Pi..?

How can a number with an endless array of seemingly random decimal places with no pattern whatsoever be so useful to maths? Take Pi for example: Have you ever wondered how your calculator actually comes up with the number in its tiny computer brain? How did we even calculate it in the first place?

GEOMETRY

Some of the first methods used to calculate pi were geometrical. 

Although Pi is equal to the ratio of the diameter to the circumference of a circle, you may be thinking why don't I just measure the diameter, work out the circumference and then calculate the ratio. If only it were that easy :} Length's are continuous, which means if you measure something, your ruler might say it is 20cm, a more accurate ruler would tell you it was 20.3cm, a more accurate one would tell you 20.34 and so on.. forever, so it wouldn't measure a definite value. And anyway, if you didn't have pi to start with how would you calculate the circumference of the circle to begin with. 

In 250bc, Archimedes found upper and lower bounds of pi using an algorithm involving shapes. He drew a regular hexagon inside and outside the circle. then he used a regular polygon with 7 sides, then 8 sides and so on until he reached a 96 sided regular polygon. By finding the perimeters of these polygons, he proved that

223/71<pi< 22/7 (3.1408<pi<3.1429)

In 480 AD, Chinese mathematician Zu Chongzhi used a similar method on a 12,288 sided polygon, and he found  π ≈ 355/113, this remained the most accurate approximation to pi for the next 800 years (accurate to 6 decimal places).

INFINITE SERIES

The real revolution in calculating the digits of pi came with the invention of infinite series.

The series formula:

Is a striking representation of pi. But it's still not easy to compute the digits of pi since this is an infinite series

There are many representations of pi as an infinite series, Wallis's formula states:

But these methods can be just as tedious as the geometrical method, As Newton himself states "I am ashamed to tell you to how many figures I carried these computations" 

COMPUTERS

But with the development of computers in the mid-20th century, the hunt for pi was revolutionised. Simply by using a desk calculator, John Wrench and Levi Smith reached 1,120 digits. The mathematics and computing were inextricably linked, as in the same year, a team led by George Reitwiesner and John von Neumann used  an inverse tangent infinite series achieved 2037 digits with a calculation that took 70 hours of computer time on the ENIAC computer. 

The record, now always relying on an inverse tangent series, was broken repeatedly until 1 million digits was reached in 1973. But the current record.. by computer scientists Alexander J. Yee &Shigeru Kondo is 10 trillion digits!!

But really.. what's the use for calculating all these digits of pi? Did you know that you only need 39 digits of pi to calculate the circumference of the entire observable universe within the width of one hydrogen atom. Yep. 

To conclude. my best guess is that your calculator just stores the digits of pi to 9 places. 

Also check out this video where Matt Parker calculates Pi with Pies

Admin O.

Why can't you divide by 0?

We've all been told in school that you can't divide a number by 0, but why is this exactly? Even when you try to divide a number by 0 on your calculator, you get an error. Lets look at an example:

'What is 6 divided by 2?' You can rephrase this question as: 'what number, when multiplied by 2, will give me 6?' Here, the answer is obviously 3. There's only one answer, it couldn't be anything else- can we say the same for 6/0 ?
Lets do the same sort of thing: 'what is 6 divided by 0?' Well, we can rephrase this as: 'what number, when multiplied by 0, will give me 6?' That's.......... hmm........ D: 
Clearly a problem here. '0 x ? = 6'. Apparently, there are no solutions for '?', so you can't divide a number by 0. How about infinity? If we say '?' = infinity, does that solve the problem? If we look at the hyperbolic graph of 'y=1/x', we find that, as we get to x=0, one side of the graph sort of 'shoots up' towards positive infinity, whilst the other side 'shoots down' towards negative infinity.

However, even if we say it's positive or negative infinity. If 0 x infinity = 6, we could say the same for ANY number divided by 0, e.g. 4/0 = infinity, so 0 x infinity = 4. In fact, using this idea, you can say that 0 x infinity = any number, which doesn't appear to be particularly logical. Or does it? Infinity is a very abstract concept, far beyond the scope of human comprehension; infinity doesn't behave like a number, and it's properties are mostly unknown, so we can't say whether this is a logical idea or not. Basically, we're not sure, so we say it's undefined. Anyway, if you look at this graph, you never actually get to x=0, so how can we define what it is we get when x=0, when this value doesn't even exist on the graph? Or is that what infinity is? Is +infinity the sum of all the positive numbers, and is -infinity the sum of all the negative numbers? Even then, there are complex versions of infinity to deal with (to do with complex numbers, involving the square root of -1). Confusing indeed.

So, how about 0/0? This causes even more headaches, because there are so many more things to consider here. You might say that any number divided by itself equals 1, so 0/0=1. However, there's a VERY good reason why mathematicians don't agree with this. Let's assume that 0/0=1. Then:

               2 = 2x1 = 2x (0/0) = (2x0) /0 = 0/0 = 1                Therefore, 2=1

Ah.

As you can see, '0/0=1' clearly doesn't work, as we could then make any number equal to any number. So, why don't we say 0/0=0? Surely, 0 divided by any number is 0? Well, lets use same idea we used for 6/0:
'what is 0 divided by 0?', can be rephrased as: 'what number, when multiplied by 0, will give me 0?' Hmmm..... that's........ EVERY NUMBER, not just 0. Therefore, we can say, for example, that 0/0=3, and 0/0=5. Hence, 3=5. In fact, every number=every number. 

Doh.

Another problem. Many mathematicians like to say 0/0= infinity, but, even then, they have to be careful, as infinity is definitely not a number, and must be treated with caution. Indeed, you can't even define infinity as a certain entity, as there are different kinds of infinity, as explained above.

So, basically, dividing by 0 makes no sense at all and it would be illogical to give it any sort of value. We just have to live with the fact that it's 'Undefined' and always will be, or else the entire mathematical system would fall to pieces.

If you want to read about dividing by 0 in a bit more detail, you can visit the Wikipedia page: http://en.wikipedia.org/wiki/Division_by_zero 

see ya!  :)
Admin. H

Maths: Our flawed system

When one looks at maths, one usually thinks of an absolute, a truth, undeniable proofs and facts and calculations. 

We often think of every sum, calculation and equation as unmoving, unchangeable or completely perfect.

For example:
1 + 1 = 2

This is an example of a mathematical FACT. We know its true and we know that the answer cannot be different.


Another example goes a little something like this:
1/3 + 1/3 + 1/3 = 1

Again, mathematical fact. Truth.

BUT wait, what happens if we convert these fractions into the decimal system?
we get this...

0.333.... + 0.333... + 0.333... = 1

And as quickly as that we are met with an obvious flaw.
The answer to this equation Should be 0.999....  right?

OR should it?

The 'truth' is that 1 and 0.999.... are both correct answers to that equation. Not only that, but;

0.9999... IS  1,
(there are various proofs for this, you can search them up if you want.)


But it doesn't make sense!?  
0.999... < 1   
0.999...  = 1
Well, ya just gotta get used to it!  There will be many problems like this when you dive  deeper and deeper into maths, and the fact is that we don't have an absolute answer for everything.

But I think that's good. It gives maths personality, a sense of wonder around it. It reminds us that maths is limited by our minds and cannot go beyond it, since we are the ones to conceptualise it. It reminds us that maths is far from complete, far from perfect and it gives us the motivation and drive to get to as close to perfect as is possible.

BY highlighting this one simple 'flaw' in mathematics I hope I have shown you that maths is a lot more complicated than absolute facts and irrefutable claims. I hope I have shown you that maths is not 'finished' and that there is always more to discover, always more to learn and most of all, there is ALWAYS room for improvement.

Thanks for reading!

HK

Is maths 'cool'?

How many times have you heard someone say: "Ah maths! Hate it. Never understood it. Not for me". 

Unbelievable, right? Why is it so cool not to like maths? Why is it cool to be afraid of it? I mean you don't hear anyone saying the same about subjects like English, Geography or History. But they're really just as involved in our daily lives as maths!  

Think of it like this... A mathematician probably knows Shakespeare's work pretty well, I mean we're all familiar with a number of his plays and poems... But how many poets are familiar with the work of Leonhard Euler, Carl Gauss or even Sir Isaac Newton? The funny thing is.. maths is the only subject that is actually true.

Think about english. We write about our opinions on pieces of work, we try and understand the intentions of the authors themselves through careful analysis, yet often you'll find if an author writes something like "He wore a black tie", an English teacher might say "it depicts the bitterness and harshness of the deep abyss created in the heart of the character while underlying his need for deception in the outside world", when in reality, all the author really meant was that he liked black ties.

It's even worse for the sciences. Our understanding of the world around us is constantly changing and improving, in fact, physics classes taught at the start of the twenty first century wouldn't have even known about the existence of atoms, something so basic yet fundamental to our understanding of the universe today. It's not just adding to what we know, but changing what we thought was right beforehand!

But basic principles of maths will never change. It's an undeniable force of our world that has been, and will be used for developments in so many fields of work.  The simple fact that you're reading this article on a computer or smartphone that required an advanced level of mathematics to create is an homage to the feat of many mathematicians of centuries before us.

Of course, as Hannan mentioned earlier, mathematics isn't quite 'perfect', and there is still always room for improvement, but I don't understand people who think things like trigonometry or calculus are almost 'from a different world', No, its not. It's maths, it's true, it's useful and it can be used almost everywhere. Mathsassins, we can't let this problem seep through society! 

A History Of.......Pythagoras

In order to truly appreciate the magnitude and beauty of mathematics, it's important not only to learn about the maths itself, but also the people who took great strides in developing our understanding of mathematics. As the title suggests, this post is all about Pythagoras (not so much the theorem, but more the man himself):

Who Was Pythagoras and Where Did He Come From?
Pythagoras was Greek, born on the island of Samos in North-east Aegean around 569BC (exact date is unknown). He studied Philosophy mainly, though also attended lectures on Cosmology and Geometry. His later life was highly eventful, particularly upon being captured by Cambyses II, the King of Persia, when he travelled to Egypt. It was soon after this that he was exposed to the world of mathematics...

The Pythagoreans
After his capture, he was taken to Babylon as a prisoner, where he learned Babylonian mathematics (as well as musical theory). Upon committing his life to mathematics, he later founded the school of Pythagoreans in Crotone, Italy. It is this mysterious cult of the Pythagoreans for which he is best known, for they had very different beliefs to mainstream society. Not only were they vegetarians, they were strictly against eating, or even touching, any sort of bean, probably because they thought beans were created from the same material as humans. They walked bare foot, and, most importantly, believed that the universe was mathematical, and that all numbers and symbols had spiritual meaning. This belief was based on the idea of Platonism, which suggests that all numbers exist, though not in the same space-time dimensions as our universe. They also strongly believed in the idea of certain humans having a higher level of soul purification, and thought pure mathematicians, and those who merely contemplated on life, lived in the highest plane of existence, or had the purest souls.  

Pythagoras's Theorem and A Proof For It
Many mathematical theories were associated with being discovered by the Pythagoreans, though the most famous, of course, is Pythagoras's Theorem, which you most likely first came across in school. For any right angled triangle with sides 'a', 'b' and 'c', where 'c' is the hypotenuse (longest side): a^2 + b^2 = c^2. Pythagoras probably didn't have a proof for this (most likely he just believed it to be true), but many proofs have been found. The proof I've explained below is one of my favourites:

As you can see, there are four identical right-angled triangles in this square, arranged so that they form a smaller square inside, where all of the sides are the Hypotenuses of the four triangles. There are two ways to calculate the area of the large square:

1. The obvious way is to square the length of a side. Each side has length 'a+b', so the area is '(a+b)^2', which expands to give: 'a^2 + 2ab + b^2'

2. An alternative method is to add the areas of the four triangles and the smaller square. Each triangle has an area of 'ab/2', and the square has an area of c^2, so the total area is: '4(ab/2) + c^2', which simplifies to: '2ab + c^2'

Now, these two calculations both give the area of the large square, so we can equate them (make them equal to each other): a^2 + 2ab + b^2 = 2ab + c^2. 
'2ab' will cancel on both sides, leaving:
a^2 + b^2 = c^2    

The Death of Pythagoras (According to Legend)
Since Pythagoras lives so long ago, it is near impossible to be sure of certain details, such as how he died. According to legend, enemies of the Pythagoreans set fire to his house, forcing him to flee for his life. He soon came upon a bean field, and his belief that beans should not even be touched was so strong, he would not enter it. He declared that he would rather die that enter the field, and thus allowed his enemies to capture him and slit his throat. So the life of Pythagoras came to a sorrowful end; though the legacy he left behind shall live on until the end of time.

Thanks for reading everyone, hopefully it was an insightful experience for you. If you liked it, or if you have any queries, make sure you comment in the comments section below. Hopefully, within a couple of weeks or so, I'll have done another post about Leonard Euler. Keep checking back here for more awesome stuff about maths. Until next time! 
Admin H.   

Graham's Number

Anyone with arithmophobia (a fear of numbers), get away from here NOW, becuase the number I'm gonna talk about is the king of massive numbers. Of course, there's no limit to how big a number can be, but Graham's Number is possibly the biggest number ever to be used constructively (as in, it's actually been used to solve a mathematical problem). Before I show you Graham's Number though, think about this, as it'll help you to get a slight idea of how big a number this is...

Your brain is capable of storing hundreds of millions of GBs of data. All the information you've ever received since you were born is stored in your brain. Sure, you forget things, but something can happen which jogs your memory, and those forgotten memories seem to 'come back'- so they're always stored in your brain. Basically, your brain is pretty impressive right? Well, Graham's Number is so unimaginably massive that, if you tried to store it in your brain, your head would collapse into a black hole. No exaggerations, it's a fact. So, Graham's Number...

To understand this, you need to be familiar with arrow notation. Since there's no way I can use arrows here, I'll use dashes (/). Basically, 3^3 (3 cubed), can be written as 3 / 3 ('3 arrow 3'). Also, 3^3^3 (3 cubed cubed- 3^27) is written as 3 / / 3. Got the idea? Okay, here we go...

Start with 3 / / / / 3. This is already an insanely large number. It's so large that your calclator can't even process it. Try typing it in and it'll come up as an error. Lets call this number g1.

Now, lets do a similar thing, although, this time, there are gonna be LOADS more arrows. In fact, there are g1 number of arrows. O______o   Mmhmm, that massive number from before is how many arrow we now have: 3 / / /............................ / / / 3. This is g2.

It doesn't end there. g2 is unimaginably vast, though it's not even close to Graham's Number. We continue in this way... so g3 will have g2 number of arrows. This goes on all the way up to.........
g64, which has g63 number of arrows. <= Graham's Number. Pretty big eh?

You might be thinking, 'how the heck could that actually be a useful number?' Well, it's a pretty complicated problem that's related to Ramsey Theory and a Hypercube. I won't talk about it here- I'm sure Graham's Number is already enough to mess your brain up. If you're interested though, read up about it, it's pretty cool. If you like our blog, make sure you follow it; and tell all your friends about it. Admin H.  

Degrees vs Radians

For those of you lucky enough to be taking maths at as level, you'll have come across a lovely new way of measuring angles - Radians. It really makes you think why did we choose to be using degrees for so long if use of radians at 16 was inevitable. I mean 360 is a pretty random number.
Why 360? well we don't know. Some people think it's because ancient astronomers thought that there were 360 days in a year, yes, stupid. So why did we carry on with it? Why is it when we found out there were actually 365 days in a year we stopped using degrees? well 360 is actually a very useful number, lots of numbers go into it (oi oi), like 1,2,3,4,5, and 6, this also links to the use of counting with sexagesimal numbers (oi oi) which basically means counting in base 10. But that's a whole different story. In short, it makes 'sense' to use radians rather than degrees, but we find it harder because we've been using degrees for so long.
Admin O.