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!

Happy 3rd Birthday!

Physics, Life and Everything Else was created 3 years ago today! Long may this blog continue to prosper!

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

Relative Matters

While reading about the amazing NASA New Horizons Space Probe (which is due to arrive at Pluto in less than a year!) I came across a webpage tracking the movement of the probe. On the webpage (http://pluto.jhuapl.edu/mission/whereis_nh.php), the software NASA uses to produce nice visuals of where the probe is and other information about it includes a special statistic: heliocentric velocity.
Now, initially I didn't know what this was, but looking at the description given:

Heliocentric Velocity. The current position graphic also notes the spacecraft's heliocentric velocity - its speed with respect to the Sun - in kilometers per second. One kilometer per second is equivalent to 0.62 miles per second, or 2,237 miles per hour.

This might seem like a pretty normal statement, but think about it carefully. If there has to be a central reference point for this probe's velocity to be measured from, then that must mean that all velocities are relative (which according to the theory of relativity, they are). This must also mean however, that the sun has a velocity, and that velocity is relative to other large stellar objects, or even things as small as atoms perhaps, which have a definite position in our universe. This makes sense, in earthly terms, where we measure our velocity relative to what can be regarded for all purposes as the stationary ground.
But, say that there existed just one object in the universe, with no other atoms around to help measure its velocity. Would that object ever have velocity? Actually, wouldn't that mean that even if a force was exerted on the object (somehow without another object causing it) that the object would seemingly simply be as good as stationary, since there would be no reference points to measure any acceleration in its movement from. Do not be fooled by the fact that if you were standing next to the object, that you would see it moving away from you, since this assumes that you would also be in this completely empty universe, meaning that there are now two objects which can be measured relative to each other (you and the object).
But of course, when I said that a force was caused, and implied that this happened without another object causing it, then this would be impossible. In fact, the only way for that object to have a force effected on it, would for it to split in two (in an unexplained and spontaneous explosion) and so resulting in two parts of the object having pushed away from each-other with equal force. Of course, we could not actually measure how large that force was in terms of our standard units since that would mean that we were there as well, and would be measuring the forces relative to the current system of our universe, which is wrong since the single object (now dual-object) universe is a different universe with different physical rules of relativity - our universe contains a complex set of objects relative to each other not comparable to that universe.
The way our universe is comparable to that single-to-dual object universe is that they seem to both have started in the same way. Perhaps our universe started as a very similar object. Of course, since that object was the only thing that existed, there was nothing to compare it to and so it existed relative to itself both in size and motion: meaning in fact that it had infinitely small size and motion (or infinitely large depending on how you want to think about it). This concept seems similar to the concept of a singularity - the thing that our universe is thought to have evolved from. Perhaps physical laws and rules emerged at the beginning of our universe in a similar way to the object splitting in that alternative universe, and did so for no apparent reason (this concurs with the spontaneous expansion idea of the Big Bang Theory). Of course, maybe I can now attempt to explain the complicated notion of 'non-existence' outside the universe: perhaps this simply implies that outside the boundaries of relative relationships between matter in our universe, nothing exists since there is simply nothing to measure it relative to, since that space is immeasurable relative to objects in our universe and we cannot simply get out a ruler and say how large objects outside the 'boundaries of the universe' are or how fast they are travelling.
This relativity might also explain (in a convoluted manner) the deveptively-simple, ever-so-weird concept of gravity. The explanation might be that gravity is simply a result of the relative equilibrium of all the particles of the universe having been disturbed by that unknown force which caused the Big Bang to begin. Ever since then, there has been an imbalance in that equilibrium, however since the universe is so complex and matter-full, the particles were unable to regain their original structure (the structure they held when they were a singularity) and have simply been trying to "locally compensate". What I mean by this is that instead of the whole network of the particles in the universe simply returning to their original relative positions, they have formed 'mini equilibria' in certain areas of the universe, such as on that lovable place we like to call earth, or even in our galaxy as a whole. Were every single particle in the universe to suddenly become aligned in such a way that they might be able to come together again to reform overall equilibria, that might be more favourable, and perhaps it has happened before (a la the theory of repeating universes: where universes continually form and collapse upon themselves to form cycles of singularities). Perhaps, this will happen again in the future? Nobody knows whether it will however, and if it will, when it will happen? Or is time, in this context, strangely relative as well? I leave that question to you.

Credit to:
http://pluto.jhuapl.edu/mission/whereis_nh.php
http://www.nasa.gov/mission_pages/newhorizons/main/#.VBSbcMJdVA0

Do you believe in fate?

We oftentimes hear about the suggestion that there is such a thing as "fate", "destiny", a "plan" for humankind. These words are tossed around in everyday speech in a myriad of different contexts, whether it be the often religious notion that God has a plan for humanity and that things were set out for us from the very start or perhaps the astrological notion that the stars dictate the way different sorts of people will act each day due to various 'correlations'. Personally, I don't put much faith in astrology, but others do and recently I have come to see reason for them to do so. Why, you may ask, would any extra-terrestrial happening dictate what I might feel like eating for breakfast tomorrow. Well, perhaps these things don't affect each other directly. It is the notion of cause and effect and there is often a misnomer about what people mean when they say two different events are related to each other. In the case of the positions of the planets and the stars, what is to say that certain movements of these objects in space could actually be related to the actions of is on earth, not because they themselves caused them, but because there is a meta-cause of both these things. It is the same problem as the idea of the number of drownings going up at the same time as ice cream sales. This does not mean that one causes the other, but that a higher-order cause was involved in bringing about both, namely the fact that both these statistics peak in the summer because more people go to the seaside when the weather is hotter and swim as well as being more inclined to buy a nice, cold ice-cream cone.

Consequently you might question what this higher-order cause might be, and it is a very good question to ask. Perhaps it is a very complicated cause to do with time. Time is one of the only things that links everything in the universe with everything else, among other theories of humans such as the idea of the three spacial dimensions. Perhaps, the pseudoscientific notion of biorhythms comes into play and means that each month the stars will be in certain positions and at the same time, purely on the basis of the regular rhythm of life, you will also be full of a certain hormone which makes you more likely to feel a certain way. In this sense looking at the stars to determine our fate is completely redundant, we may as well make less work for ourselves by analysing what time it is and our past history to determine a more accurate and less confusing view of why we might feel a certain way, rather than simply because 'the moon has passed close to the line of the equator'.

Everything in the world, and in the universe as a whole for that matter, obeys a regular rhythmic pattern dictated by the forces and objects around it. These ideas stem from the most basic physical principles we regard as 'most likely true based upon our current knowledge', and were discovered by scientists like Newton and Einstein among many many others. This is why we can predict the position of the earth at a certain time, or the position of another planet relative to the earth at that time. We would see it as strange to think that tomorrow Mars would suddenly, without reason, be moving much faster around the sun than it was today. We would have to see evidence for this, otherwise we would have to probably rethink the most basic, axiomatic principles governing the physical universe.

In fact this idea of these firstmost principles is very interesting. It is strange nowadays to use the word 'theory' to describe such things as gravity and atoms for example, since these 'theories' have become so well accepted, and seem so true that they are now almost always referred to as 'facts of life'. But of course, they are facts, but only in the realms of our own understanding. We have built other theories upon them and other theories came before them that they had to in some way agree with in order to be seen as credible. But the limit of our ability to test theories we make to determine whether they are true or not comes when we hit the barrier of what we are able to observe by any means about the universe around us. If we could in some way 'observe' time in some way then I am sure we would have a very different system of physics at the present moment. However much you think about it physics is at its roots a human explanation for something we do not fully understand and possibly never will. Just because something seems to fit, however well, with our understanding, in actuality it is never good enough to be regarded as real 'fact' since somewhere along the line, often at the very beginning we had to make an assumption, or a 'given statement' such as "all things fall down on earth". What is to say that they do? All that we have to observe is five basic senses at the end of it (not even in agreement with the number of recommended senses for something to be objective and not simply down to mere coincidence), and to be honest those alone will never be good enough to say for certain that all things fall down, or in fact that the objects and systems we interact with in everyday life are really what we see them as. At the end of it they are only viewed from our biased points of view. So where does the real truth lie? Well, in actual fact until the real truth - if there is one - is discovered for certain we have to believe every theory to be true in its own right, however crazy it might sound, since at the end of it everything is still only subjective to how credible we as a population think it to be, and that never equals certainty. For now however, I suppose our theories about life, mathematics, the universe and everything are all in actual fact true, since the concept of truth is, if you think about it, as much of a human invention as anything else. It belongs to the concept of logic and lies within the same thought system as Newton's laws of motion, Einstein's theory of relativity and the laws of thermodynamics and so all these things are true in that sense.

Coming back to my original idea about fate, there is a flaw in cause and effect - that is expressed as the very weird and wonderful notion of the human brain and consciousness. If everything, as most people believe it does, follows the currently accepted physical laws, then true randomness is in theory impossible, since everything must follow a rigid and structured system. By these premises that must mean that human thought is simply a direct result of the atoms and chemicals in our brains interacting with electrical signals in a complicated yet theoretically predictable manner meaning that fate has always been decided for us. This would also mean that if we were able to computer-generate a model of a human brain at a specific moment in time with every detail included, and programmed it using all the physical laws then this should mean the brain would work almost exactly as that of the person who's 'brain snapshot' was taken (this is a similar notion to modelling the atmosphere to predict the weather), give or take a few differences resulting from the absence of the external factors the person was subjected to: stimuli. But these differences are of course very important since in the long run they might add up to make for completely different paths of events. Of course, these stimuli would theoretically come from other predictable physical events and objects causing them which in theory could also be programmed into the computer. But of course these stimuli were in theory caused by other stimuli and so on and so forth until eventually you realise that to obtain an accurate model of what that person's brain would be doing at a certain time you would need to program the entire universe into the computer. Now, here comes the big problem - what about the computer itself, it of course is an external stimulus and part of the universe, so how do we program it into the equation? Well, as you may have guessed you would now need an infinite number of computer simulations each inside of one another to do this, much as you get with two mirrors placed in front of each other forming an infinite number of images within one another, which of course is impossible. But perhaps there is some truth in this: perhaps we ourselves are just one of an infinite number of sub and meta universes each inside each other in a never-ending chain. However, as we have seen with this theoretical computer simulation of the universe, there is perhaps a beginning. What could this beginning be? God perhaps? Well, again we are jumping to conclusions - my theory could possibly be false from it's very origins and therefore further reasoning would be futile, but it could just as well be true for all anyone knows. In a sense it is actually true by definition in the strange way I described before - true just as every other theory is. It just goes to show that seeing isn't actually believing, but the other way round. If you believe something, then it true by your own definition since the universe is a very, very relative place. What might be true for you may not be true for others. Because of my computer argument I can see how fate might simultaneously exist and not exist. To predict it all you would need would be a brain of infinite capacity - God as some might describe it. You might say this is impossible, but of course so was motion according to Zeno of Elea in his Achilles and the Tortoise paradox. What is impossible for someone may not be in another situation or train of thought for reasons that people are still trying to understand. That, I believe, is one of the beauties of life and consciousness.

Credit to Wikipedia for this image of an extract from the page on axioms... Just something to think about, is this truly true?

Credit to Godel Escher Bach among other things for inspiring me to write this article.