Mathematician of the week: Hubble

Mathematician of the week:

Edwin Hubble: 1889 -1953

Hubble was born on the 20^th November 1889 In Missouri and apart from changing our view of the universe he was actually rather good at sport! He held a state record for the high jump and he also played basketball for the University of Chicago.

He went to Oxford to study law and it was his return to America which heralded the start of his work into astronomy. He helped determine what galaxies are during the 1920’s and he also measured the distance to the Andromeda galaxy in 1924. This also helped to form the conclusion that it was a similar size to our galaxy!

Hubble then began to measure the distances to other galaxies and he realised that their apparent brightness was an indication of their distance from us. Now another smart thing that Hubble calculated was the speeds at which these galaxies were travelling at. He performed this calculation by using the Doppler shift of the light emitted by the galaxies. You can think of the Doppler shift as if you moved a sound source away from you the sound becomes lower then as a light source moves away the light becomes redder.

Hubble then found that the amount of redshift was proportional to its distance. This means that galaxies are moving away from us and each other as the universe expands! Brilliant! (this happened in 1929)

The Hubble telescope bears his name in recognition of his fantastic work.   

Maths fact!

Right everyone I know that we haven't had a fact in a while so...........

The sum of the reciprocals of all prime numbers diverges:

This was proved by Leonhard Euler in 1737!

This also links to the another fact that there is an infinite number of primes (two facts for the price of one!) which was proved by Euclid (300 B.C) way before Euler was even a twinkle in his mothers eye.

Problem 4

 

Here is an nice gentle problem to kick off a new season of problems. I found this one from a British Mathematical Olympiad (1993)

 

Find the first integer n > 1 such that the average of

 

12 , 22 , 32 , . . . , n2

is itself a perfect square.

Have a go you might surprise yourself!

Wolfram Mathworld

Wolfram Mathworld a fantastic resource for all you mathematicians out there! I have spent a many hours just searching for some of my favourite topics!

It is extremely easy to use and it offers the user clear definitions. It has a very simple interface where you can search for topics or just look around at your own pace. You really must try it! (I sound like some tedious advert!)

Try it out at:

http://mathworld.wolfram.com/

Maths Fact!

The set of numbers {1,2,...,n} has the property that the sum of its cubes is the square of its sum!

1³ + 2³ + ... + n³ = (1 + 2 + ... + n)²

 

 

Tell your friends and family. They will like you for it!

Hardy: A Mathematician’s Apology

Hardy: A Mathematician’s Apology

You rarely get an insight into the workings of a mathematician mind. However in, A Mathematician’s Apology G.H Hardy gives an excellent account of what mathematics is. Its usefulness (or lack of) and how it is more than just a science it is a creative art!

The apology was written as his powers as a mathematician were on the decline. This detail offers the reader an insight into how Hardy viewed his career as a mathematician and how he viewed some of his contemporaries. He mentions his work with Littlewood, his discovery of Ramanujam and his time at Oxford and Cambridge.

Hardy gives a brilliantly composed account of ‘real’ mathematics and this comes from his background in pure mathematics. He was known as ‘a real mathematician’ and this style of thinking comes across in the well-constructed, eloquent argument he proposes for his subject. His dislike of war and the lack of involvement that mathematics has had upon it is displayed in the book. He acknowledges the use of applied mathematics on ballistics however he hardly ranks them as ‘real’. He further states that the mathematics involved are “indeed repulsively ugly and intolerably dull.” You can’t help but like the language he uses.

This book is a must for anyone (not just mathematicians) as it is not only fantastically written but it offers an insight into one of the finest mathematical minds of the twentieth century.          

Maths LoL

Problem 3

Problem 3: You find a body! (Well that's your story to the police anyway)

A body is found at midnight, on a night when the air temperature is 16 degrees C. Its
temperature is 32 degrees, and after another hour, its temperature has gone down to 30.5
degrees. Estimate the time of death.

All problems are also found on the problem page (maths problems not people problems)

Mathematician of the week August Möbius

August Möbius

1790 – 1868

August Möbius was born in Saxony (now more commonly known as a part of Germany). He was an only child and an interesting fact to mention straight away is that his mother was a decedent of one Martin Luther!

Möbius was home schooled until he went to college. Once he had completed his studies at college he entered Leipzig University in 1809. At university his family wanted him to study law (and here comes the but.......) but Möbius found greater pleasure in mathematics (obvious), astronomy and physics and so he changed courses. Moving to 1813 Möbius studied under Gauss who was working at an observatory in Göttingen. So Gauss was able to coach Möbius in astronomy but being the greatest mathematician of his day Gauss was able to give Möbius an excellent mathematical education as well.

The Prussian army tried to draft Möbius to which Möbius was extremely annoyed. I am sure there’s a quote from him in response to being drafted saying something similar to, “no one will be safe from my dagger if they suggest such a thing”. He managed to avoid being drafted and he was given the chair of mathematics at Leipzig University. Apparently he wasn’t a good lecturer (we can’t be perfect all the time) but he was an excellent researcher which led to a promotion to a full professorship.

Der barycentrische Calcul was a classic piece of work on analytical geometry where he introduced the idea of a Möbius net. His name is also attached to the objects Möbius function and Möbius inverse formula these two terms came in later works.

Möbius was also interested in topological ideas and an illustration of this was a problem he posed about five sons to a king.

The problem is as follows:

A king has 5 sons each of which will be given a part of the kingdom when he dies. Can the kingdom be divided in such a way that each region has a common border with the other four?

I will leave you to work it out!

He is also known for the Möbius strip which is a two dimensional surface with only one side. You can make one taking a strip of paper twisting it by 180 degrees and then joining the two ends together. You can then dazzle your friends with your amazing piece of paper!

Mathematician of the week Alan Turing

Alan Turing: 1912 – 1952

Alan Turing was a highly talented mathematician who was born in London in 1912 and by all accounts he was average when he was at school (all the best people are). He was criticized for his poor handwriting and for following his own ideas and methods rather than that of his teachers. How often do we hear that old chestnut from teachers? However whilst he was at his school he managed to win all the prizes in mathematics. (1926).

Now what the teachers did not know that Turing was partaking in a little outside reading and this is where he gained his knowledge at an early age. I think this is evidence of the power of just taking a chair and having a read every now and again.

Let us move forward to 1931 and the young Turing went to Cambridge to study mathematics. He graduated in 1934 and was elected as a fellow of King’s College for work, which helped to prove results in probability theory (central limit theory). So far Turing had just been working in the area of probability. It was 1936 that Turing moved into what would be he foundations of what we now call computer science and he published On Computable Numbers, with an application to the Entscheidungsproblem. It was this paper that Turing introduced us to the “Turing Machine”, (he obviously didn’t call it this) essentially the machine could write or delete a symbol on a tape. In doing so it would be following an algorithm and so change state (start with one result and end up with a different result).

It is now 1939 and with the outbreak of the war Turing moved to Bletchley Park where he was involved in code breaking of the Germans. In doing so he received the O.B.E for his contributions.

1948 and Turing is now in Manchester where he had been invited by Newman. It was here that Turing produced work into computing and decidability. He was elected to the Royal Society of London in 1951 for his early work with Turing Machines.

We can’t have a Biography of the gentleman without a mention of his arrest in 1952 for a homosexual affair he was having. He actually handed himself into the police as has was threatened with blackmail. He offered no defense as he said he was doing nothing wrong (how times have changed for the good).

He was also at this point working for GCHQ. Who due to the circumstances of his arrest stopped his security clearance. We are now in the cold war and the strange circumstances of his death. Turing was conducting an electrolysis experiment and eating an apple (you can see where this is going). He ate half of his apple and died. Upon inspection potassium cyanide was fond on the apple. It was said he knew it was there.

Well that was Turing the father of modern computer science. Apparently he was also a bit of a runner! Good man.

All the usual apologies for mistakes and things like that.

Problem 3

Problem 3: This one is all about counting and choosing

You are trying to escape from a prison (you where framed of a crime you didn’t commit)

A security system uses passwords of the following form.

Two letters followed by two digits. (Letters are a,......, z and Digits are 0, 1, 2,..., 9)

How many different passwords are there?

i). Assume you could have repeats e.g. aa22

ii). Assume no repeats

Problem 2

Mathematician of the week

Pierre-Simon Laplace

1749 -1827

Laplace was born Normandy to a family who were relatively well off for their situation. Father was a merchant and his mother was from a well off farming family. Initially Laplace went to a priory school till the age of 16. He then enrolled at Caen University to study theology (this is becoming a recurring theme in our mathematicians) however during his time there he discovered his talent for mathematics (this keeps on happening too). Credit for this discovery partially goes to his teachers there.

As soon as this talent was discovered he left the university and went to Paris at this point he was only 19. When he arrived in Paris he was introduced to d’Alembert (he is well known for studying differential equations). Laplace was then tutored by d’Alembert and he assisted him in finding work.

 His early work in Paris during the early 1770’s was into difference equations and maxima and minima. He also read papers on these two topics to the Academy during this time. He also tried to gain a position into the Academy and he eventually succeeded in 1773. It was also around this time where he would begin the study of planetary motion, differential equations and probability.

In the next stage of his career during the 1780’s Laplace increased the depth of his results and gained his reputation. However he also considered himself the best mathematician in France which didn’t go down well with his colleagues (even though he was probably correct!) He was also appointed a position in the French army examining cadets and he actually examined a young Napoleon!

A point of note is that in the mid 1780’s Lagrange came to Paris to work with Laplace and although there was a rivalry the two men seemed to get along (obviously for the love of mathematics).

This now brings us to the 1790’s (yeh the French revolution!) and the time when the Academy was suppressed. Laplace then left Paris with his family and he gained a position training school teachers however this position didn’t last for long (his courses were a little advanced). Laplace then returned to the reopened Academy where he presented his nebular hypothesis. This was a modern view of the evolution of the solar system (a large cooling, rotating cloud of gas into what we see now).

In and around 1799 Laplace published some of his greatest work in planetary motion. He was able to prove that he planetary orbits are stable and self correcting. Moving into the 1800’s Laplace began work in probability and again published papers in applications of probability and errors to name a few.

So in summary:

·         Laplace was a very savvy politician and this enabled him to reach some of the positions had during his life.

·         The Laplace equation is named after him (highly important partial differential equation)

·          He was a little arrogant (and why not? He was good!).   

Mathematician of the week

Leonhard Euler

1707 – 1783

Where to begin with this gentleman? Well he was born in Basel and when Euler was still young his father trained him in some elementary mathematics (his father had been friends with another mathematician known as Mr Bernoulli). His father actually wanted him to follow him into the church and so he sent the still young Euler to Basel University where he began a general education. However (luckily for us) he was far more interested in mathematics and with the help of that Mr Bernoulli he managed to change courses and purely study mathematics.

Now towards the end of 1720’s (he was around about 19!) Euler completed his studies at the university and he applied for a position in the physics department at Basel. However he was unsuccessful and so off he went to St Petersburg. During his time in St Petersburg Euler served in the Russian navy. He managed to leave the navy due to his rise to professor of physics (well done to him).

Euler worked in St Petersburg until around 1750 and then he left for Berlin (Russia wasn’t the best place for a foreigner at this moment in time). He stayed in Berlin for around a decade under the employment of the Academy there. He did return to St Petersburg after this and this is when he lost his eyesight and he was totally blind by the 1770’s. This did not stop his exceptional output as he published half of his overall work when blind! (You can’t even imagine this!) Euler eventually died in 1783 although the Academy in St Petersburg continued to publish his work for the next 50 years.

His body of work is massive no other mathematician has come close to this. He developed: mathematical analysis made contributions to number theory and he solved the Basel problem (what is the exact solution of the following sum? ∑ (1/n²) ). He also made contributions to calculus and geometry. He investigated planetary motion and the three body problem. He introduced i for the square root of -1, f(x) to represent a function, e for the base of natural logs, π for pi and ∑ for summation and even more notation!

I could easily go into more detail and list more topics and notations but this would take me half a lifetime!

I can say I have not given this man the justice he deserves in this post. (I would end up writing a book and I don’t want to!) He was one of the greatest mathematicians ever. Well done Mr Euler what a lad!

Again I apologise if I have missed any obvious achievements or made any glaring mistakes.

Problem 1

Problem 1: A little sailing trip! (answers on a postcard)

You are sailing due south on the sea. You see two lighthouses and you are perpendicular to the first lighthouse. The second lighthouse is 10km due south form the first lighthouse (you find this distance from your chart). You then decide to measure a bearing from your current location to the second lighthouse and find it to be 214⁰. Sometime later you pass the second lighthouse and again you measure the bearing from your new location to the second lighthouse and it is now 283⁰.

Assuming you have travelled in a straight line, what distance have you travelled between the two measurements?

Mathematician of the week.

Augustin Louis Cauchy

1798 – 1857

Cauchy was born in Paris. However due to a certain historical event (the French revolution) his father decided to move to a more quiet place called Arcueil.

They swiftly returned to Paris (obviously Arcueil was to quiet). His father then began to tutor him and two gentlemen named Mr Laplace and Mr Lagrange visited from time to time, which was nice of them!

When the young Cauchy grew up he worked for Napoleon on engineering projects. This did not stop him from entering into mathematical research of polygons. This research was probably a reflection of his current situation. He returned to Paris due to an illness. People are of the opinion that this was a mental illness. On returning to Paris he rested and he then produced work on functions.

Cauchy was Catholic (not that being Catholic is a bad thing) and it seems that his views on Catholicism and his manner with his colleagues might have annoyed them.  One example of this is when a colleague approached Cauchy to ask him a question. In response to this Cauchy just pointed him in the direction of his new book and then walked off! I don’t know about you but sometimes I would love to do that. He also worked for Charles X (In Prague) tutoring his grandson. Apparently in his frustration with the prince he would scream at him. It does seem however that his manner did cause him to be passed over for many academic positions. 

Moving back to the mathematics Cauchy produced a vast amount of work during his life well over 700 papers. His major works were into analysis, divergence of infinite series and differential equations (apologise if I have missed anything).

He also researched the theory of light in which he developed new mathematical techniques. (I think it was Fourier transforms?)

Some terms in mathematics that bear his name are:

·         Cauchy integral theorem

·         Cauchy - Riemann equations

·         Cauchy sequences

This is just a very short account of Cauchy so apologies for any mistakes, inaccuracies or omissions. He was a genius and a bit of a character which can only be a good thing and for some reason I do like him even though he probably wouldn’t have liked me.

Welcome

I have always liked maths and have studied it for a number of years. I figured I would start this blog as a portal for my love of the subject. Book reviews, great reference papers and just general maths chat. I may also throw in some exciting aspects of my life along the way.