Archive for the ‘Math in society’ Category

Practical Uses of Blockchain Mathematics

To the extent that people consider blockchain technology it’s in combination with the bitcoin. Blockchain is actually a record keeping instrument that is versatile and has uses well beyond that of digital currencies. However Bitcoin’s libertarian mechanic – a validation network without a central authority in office – is not a part of blockchain and might be discarded for government uses of the tool. As a permanent ledger of transactions, blockchain serves as the core component in its bitcoin application. Computers generate them however they still have some serious horsepower required to hash transactions and complete the calculations. Every ten minutes or so, a brand-new block gets added which everybody can see. There isn’t any authority that issues them, Deputy U.S. Chief Technology Officer Ed Felten said.

One of the best explanations I’ve found about blockchain is actually a short video on the BBC website which you can find here – If you’re interested in the more specific subject of digital currency, the financial programmes all now cover the rise of Bitcoin, Ethereum and the other currencies. These are all accessible online however you may have some issues if you’re outside the UK. Using a VPN used to work well to access the BBC but then they started to get blocked, update report here.

Although digital currencies are driving forward the use of blockchain, there are significant developments in other areas too. Agency or A business may use that concept as-is, or it might insert itself. The U.S. Postal Service, for example, has spread the idea of maintaining and producing a Postchain platform. Institutions such as Goldman Sachs are currently researching using blockchain. Felten advised this Information Security and Privacy Advisory Board in June the blockchain could enable contracts that were complicated removing the possibility of human error to be released months or years after establishment.

Travis Hall, a policy analyst in the National Telecommunications and Information Administration, stated government agencies may use blockchain for a slew of instruction activities, such because keeping voter and health care records up-to date, controlling your stresses property titles or monitoring certificates and authentication for Web of Things devices for cybersecurity functions.  There are dangers in this though not withstanding the fact  that many people are able to hide their locations by using devices such as online IP changers just like these.

Since Cryptography underpins the entire series of records, keeping this security of cryptographic keys will be essential if authorities plan to rely on blockchain, Felten stated. Rew Regenscheid, a mathematician at this National Institute of Standards and Technology, stated future cryptographic solutions could offer multiparty digital signatures and privacy enhancements.

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Probability Theory – An Introduction

Specifically probability theory is a distinct branch of mathematics. Most people’s first introduction revolves around some simple random device such as tossing a coin or rolling a dice. It’s the easiest way to start to understand a subject that although simple in concept can get very complicated in the detail. So returning to that dice, if you roll the die in a way that is totally arbitrary, the likelihood of getting any of the six faces is one out of six. It’s very likely that the initial study of Probability theory grew from issues encountered by 16th century gamblers.

What makes probability theory is that it may be used to ascertain the outcome that an airplane will crash that the lottery will be won by someone. Problems inspired theory and some of the most important came from Geroloma Cardona who was an Italian mathematician working in the 16th Century. Cardano’s work even received little attention and had little effect since his manual didn’t appear until 1663 in print. The philosopher and mathematician Pascal became intrigued that he started studying problems. He discussed them with another mathematician and they laid the basis of probability theory.

It’s an important branch of mathematics and sometimes difficult to find useful information on the more advanced statistical components. In decades gone by many maths students certainly in the United Kingdom looked to late night televised lectures from the Open University. Nowadays there is plenty of material available online although the BBC is still a great resource for maths, you can watch most of the programmes on the BBC iPlayer and download them like this.

Theory is concerned with determining the connection between the number of the number and times some given event happens. Probabilities could be done in two manners: empirically and theoretically. By supposing every event is equally likely, the likelihood that the coin ends in the head is or 0.5. The likelihood is then equal to the amount of minds really found divided by the total amount of flips. Probability is always represented as a fraction, for instance, the amount of occasions a 1 dot turns up when a die is rolled or the amount of occasions a head is going to turn up when a cent is flipped.

Therefore the likelihood of any occasion always lies somewhere between 0 and 1. In this range, a likelihood of 0 implies that there’s no likelihood at all the given event’s occurring. A probability of 1 implies that the specified event is certain to occur. Probabilities might or might not be dependent on each other.

Further Reading:
BBC Detects VPN Programs –

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Common Childhood Problems with Maths

During an early phase of a child’s growth, mathematical issues or disabilities could contribute to enormous fear in their own minds. There are lots of unique skills accountable for natural or psychological deficits that could result in your kids requiring mathematics homework help. They could have a negative influence on your child’s cognitive advancement. Let’s attempt to gouge these:

Confusion Managing amounts

Universe fundamental computations like division, multiplication, addition, and subtraction result in math on the entire world. Memorizing and remembering straightforward equations, according to those essential operators, is crucial and where many children falter. These basic calculations develop advance mathematical understanding. It may possibly be prevented by employing an internet math tutor from throughout the world which could come for affordable and possess a comprehensive understanding that could prove very helpful for your children.

Difficulties with computers

There are lots of children around who have great understanding of mathematical theories, but implementing exactly the exact same on computers prove to be a massive undertaking for them. Errors associated with writing numerical, estimating amounts, or misreading personalities, are normal. They wind up fighting doing the fundamental jobs. They surely require help in the shape of online mathematics tutoring at the place where they can learn blending computing and mathematics to enhance their mathematical thinking skills.

Correlating math with reality

This really is among the most frequent issues faced around by children. It’s the inability to associate the realistic way of mathematical theories. Understanding symbols, formulas, shapes, etc., and recalling the same in regards to employing in real life, is 1 hell of a project. It’s the obligation of parents or perhaps somebody who can help your children with math homework assistance, so that actual life program becomes possible with those children.

Putting connections between mathematical entities

It’s hard for children to establish relations between multiple mathematical situations, scenarios, interdependencies, etc. that is essential to make matters relevant. Even an internet math tutor ought to be able to understand that the limitations lying at the mindsets of these children. The mathematics abilities ought to be anchored so they can comprehend items readily.

Learning math wrong way

To get a huge majority, talking, writing, and reading, math itself is the largest hurdle. They confront loads of issues in pronouncing mathematical conditions, facets, terminologies, and much more. They can not comprehend verbal or perhaps written explanations. They find it challenging to interpret things too. Online math tutoring can definitely help sort out these problems. But, in addition, it depends upon how well the children take things favorably.

Useful Links:

BBC BiteSize on Mathematics –

Accessing BBC iPlayer from France and Abroad –

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Has Someone Actually Solved the Riemann Hypothesis

Sometimes a paper comes along which can breath new life into a subject or problem long thought unsolvable. This year a trio of mathematicians looks like they’ve done just that in offering a new tactic to solve the ‘greatest unsolved problem in mathematics – the Riemann Hypothesis.

This paper has just been published in a maths journal called Physical Review and suggests that the analysis is proven correct then it can also be used to prove the Riemann Hypothesis.

 Prime Numbers

Predicting Prime Numbers

For those whose lives are not centred around mathematics this might sound a little obscure.However for mathematicians it represents fame, success and of course cash.The solution to the Riemann hypothesis is one of the seven Millennium Prize problems which cover the most difficult problems in maths.   For more information on this prize have a look on BBC iPlayer where there was a recent maths documentary, this link shows how to access it from outside the UK.  Every one of these problems comes with a one million dollar prize for a solution.

This hypothesis is names after the German born mathematician Bernhard Riemann.It’s such an important problem because it offers a method to understand the distribution of prime numbers. If a method was found it would completely revolutionise mathematics.Being able to work out how may prime exist in any given situation would make many branches of the science much, much easier.

So where is this solution hidden, well it is suggested it lies in quantum mechanics.

An amazing statement from this paper proposes that quantum mechanics could solve the Riemann Hypothesis. This difficult area of physics usually used to try and make sense of some of the smaller scales in nature.

So what’s in the paper? Well the authors have suggested that the existence of a quantum system of energy corresponds to the proposed conditions in the Riemann Hypothesis.They have also defined a specific variable called the Hamiltonian Operator as the crucial part of this system.

If this all works out then the method effectively reduces the huge problem of the Riemann Hypothesis down to the level of the Hamiltonian Operator. A mythical problem that was almost deemed impossible to solve suddenly becomes much closer.The paper is only in the first stages though and peer review is next which could take some time.

But it certainly has created some excitement for anyone who has even a passing interest in mathematics.

Further Information: BBC News Streaming

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Should We Follow Japanese Maths Teaching?

Improving mathematics standards in the US has been a common goal since the Common Core State Standards were introduced. Unfortunately if we use the National Assessment of Educational Progress figures there have been no improvements in maths at all during the last few years.

This has pushed American educational experts to look overseas for some inspiration, and they think they’ve found some answers in the way the Japanese teach maths. Unlike the traditional US strategies which focus on memorizing, the Japanese method will focus on solving mathematical problems.The method is Sansu arithmetic and it actually aligns quite neatly with the US Common Core so it wouldn’t actually be that difficult.

It’s actually somewhat ironic that the method adopted by the Japanese was actually first identified in the USA. The National Council of Teachers of Mathematics investigated this method in the 1980s however it was never officially adopted.And so, some 40 years on our children are still learning mathematics by memorising from simple sheets.

The Japanese instantly recognized the benefits to students of being able to create their own solutions and methods for solving problems.  It means that in classes, you can create a context and student will learn much more easily than simply learning by rote from a sheet of paper.  A similar method has been tried in some European countries including in Germany, article on BBC about German education on BBC iPlayer, access here.  The focus is to learn and interest students by a concept called hatsumon (addressing a concept through questioning). If it taught well, students will see their learning benefit them in real life situations.Which also helps promote elements of self confidence and some enthusiasm for the subject.

Lessons are created by individual teachers who then try the lesson in front of students and other teachers including a professor. The lesson is then discussed with the teacher so that feedback can be given and modifications made if necessary.If there is no feedback teachers are forced to make their own decisions on the quality of their own lessons.

These might seem like simple, common sense steps but unfortunately they are missing in many Western classrooms.There are many critics of the Common Core standards despite being endorsed by most educational organisations in the United States. One of the issues though is that teachers are given very limited training in the method.

A shortcoming which we’ve seen before. In the 1960s, there was a big push to introduce ‘new maths’ to push onto the space age. There was much enthusiasm but little change simply because nothing was invested in training the teachers in new methods.

There is a feeling that maths teaching in the US should change but a lack of direction and funding to implement this change. Japanese teachers get much more support whenever new methods are implemented across any area of education.Which in turn is usually reflected in the scores of students in international comparison tables.

There is no reason to accept that American students should be worse at maths. However, the fact is that a global economy demands certain skills and mathematics is at the top of that list.

Further Reading

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Mathematicians and the Lottery

Do mathematicians play the lottery, it’s a difficult question as I suspect many would be hesitant to admit it. Although ly one thing is certain, mathematicians do get asked how they should play the lottery.   Even though you don’t usually see an endless parade of maths professors walking off with  the jackpot.

However although a guaranteed win strategy is unlikely, most forms of gambling have sensible tactics which can at least maximize your chances of winning (although minimizing losses is probably nearer the mark).    So can you use statistics to predict which numbers to choose?   Again unfortunately this is unlikely and any essay entitled how to win the lottery should be treated with a large amount of caution.

The safest mathematical response to the question of how to increase your chances of winning is this – buy more tickets.   No-one can dispute  that the more tickets you have the greater your chance of walking away with the jackpot.    Although these lotteries are never meant to be a proper gambling choice, the chance of winning is so remote that you would have to buy an awful lot of tickets to be in with even a reasonable chance.    The ticket is more a chance to dream than a realistic one of winning having said that over 1500 people have won more than a million pounds in the UK lottery for instance.

The UK lottery requires you to pick 6 numbers from a choice of 1 to 49.  You win a prize if you match three of those numbers,   the more numbers you match the more you will win.  The jackpot is normally paid out for all six numbers although this will be shared by anyone else also picking these numbers.   There is an additional bonus number which can be used if you get the first 5 which also is likely to pay out a big prize.

We can assume that most national lotteries are completely fair and random, after all there is no incentive to cheat as all the money is paid out.  Most of the European countries have lotteries and most you can play online, you may need to change your IP address like this to an Italian, Spanish or French one though to get access.

Of course with the lottery we are plainly in the field of probability and statistics, because it’s ultimately all about chance.   However if you’re aim is to maximise your chance of winning a large jackpot (rather than just picking the correct numbers), then there’s an important fact to remember.

The amount of the jackpot a winning ticket will benefit from depends on how many people picked those numbers.   Here there is a fundamental fact that will help you narrow down your choices slightly.  To maximise your chances of ‘not sharing’ then you need to avoid a certain group of numbers as much as possible.   The reason is that many millions of people base their numbers on birthdays, so the numbers 1-31 will be much more likely to selected than those of 32-49.

John Jones

Author of Watching BBC Abroad in 2017

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Bertram Kostant, 88 Dies in Massachusetts

Bertram Kostant was one of the world’s foremost mathematicians.  The Professor emeritus at MIT died last month aged 88 years old in the Hebrew Senior rehabilitation Center in Roslindale.

Kostant held the post of Professor of Mathematics at MIT from 1962 until 1993 when he officially retired however he continued to lead an active life engaging in research, travel and he even continued to lecture at Universities across the world.

Over 60 years, Lostant published well over a hundred papers and he was responsible for some diverse and inspiring ideas within his core subject mathematics and theoretical physics.   he was born in 1928 in Brooklyn, New York and graduated from Peter Stuyvesant HIgh School in 1945 at the end of the war.  He first started studying chemical engineering at Purdue University but later switched to mathematics.  He was mainly inspired to this which by the lectures of Arthur Rosenthal and Michael Golomb who had emigrated from Germany.  He graduated with a bachelor’s degree in mathematics with distinction.

Later in his career he was awarded the Atomic Energy Commission Fellowship for graduate studies at the University of Chicago.  It was here that  he began working with some of the premier American mathematicians like Marshall Stone and Paul Malmos plus many others.   It was only in 1951 that Kostant received his MS and three years later his PhD with a thesis entitled -” Representations of a Lie Algebra and its enveloping algebra on a Hilbert Space “.

Kostant spent many more years in Chicago until he joined the faculty at MIT in 1962 which was to be the last move of his academic career.  His earliest lectures were focussed around his Lie Theory and over the years he mentored many influential mathematicians like James Symons the differential geometer.

Over the years, he has received many awards and was often cited by colleagues in other research areas.  His theories were often discussed and indeed you could often hear his theories discussed on TV particularly on the Open University broadcasts on the BBC.  There is rumour that some of these will be made available from the archive although you will need a VPN to access BBC iPlayer abroad like this  from outside the UK.

In May 2008, the Pacific Institute for Mathematical Sciences hosted a conference: “Lie Theory and Geometry: the Mathematical Legacy of Bertram Kostant,” at the University of British Columbia, which covered his work and celebrated this life when he was 80 years old.  In the second half of 2012, he was finally elected to the inaugural class of fellows of the American Mathematical Society. When in las June, Kostant traveled to Rio de Janeiro for the Colloquium on Group Theoretical Methods in Physics, he was also to receive the prestigious Wigner Medal, “for his fundamental contributions to representation theory that led to new branches of mathematics and physics.”

James Hawkins

Author of BBC in Ireland, history of the broadcaster, 2016.


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How to Watch BBC Iplayer in USA

Not strictly a maths post, but more a need in order for me to watch a programme being rebroadcast and thus watch BBC Iplayer in USA.  It was released a year or so ago, but I have been tipped off that it is being shown again and so for US citizens possibly a chance to catch it on the IPlayer application.  The series is called The Story of Maths and is presented by Marcus du Sautoy, it chronicles the history of maths through the ages and I am expecting great things.

But first the problem – BBC Iplayer doesn’t work in the US as when you try and access it does some sort of lookup and blocks non-UK addresses.  So I did some reading on geotargeting, proxies, VPNs and IP addresses and finally found a way to bypass these blocks and watch whatever I like.

So here it is –

How to Watch BBC Iplayer in America

Now I don’t know if you’ve tried at all but when you visit the BBC Iplayer abroad, if you’re not in the United Kingdom, you get redirected to the Radio section (which you can use).  However if you persist and try and watch something on the main BBC site – you get the following message below.

This happens whatever you try to watch and it’s purely down to the location registered to your IP address.  At this point of course many people give up or try but it’s actually not that difficult to bypass these blocks.

Firstly you need to know what’s happening – when you connect to the BBC website your IP address is recorded.  This is the unique number that is assigned to you by your ISP when you connect to the internet, everyone has one.  Next this address is checked against a large database which contains all assigned IP address ranges and which country they’ve been allocated to.  So if you’re in the US or using a USA Proxy then you’ll have an American assigned address, explained here.

That’s pretty much it, if your IP address is registered anywhere other than the UK then you’ll get the above message. So in theory all you have to do is change your IP address so it’s a British one and not American registered.

Now you can’t change the initial address you’re allocated because a US ISP can only assign the addresses it has been allocated.  Occasionally you might get lucky and an address range is incorrectly listed in  the database – but that’s a bit of a long shot.

The trick is to use a proxy or VPN service to connect through.  This is a server that sits between you and the website you visit, it simply forwards all information acting like a middle man.  However if this server is based in the UK, the BBC application will think you are also based in the United Kingdom.

So here’s the application I used, there are lots of others available but I used this one because it’s one of the cheapest and I know someone who worked on some of the code so I know it’s a legitimate company.   Here’s the screenshot from the program I use called Identity Cloaker.


This is the screen where I select which server to connect to, they have lots of countries but I need a UK one. After clicking on a UK server I then can go back to the BBC Iplayer site and try again.  This time I connect the site sees a UK based IP address (from the proxy) and so allows me to watch whatever I want.

In fact you can even disconnect the proxy after the program has started and it will still work as the IP address is only checked at the beginning of the show. So that’s it really – how you can access BBC Iplayer in America! I’m really pleased with Identity Cloaker,  it’s super fast and very easy to use – you can get a 10 day trial here if you want to try it.  My only issue is that there’s no sign of The Story of Maths yet so I have to keep checking!


Additional Reading

Updated software to bypass Netflix blocking VPN although this is currently only for the US version.

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One Variable Hypergeometric Functions

For more than 200 years some of the greatest mathematicians in history have studied the hypergeometric functions in their one variable context.   You’ll find them covered in all sorts of mathematical works ranging from Euler to Riemann. However it was Barnes and Mellin who initially studied the integral representations whilst Goursat investigated their special properties.

It can be difficult to classify the different type of hypergeometric functions however there are three main categories:

  • Functions defined as integrals.
  • Solutions to systems of differential equations.
  • Functions to series whose coefficients satisfy specific recursion properties.

Of course readers are probably familiar with the Mellin Barnes integral which defines these specific integrals. One variable functions have been understood for many years but mathematicians have now developed these approaches to cover multiple variables which of course yield slightly different results.    There are some interesting programmes on the UK Open University covering this, which may be available online – this link helps access the BBC since it was blocked abroad.

So while the study of the single variable functions is well over two centuries old, the study of the multivariate function is more recent. There has been something of a resurge in interest in the study of hypergeometric functions particularly in the last few years. The interest mainly is around the connections between hypergeometric functions and other areas of mathematics.

These include interesting areas such as investigating connections with such areas as algebraic geometry, symmetry and number theory for example. One of the key developments has come in the 80s and 90s through the work of Kapranov and Zelevinsky among others. In recent years this has been expanded by Saito, Sturmfels and Takayama.

Further Reading

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The Breakthrough of Irrational Numbers

Many centuries ago, Pythagoras was one of the first people to make the claim that the Universe was governed by numbers.  However at the time, numbers were understood to be whole numbers and ratios of these numbers (fractions).   However mathematics had a surprise in store for this famous mathematician – there are numbers which don’t come under either of these two categories.    They are called irrational numbers and they can be difficult to comprehend as they cannot be written down as decimals or even as recurring decimals.

For example you can in fact easily define an recurring decimal in a relatively straight forward way.   Even infinitely recurring decimals can easily be expressed as a fraction if needed.  However if you try and express and irrational number as a decimal you end up with a number which continues indefinitely without a regular or consistent pattern. The concept of these ‘irrational numbers’ was a huge breakthrough for mathematicians who were trying to look beyond whole numbers and their associated fractions.   They were considered a discovery though as Leopold Kronecker stated in the 19th century – “God made the integers, all the rest are the work of man.”

The most famous irrational number is of course, Pi.  In schools it is usually approximated to a more friendly 3.14 or 3 1/7 however the true value of Pi is nearer 3.14159265358979323846 however even this is only an approximation of it’s true value.   In reality Pi can never actually be written down exactly as the decimal places go on forever without and consistent pattern.  It still useful though and these numbers are frequently used as RNG seed numbers for things like encryption and ciphers like this Smart DNS software here.

Here’s a starter of a few hundred decimal places for Pi-

3.141592653589793238462643383279502884197169399375105820974944592307816406286 208998628034825342117067982148086513282306647093844609550582231725359408128481 117450284102701938521105559644622948954930381964428810975665933446128475648233 786783165271201909145648566923460348610454326648213393607260249141273724587006 606315588174881520920962829254091715364367892590360011330530548820466521384146 951941511609433057270365759591953092186117381932611793105118548074462379962749 567351885752724891227938183011949129833673362440656643086021394946395224737190 702179860943702770539217176293176752384674818467669405132000568127145263560827 785771342757789609173637178721468440901224953430146549585371050792279689258923 542019956112129021960864034418159813629774771309960518707211349999998372978049 951059731732816096318595024459455346908302642522308253344685035261931188171010 003137838752886587533208381420617177669147303598253490428755468731159562863882 353787593751957781857780532171226806613001927876611195909216420198938095257201 065485863278865936153381827968230301952035301852968995773622599413891249721775 283479131515574857242454150695950829533116861727855889075098381754637464939319 255060400927701671139009848824012858361603563707660104710181942955596198946767 837449448255379774726847104047534646208046684259069491293313677028989152104752 162056966024058038150193511253382430035587640247496473263914199272604269922796 782354781636009341721641219924586315030286182974555706749838505494588586926995 690927210797509302955321165344987202755960236480665499119881834797753566369807 426542527862551818417574672890977772793800081647060016145249192173217214772350 141441973568548161361157352552133475741849468438523323907394143334547762416862 518983569485562099219222184272550254256887671790494601653466804988627232791786 085784383827967976681454100953883786360950680064225125205117392984896084128488 626945604241965285022210661186306744278622039194945047123713786960956364371917 287467764657573962413890865832645995813390478027590099465764078951269468398352 595709825822620522489407726719478268482601476990902640136394437455305068203496 252451749399651431429809190659250937221696461515709858387410597885959772975498 930161753928468138268683868942774155991855925245953959431049972524680845987273 644695848653836736222626099124608051243884390451244136549762780797715691435997 700129616089441694868555848406353422072225828488648158456028506016842739452267 467678895252138522549954666727823986456596116354886230577456498035593634568174 324112515076069479451096596094025228879710893145669136867228748940560101503308 617928680920874760917824938589009714909675985261365549781893129784821682998948 722658804857564014270477555132379641451523746234364542858444795265867821051141 354735739523113427166102135969536231442952484937187110145765403590279934403742 007310578539062198387447808478489683321445713868751943506430218453191048481005 370614680674919278191197939952061419663428754440643745123718192179998391015919 561814675142691239748940907186494231961567945208095146550225231603881930142093 762137855956638937787083039069792077346722182562599661501421503068038447734549 202605414665925201497442850732518666002132434088190710486331734649651453905796 268561005508106658796998163574736384052571459102897064140110971206280439039759 515677157700420337869936007230558763176359421873125147120532928191826186125867 321579198414848829164470609575270695722091756711672291098169091528017350671274 858322287183520935396572512108357915136988209144421006751033467110314126711136 990865851639831501970165151168517143765761835155650884909989859982387345528331 635507647918535893226185489632132933089857064204675259070915481416549859461637 180270981994309924488957571282890592323326097299712084433573265489382391193259 746366730583604142813883032038249037589852437441702913276561809377344403070746 921120191302033038019762110110044929321516084244485963766983895228684783123552

and it keeps on going!

Euclid was one of the first mathematicians to really tackle the issue of irrationality in numbers in the 10th volume of ‘Elements’ where he tried to prove that there could be a number which couldn’t be expressed as a normal fraction.   He however started with something different from Pi and used the square root of 2, ie the number which when multiplied by itself is equal to 2.  His proof involved first assuming it could be written as a hypothetical fraction and then started to simply the expression.

However during his proof he demonstrated that unlike a normal fraction which can be simplified to a single form.  The fraction which was representing the square root of two could be simplified over and over again, in fact an infinite number of times, basically it could never be simplified.   This was his proof that this number was indeed irrational and the hypothetical fraction he used, could not in fact exist.

John Steadway


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