Michael mentions a case study where a child is told that a woman called Sally puts an apple in a box, inside a room with a box and a crate, and then leaves the room. Then another woman called Susie comes into the room and takes the apple from the box and puts it in the crate. When the child is asked where Susie would look for the apple if she came back into the room, the child says the crate - although it has been made clear that Susie was not in the room when the apple was moved and therefore would not know that the apple had been moved through simple logic.
VSauce's Michael suggests that this is due to children not being able to perceive that others might have different perceptions of reality to ourselves, and that everybody sees everything in the same way. He even suggests that although we can hypothesise about animals not seeing the world in the same way as us, they cannot do the same with regards to us perhaps. Whether  this is due to their having different brain structures to ourselves or due to them being able to experience the world using different - or slightly-altered versions of, - senses to ourselves (for example a butterfly being able to see the UV-section of the spectrum) we do not know.
In fact, perhaps (and this is just my opinion) it could be that children are born into this world with an all-knowingness that results from the essence of their souls - their very being itself. Before we are born (that whole topic is something best left for another post (or book)) it might be that we can simply experience every sense, even the ones that we might not think exist, and that due to this overwhelming all-knowingness we have none of the prejudices created through ignorance. Perhaps it is true that total ignorance is bliss, but some ignorance creates relativity. So, if you knew nothing about what this universe actually was, you wouldn't exist in any sense (until somebody conceived that person who knew absolutely zero in a thought or blog-post perhaps), but if you know everything through being able to experience everything through every sense possible, then you are never going to see anything relatively, since all the relative senses are also available to you and you will be able to see how they all relate to each-other: you must be omnipresent essentially. A being exhibiting such traits might be described as 'God' depending on the relative culture or faith-group you originate from. But of course, we know not everyone has the same senses - firstly because we all experience the world through different senses, secondly because the actual absence of sense is a sense within itself. Have you never wondered why as a child you were so much more creative? In fact in a test conducted, 98% of five-year olds were technically classed as creative geniuses (Genii perhaps? This word is very odd, but then spelling it as it is normally spelt enforces the concept of old Latin words becoming Anglicised through adding the ending 'es' for a plural and is therefore probably correct. That is how English originated anyway!), whereas just 2% of adults were in a similar experiment. These figures speak for themselves. Why though? Is it because society forces us to think in a certain way? Or is it perhaps because the five year olds had no tangible way to relate to the world. It has in fact been shown that the further away you are from something mentally or physically - the more creative you become in terms of describing or using it - the purest form is an abstract concept: the very definition of the intangible and the pinnacle of creativity itself. It is ironic since intangibility is in fact an abstract concept in itself. It seems to be that intangibility and abstract concepts are things which can only be described in terms of themselves: something which I will come on to later. In fact, the YouTuber Tommy Edison, who has been blind all his life is baffled by the concept of colour - however, perhaps this complete and utter confusion is part of what it takes to understand the world. People who can see can never experience what it must be like to have never know colour - since they will always have the concept clearly written into their memories. It is interesting - since it becomes a paradox to be able to understand the whole universe: you would need to be able to experience both all and none of the senses at the same time, which to the human brain is an impossible feat. Perhaps however, we only consider this an impossible feat because of the prejudices of the rules laid out for us by common 'logic'. What is to say that nothing and everything can't exist simultaneously. Isn't existence itself just another concept we talk about but never really understand?
Back to relativity and the idea of something which can only be described in terms of itself. In fact, everything we have ever known or will ever know fits this bill. You might say that this isn't true, but by definition it both is and isn't, depending on your relative standpoint.
Everything we know loops in a circle coming back to itself: the universe, logical arguments, physical concepts... the list goes on and on. They are the strange loops that Hofstadter talks about in Godel, Escher, Bach. In fact, take the concept of physical mass for example. What is it really? It is not determined by anything other than how much force needs to be applied for a set interval to make it start moving a certain distance per unit time. But of course, what is force defined in terms of? Mass and acceleration of course. Nothing else. We can only find out the mass of objects by comparing the relative properties of different objects - relative to each-other. It seems almost paradoxical, but it works - the universe itself is a paradox.
By the way, with reference to butterflies, how do they experience the colour given by the ultraviolet region of the spectrum? Is there not a clearly defined number of colours that exist - those that a healthy human can see? How can a butterfly see all those colours, and then some? How does the butterfly's brain process all those different colours. You then come to realise that, in fact, colour is merely an illusion created by our perception of reality - it is relative to the person experiencing the colour. Very much 'beauty is in the eye of the beholder'-style. In fact, as has been mentioned in physical journals millions of times - the universe itself is all relative, depending on your standpoint. Nothing is ever the same to any particle in the universe - it cannot be by logical reasoning, since relative to any other particle, its physical states (mass, energy, velocity, direction) can never be the same, otherwise it would simply BE that particle (what would differentiate the two?).
Hope you enjoyed this post - there are a lot of things I wanted to say, but lost track of going off on lots and lots of different tangents. Perhaps though, I'll come back to them in more detail in later posts.

Relativity - MC Escher

Zero divided by zero?

What is the value of 0/0? At first glance you might think that this actually has the value of 0, considering that if zero is divided by anything it is still zero! However, this isn't actually really true as you will soon find out.

Consider the basic algebraic principles: they state that if one number or expression is equal to another, then you may apply basic arithmetic operations to one side as long as you mirror them on the other. For example, the expression 7=14/2 is mathematically true. Multiplying both sides by 2 will give you 7x2=14 which of course is also true according to our number system. More generally, if a=b then ac=bc as well.

We can also apply this principle to the expression 0/0. Say, we are assuming that we do not know the value of 0/0, so we will set the expression equal to the letter X. So now X=0/0. Applying the principle that multiplying zero by both sides will result in an equal expression, we do so, giving us Xx0=0. But wait a minute; the value of any number multiplied by 0 is equal to 0, so can't X take on an infiinte set of values, not just 0? Strange as it may seem this is mathematically true, however we can't simply leave the definition of 0/0 as "all real or imaginary numbers", since this goes against the rules for forming legitimate funtions.

If we define the function f(X)=X/0, we should be able to set Y=X/0. However, this means that again Yx0=X, which as we have seen means Y is equal to any possible number. This cannot happen however, since the function isn't valid. It is not a many-to-one or one-to-one function. It is in fact a one-to-many function meaning that in our system of mathematics it is undefined due to the fact that for the value X=0, we get more than 1 equally likely result. So, in actual fact 0/0 is not 0, it is undefined.

There are a few strange things to note however, about multiplying and dividing by 0. The first of these things is that if Y=X/0, then if X takes on any value other than 0, the resulting value for Y is infinity.

So for example, if X=3, then Y=3/0, which of course is infinity. This becomes stranger however when we take Y as infinity in terms of the algebra.

if ∞=3/0

∞x0=3

and

0=3/∞

Now, that last statement makes sense: if 3 is divided infinitely, then it will approach 0. This at least seems feasible in terms of convergence to a limit. However, what about the statment ∞x0=3. Haven't we already assumed that any number multiplied by 0 is equal to 0? Infinity however multiplied by 0 seems to be equal to an infinite number of differnent numbers, as with the value of 0/0. So does this not mean that since both expressions are equal to the same vast range of different values, we can set ∞x0=X, where X comes from the expression Xx0=0, so X=0/0 and has a range of different values, just as ∞x0 has. However, since both expressions may take on a range of different values, then at the same time as ∞x0 is equal to 3 in one case, X may be equal to any number, for example 4, in the other. We then come out with the strange result 3=4. Now, this proves at least that many to one functions cannot exist at the same time as our normal number system, since strange results occur. The only solution to the problem is to set 0=∞. Now everything begins to seem more rational. By saying this we can now say that ∞=3/0 is not valid in the first place, since any number is 1 divided by itself, and that value only. So the only acceptable value for ∞=X/0 is X=0 (or X=∞ since we are now assuming 0=∞). This now becomes a much simpler 1 to 1 function, which is acceptable in mathematics.

Our other problem of Xx0=0 now becomes one which is much easier to solve. If 0=∞ then Xx∞=0 and the only value that X can now take is 0. This also means however that strangely ∞^2=0.

In summary, the strangeness of the whole dividing by zero thing, seems to have a logical solution: that 0 and infinity are one and the same. If you have read Godel, Escher, Bach by Douglas Hofstadter, you will be familiar with the term strange loop - and this is one. A strange loop is essentially a figure which comes back to its original value or state, as you get further and further away from the original: a strange sort of paradox. It is a weird concept, but one which perhaps could be used to explain why the universe seems to curve in a strange way I have realised: since it has been said that the universe's conept of space is undefinable in physical terms since if you travel infinitely far away from the place you are at then after an infinite distance you will come back to position 0 (or back to where you started). But of course it would be impossible for us to ever reach that infinite distance, since we would need to be able to travel at an infinite speed, and one thing is for certain: the speed of light is the limit.

Drawing Hands by M.C.Escher - a drawing which embodies strange loops