The number 1 is really great. You can do a lot of great things with it, for instance: if you take the 18947th root of 1, you get 1. how onederful.
1 raised to the power 1, is also 1. any number raised to the power 0, is 1. Even 0 itself. That’s right, 1 is so fundamental and so powerful that raising nothing to the power of itself, is 1. Conversely, when we take the natural log of 1, we return to 0. 1 is mysterious. Further to this, when we find the factorial(!) of 0 (where the factorial of a number is found by multiplying that number by every number before it until you return to 0) we get 1, such that 0! = 1.
Arguably one of the most famous identities in mathematics, Euler’s identity, shows how some of the most exciting numbers in maths all relate back to 1.
If we take e, a transcendental number, which is the base of the natural logarithm, and an important number in the study of continuous compounding interest; raise it to the power of i, the square root of negative 1 (which makes the life of an engineer a lot easier), and π, the ratio between the circumference of a circle and its diameter, we get -1. It’s all marvelously coincidental.
Historically, the number 1 is so important. When humans, or anything that even bore resemblance to humans, started counting things, they started by counting 1 of something. We’d be so lost without 1.
1 helps every other number to identify itself. You can multiply any number you like by 1, and you’ll get that same number. It is so useful, versatile, and beautiful.
I love 1, and so should you.
Check out this video from Sal Khan about the identity property of 1. I understand it's basic, but I think that all too often we fail to think about how useful fundamental knowledge can be.
Identity Property of 1: Identity Property of 1
My Feynman Lectures have arrived!
So I recently turned 18, which is exciting. For my birthday, my fabulous brother and my as-of-fairly-recently-sister-in-law sent me a £100 Amazon gift card, which is also exciting. I suppose I could have saved the money and used it tactically throughout the year, but what would be the fun of that? I decided to splash the whole amount out on The Feynman Lectures on Physics boxed set. While it’s true I could have bought the three volumes in paperback for a fair amount cheaper, I wanted the nice little box of hardbacks. I mean, these books are to physicists what Shakespeare’s Works are to English Literature students. It’s all very exciting.
I was so happy to receive my amazon parcel today, that I guess I wanted to post something about Richard Feynman, and why he is so great.
Over the summer holidays, I took out ‘Don’t You Have Time To Think?‘ which is a compilation, put together by Feynman’s daughter, of many of the letters that Richard sent and received over the years. It includes sensitive letters to his mother, and to his wife- as well as notes to people of high authority, which often upset them. Coming away from this book, I found the man to be something quite marvelous.
I can’t say I read all that much, but there was just something about this man that was all very exciting.
Richard Phillips Feynman was a remarkable physicist, educator, and bongo player. He was eccentric, as any real physicist should be, as well as massively innately clever. He loved what he did.
Feynman was the pioneer of a ‘little’ concept known as nanotechnology (I sure hope that pun translates), as well as quantum computing, a field which is rapidly expanding as of late. At one point in his career, Feynman decided to refrigerate some helium to see what would happen. When he supercooled some Helium, he discovered a property of the fluid that he coined as ‘superfluidity’. That is to say that, when cooled to a relatively very low temperature, helium appears to not have any viscosity.
The most important piece of work of Richard Feynman, in my opinion, is his diagrams. Anyone that has studied Physics past the stage of GCSE will have drawn one of Feynman’s famous diagrams.
Beta Minus (B-) decay
Feynman was awarded a Nobel Prize for Physics, jointly with two friends:
“for their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles”.
This included, of course, his diagrams. Feynman was the first to model the idea of the ‘weak decay’, a concept which allows physicists to describe most of everything that happens in the universe, which is quite something.
There are a lot of books written both by and about Richard Feynman, and rightly so. The man was curious, clever, and charismatic. He was everything a physicist should be, and then some.
It is for these reasons that Richard Feynman has been listed in the top 10 physicists of all time by Physics World, and he is definitely in my top 5.
I was watching the BBC News this evening, and one of the main bulletins was about Global Warming. I knew I would have to brace myself for another almost meaningless news report that discusses the issue in the least scientific way possible. I always find that mainstream media almost insults the intelligence of their viewers – I am perfectly aware that the entire public does not possess high levels of scientific knowledge (nor is it even necessary), but this does not mean that everything needs to be oversimplified to the point of which, there remains little or no science at all!
I spent the five minutes just trying to understand the point of the bulletin, as far as I could tell, it was just regurgitating the same story that they normally do – ‘The Earth’s temperature is rising due to CO2 emissions caused by human activity, the Government calls for reductions in everyone’s carbon footprint’. The report claims that although temperatures have been increasing in comparison to the last few decades, since around 2002, temperatures on Earth have apparently decreased. That was pretty much it. No explanation, no real insight into anything affecting anyone, they simply produced a pretty graphic of the temperature change over the decades, and then showed some video clips of some awful weather conditions experienced in the Americas and China.
I still do not really know what the report intended to explain…
Anyway, the general public is aware of the basics surrounding global warming, and they are also aware of the simple things they could be doing to help reduce the impact of our horrible CO2 emissions. They are however, very ignorant too, and are unwilling to make the minor changes in their lifestyles. News reports from broadcasters like the BBC do not help the matter either – if they actually provided some real science bulletins, perhaps our public would respond a bit differently.
Not too sure why I turned this into a blogpost, it is more of a rant than blogging about scientific knowledge for today, so apologies. Perhaps I will research the current state of Global Warming, and return with a well-informed and constructive post!
As I walked home from my sixth form today, I started to think about the human immune system.
I thought about the fact that due to the rate of reproduction of bacteria and the fact that this high rate (relative to the rate of reproduction of humans) means that evolution in germs occurs much more rapidly than humans.
I’m not a student of biology, so my view on this might be very naive, but to me it seems that germs can recognise what humans are immune to and respond to these immunities swiftly- that’s how superbugs happen, right?
I find it quite amazing that the immune system relies only very loosely on genetics to be able to fight this ongoing battle against disease. We gain immunity to something, and it responds, and our immune system responds back. It’s quite marvelous that all this happens while we’re so blissfully unaware.
Excuse the probably massive amount of inaccuracy in this post but, I just wanted to share my thoughts.
In the past few years, people have become more and more concerned with the effect of fuel emissions on the environment. At the same time the global demand for air travel is increasing. So NASA (yes they make planes too, yes I was surprised by this) amongst others have been trying to come up with a solution that reduces emissions whilst still providing the performance needed – and as you can see from the picture, looking awesome in the process.
The Blended Wing Body with Distributed Propulsion provides a much greater aerodynamic efficiency and could reduce fuel emissions by over 75%. So what’s the problem? The technology is just not there yet. This is set three generations ahead of current aircraft and has the major engineering issues to match. For example: With Distributed Propulsion two jet engines would need to provide power to multiple fans along the back of the aircraft, this would not work for our regular resistive, power losing wires so the system would require superconductivity to ensure no power is lost. Superconductivity requires very cold temperatures possibly achieved with the use of liquid hydrogen cooled to 30oK (or -243oC). If not kept at this temperature the hydrogen could rapidly expand and blow the whole plane up.
There are many other issues but that is one of the more dramatic.
So, Will we see this plane in the future? Maybe not.
Is it a really cool idea that looks like sci-fi spacecraft? Hell yeah.
Infinity. Simply thinking about it is enough to drive people insane – no literally, Georg Cantor the man who more or less wrote the book on infinity was regularly hospitalised for his mental health.
Though that may have been more to due with the criticism he received for working with the infinite. Leopold Kronecker, another mathematician of the time, described Cantor as a “scientific charlatan”, a “renegade” and a “corruptor of youth”. There were many mathematicians and philosophers who were equally appalled by Cantor’s work but there did stand some who supported his theories. David Hilbert famously stated “No one shall expel us from the Paradise that Cantor has created.”
It is with that view that he developed a series of problems based around a hotel with an infinite number of rooms.
Here are the constraints:
The first problem says that the hotel is completely occupied but another guest arrives. How can you accommodate this final guest?
The answer is simpler than you might think.
Move all guests along one room and put the new guest into room 1. All constraints are fulfilled, no rooms have been shared or built, the new guest is accommodated and if you wanted to know where the guest who was in room 343 is now, you could with confidence say room 344.
But wait, did we not just show that ∞ + 1 = ∞? The hotel was already infinitely full and we added one more so this statement must be true.
That is by far not the weirdest thing found in these problems.
Take problem 2, the hotel is again full but this time a special infinite coach arrives with seats numbered 1, 2, 3 etc. How can these be accommodated?
Again a surprisingly simple answer can arise.
Move every individual to 2 times there room number and put the coach load into the gaps.
So ∞ + ∞ = ∞
Lets try an infinite coach with seats evenly numbered 2, 4, 6 etc.
Just half the numbers and put them in that room.
As evens represent half of all integers this also suggests ∞/2 = ∞
In fact there are many different ways of making that particular size of infinity called ‘countable infinites’.
There is a larger infinities called ‘Uncountable infinities’ and then infinitely more infinitely larger than that.
Well then. I hope that sufficiently confused everyone involved. I’m now off to go rock back and forth in a corner whilst drawing the symbol ∞ over and over again until I fall asleep.
P.S. Hi, this has been my first post. Thank you very much if you have read down this far. I study Maths, Further Maths, Physics and Mandarin at Sixth Form and I intend to study maths at Uni.
Some years ago, I read an article where a man discussed the prospect of making wine out of his diabetic grandmother’s urine. More recently, I tried to rediscover this article, but I couldn’t find it anywhere on the internet. Instead, I found one better.
James Gilpin turns urine into whisky.
Type 1 diabetes is where the body does not produce enough insulin in order to regulate blood sugar levels. As a diabetic himself, Gilpin attempted to find some use for his sugar-rich urine.
He first approached his diabetic grandmother for her urine for a trial, and worked out with her the correct way to ask an elderly person for a cup of wee. He went round communities of old people, exchanging soft toys and cushions for cups of urine.
The conveniently large sugar molecules in the urine formed large crystals, and were hence easy to remove and purify separately from the fluids. The urine is cleaned with a very similar process to that used in the purification of the water that comes through our taps.
The removed sugar molecules are added to the mash stock- to accelerate the fermentation process of the whisky- these sugars are usually made from the starches in the mash.
Once the solution is fermented into a clear alcohol spirit, whisky blends are added to add colour, viscosity, and flavour.
Gilpin bottles his product and labels it with the name and age of the urine donor.
His project, Family Whisky, is not one for the commercialisation of alcoholic produce sourced from the exploitation of the elderly. Instead, it is a project aimed at raising awareness of the complications that diabetics have to deal with when they go about their day to day business, he says:
The fact that I am associating alcohol with a severe medical condition has upset some medical professionals that I have met during the process of my project. This was a very deliberate provocation on my part as I wanted to have a dialogue with health care professionals about the real complications of living with diabetes. People still want to drink, eat unhealthy food and experience the messiness of everyday life. In my personal experience this is often overlooked by professionals who can give the impression that theses things simply should not be a part of life as a diabetic patient but they are. I am interested in finding ways in which designed systems can help overcome these very social problems.