Archive for the ‘Math in society’ Category

The Life of John Napier

John Napier was born in the year of 1550 at Merchiston Tower in the City of Edinburgh in Scotland.  He dies at the age of 67 on April 4, 1617  in his home town of Edinburgh, Scotland.  If you were to choose a top ranking table of world mathematicians then Napier would almost certainly feature in it.   He spent much of his life including his work as an Alma mater at the University of St Andrews where he was also a Doctoral adviser.  Of course as any school child will probably  know he is most famous for that little book of Logarithms.  However he was a man of many talents and he is also famous for such things as Napier Bones and the introduction of the Decimal Notation. John Napier of Merchistonalso signed as Neper, Nepair, nicknamed Marvellous Merchiston, was a Scottish landowner known as mathematician, physicist, and astronomer. He was also actually the eighth Laird of Merchiston. His Latinized name was Joanne Nepero or Joannis Neperi.  However we know him know as John Napier and is certainly best known as the discoverer of logarithms.

John Napier was featured in the recent BBC’s history of mathematics which you can still get on the BBC iPlayer for a few weeks.  If you’re outside the UK then this article entitled How to Watch UK TV from USA should help, you just need to hide your location and it all should work perfectly.

He also invented the so called Napier’s bones and made common the use of the decimal point in arithmetic and math. Napier’s birthplace, Merchiston Tower in Edinburgh, Scotland, is now part of the facilities of Edinburgh Napier University. After he died from the effects of gout, Napier’s remains were buried in St Cuthbert’s Church, Edinburgh. Napier’s father was Sir Archibald Napier of Merchiston Castle, and his mother was Janet Bothwell, daughter of the politician and judge Francis Bothwell, Lord of Session, and a sister of Adam Bothwell who became the Bishop of Orkney. Archibald Napier was 16 years old when John Napier was born.

As was the common practice for members of the nobility during that time, John Napier didn’t enter schools until he was 13. He didn’t stay in school very long, however. Little is known about those years, where, when, or with whom he might have studied, although his uncle Adam Bothwell wrote a letter to John’s father on 5 December 1560, saying I pray you, sir, to send John to the schools either to France or Flanders, for he can learn no good at home, and it is believed that this advice was followed. In 1571, Napier, aged 21, returned to Scotland, and purchased A castle in Gartness in 1574.

On the death of his father in 1608, Napier and his family moved to Merchiston Castle in Edinburgh, where he resided the rest of his life. Advances in maths – His work, Mirifici Logarithmorum Canonis Descriptio contained fifty seven pages of explanatory matter and ninety pages of tables of numbers related to natural logarithms. The book also has a fantastic discussion of theorems in spherical trigonometry, commonly known as Napier’s rules on circular parts.

Jim Hamilton.

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Deciphering the Fibonacci Sequence

You may have heard the expression, it’s certainly one of the most famous mathematical concepts – show what’s involved with he Fibonacci Sequence?.

The thirteenth Century Italian Leonardo of Pisa, better known from his nickname Fibonacci, was possibly the most gifted Western mathematician of the Middle Ages. Little is known of his life except that he has been this son of a customs official and, as a young child, he traveled to North Africa along with his father. It was here that he first heard about the Arabian mathematics. On his return to Italy, he helped to spread this knowledge through Europe, putting so in motion a rejuvenation in Western mathematics, which had lain largely dormant for centuries throughout the Dark Ages. Especially memorable was that in 1202 he wrote a very influential book called Liber Abaci, wherein he encouraged using the Hindu Arabic numeral system. Here he used the book to describe its lots of advantages for retailers and mathematicians alike across the clumsy system of Ancient Rome numerals then in use in Europe.

Despite its apparent benefits, uptake of this system in Europe was slow, and Arabic numerals were banned within the town of Florence in 1299 on this pretext they were easier to falsify than Ancient Rome numerals. Yet, common sense finally prevailed and the new system has been adopted through Europe by the fifteenth century, making the Ancient Rome system obsolete. The flat bar notation for fractions was initially first utilized in this work. Fibonacci is best known, however, for his debut in Europe of a certain number sequence, that has since become known like Fibonacci Numbers or this Fibonacci Sequence.

There are lots of explanations of this, which although initially sounding quite complicated is actually very simple.  One of the most straightforward ones I’ve heard is to be found on the BBC’s History of Maths programs – you can access this and any other UK TV abroad, from here.

He discovered this sequence – this first recursive numerical sequence known in Europe – although considering a practical problem in this Liber Abaci involving this growth of a hypothetical population of rabbits based about idealized assumptions. He noted that, after every monthly creation, this number of pairs of rabbits increased from 1 to 2 to 3 to 5 to 8 to 13, etc. Soon he also recognized how a sequence progressed by adding this previous two terms, a sequence which could theoretically extend indefinitely.

The arrangement, which had really been known to Indian mathematicians since this sixth Century, has many intriguing mathematical properties, and a lot of this implications and relationships of this sequence weren’t discovered until several hundreds of years after Fibonacci’s death. For example, this sequence regenerates itself in some surprising ways: every 3rd F number is divisible by 2, every 4th F number is divisible by 3, every 5th F number is divisible by 5, every 6th F number is divisible by 8, every 7th F number is divisible by 13, etc.

Additional: – switching Czech IP Address

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Great Mathematicians – Euclid

The Mathematics Of Euclid
The Greek Maths legend – Euclid is known throughout history as among the greatest mathematicians and in fact one of his names is the father of geometry. You might not be aware that the standard geometry all people are taught in college is actually called Euclidian Geometry. In Ancient Greece, he worked tirelessly and Euclid accumulated all the knowledge developed in mathematics during that time. From his studies he created his famous work, entitled ‘The Elements’. It is thought that Euclid likely attended Plato’s academy in Athens before moving into Alexandria, in Egypt. In this time, the city had a big library and whats more it even had ready access to papyrus made it the centre for the productions of books. The papyrus is one of the numerous reasons why excellent minds such as Heron of Alexandria and Euclid established themselves in Alexandria and the Nile delta.

Nowadays Euclid, is even more famous and his books and theorems are an essential part of any mathematics education.  You’ll find him studied at every levels from elementary to I last saw a copy in Dublin, in the University College.


Euclid’s Elements consisted of thirteen novels covering a vast body of mathematical expertise, crossing arithmetic, geometry and number theory. The fundamental arrangement of the components starts with Euclid launching axioms. From here he created 465 propositions, progressing from his first recognized principles into the unknown in several of measures, a process he called the Artificial strategy. He looked at mathematics as a whole, but was especially focused on geometry and this specific field formed the foundation of his work. Euclid’s Axioms were based upon 10 statements which could be accepted as mathematical truths. He named these axioms his postulates and split them into two groups of five, these very first set common to all math, the second specific to geometry.

To see some great explanations of these geometric principles search through YouTube or the Open University Maths lessons.  These were recently broadcast on the BBC so you can pick them up on iPlayer – this post shows how to pick up BBC iPlayer from Ireland but it should work from anywhere in the world too.

Many of those postulates appear into be self explanatory into us, but Euclid worked on the principle that no axiom might be accepted without proof. Euclid’s First Group of Postulates – these Common Notions: Things that are equal to these same thing are equal to one another. If equals are added onto equals, the results are equal. If equals are subtracted from equals, these remains are equal. Things which coincide with one another are equal to one another. A straight line can be drawn between any two points.

Any finite straight line can be extended indefinitely in a straight line. For any line segment, it’s potential to draw a circle utilizing the section as these radius and one end point as these centre. If a straight line falling across two other straight lines leads to these sum of these angles on these same side less than two right angles, then these two straight lines. These lines if extended indefinitely, meet on these same side as these side where these angle amounts are less than two right angles.

Euclid felt that anyone who can read and comprehend words could comprehend his notions and postulates but, into make sure, he included 23 definitions of common words, like point and line to ensure that there might be no semantic errors.

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Great Mathematicians – Carl Friedrich Gauss

Carl Friedrich Gauss was a dominant figure in 19th century Germany primarily because of his accomplishments in the field of mathematics. He’s famous for his monumental contribution to data, algebra, differential geometry, mechanics, astronomy and other mathematical theories among many other fields. Those Individuals who respect his work very frequently refer to him as the greatest mathematician who ever lived and in Latin this is known as the Princeps mathematicorum.

Johann Carl Friedrich Gauss was born 30 Apr 1777 from the Duchy of Brunswick from a lower class illiterate family. His mother actually didn’t record his birth date as she lacked the education to do so. In fact he had to work out his own age by figuring his birthday by himself taking traces from the times she associated with his arrival.

There’s no doubt about that the young Gauss was a prodigy from a very early age, many noticed his remarkable intellect. He was a mathematically precocious kid that he proved it over and over again. Indeed still as a teenager, he made many landmark mathematical discoveries. By the age of 21, Gauss had already wrote his magnum opus which is entitled the Disquisitiones Arithmeticae. This work of his basically altered the landscape of number theory from the years and is still utilized until this day.

The local ruler soon recognised his potential and was to become Gauss’s patron. The Duke of Brunswick found his work impressive and made a decision to send him into the Collegium Carolinum. He attended the college in the early 1790s while he studied in the University of Guttingen during the latter part of the decade.   This part of his life was covered beautifully in a documentary series created by the BBC about famous mathematicians.  If it’s still available, you might find it on the BBC iPlayer site although you’ll need to buy a UK VPN or proxy like this if you’re outside the UK.

During his studies he discovered many new theorems and transformed existing ones into a brand new functions. He created a groundbreaking discovery in 1796 which explained how a polygon can be assembled by the item of unique Fermat primes and using the power of 2. Turns out it was this gigantic discovery in the arena of mathematics that made Gauss into choosing mathematics as his main career instead of philology. Indeed Gauss made mathematics a very important part of his life right up until his death. He actually chose that his tombstone is into be inscribed with heptadecagon that was created by a local stonemason with great difficulty. Gauss embarked upon a brand new voyage with this new development in 1796. His another key work in maths was the development of number theory.

He simplified manipulations in quantity theory by making progress in modular arithmetic. Quadratic reciprocity law was demonstrated by him the same year, rendering him the first man into accomplish the task. Furthermore, he conjectured the prime quantity theorem which allows also a deeper understanding into the distribution of the prime numbers into the integers. Additionally to these self made discoveries and the many improvements to standard theories, he also collaborated with the physics professor Wilhelm Weber in 1831. They worked on the project of magnetism and came up with the representative unity of mass, time and charge.

In addition to magnetism, they made unique findings at Kirchhoffs circuit laws at electricity. They initiated the first electromechanical telegraph, connecting the institute because of physics in Gottingen with the observatory. One of Gauss’s most important and noteworthy publications is the Dioptrische Untersuchungen which was published in 1840.

Additional:

More information on watching and streaming BBC News abroad – available here

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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 – http://www.bbc.co.uk/news/av/business-38932854/what-is-blockchain-and-how-does-it-work. 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 – http://www.changeipaddress.net/bbc-iplayer-blocked-vpns/

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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 – http://www.bbc.co.uk/education/subjects/z6vg9j6

Accessing BBC iPlayer from France and Abroad – http://bbciplayerabroad.co.uk/how-do-i-get-bbc-iplayer-in-france/

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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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