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. Watch BBC iPlayer in Spain

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

Understanding Polynomials

Polynomials are categorized into various types. Put simply, as soon as a polynomial is represented in the shape of equation, it is called polynomial equation. They play a critical part in mathematics. Also referred to as factorization, this technique is mostly utilised in simplifying polynomials. You should currently be all set for subtracting polynomials.
There are several different ways of factoring these numbers based on their types and forms. When you bring polynomials, you are just likely to bring the like terms which are categorized according to the amount of terms and the degree. Such numbers are also classified as prime polynomials.


As a result of how polynomials obey the exact same rules are real numbers we could likewise do the exact same calculation in the fashion of elementary arithmetic, i.e., However many terms a polynomial has, it’s always essential to check for a best common factor (GCF) first. A polynomial is an expression with a power that’s an entire number. Polynomials are employed in a diverse selection of problems where they’re called as polynomial equations. Such polynomials are called reducible polynomial.

The potential things of the trinomial are the binomials which we may make out of these feasible things, taken in each probable order. If that’s the case, you can component out that common component. Inside this expression, there’s a typical element for the initial two terms. A function which consists of polynomial is known as a function. You simply count up how many variables you’ve got the exact same and compose the number before the typical variable part. The maximum value of exponents is known as degree of polynomial. It is also used online to define and protect internet connections for example some use it in VPNs to allow anonymous torrenting as this.

You may prefer the vertical method as you are accustomed to adding numbers vertically. So, since you can imagine, we’ve got a significant number of resulting terms! This time you should bring the numbers together as you’re finding the sum. In the overall form, the quantity of constants, on account of the term of level 0, is always one more than the level of the polynomial. In mathematical provisions, Hamming codes are a category of binary linear codes.

Since the level of the polynomial is the maximum level of all of the terms, Since there’s a single term, this is a monomial. The amount of a polynomial is the maximum level of the terms. It is the degree of the leading term. It is the highest degree of its terms. Since the level of the polynomial is the maximum degree of all of the terms, because there are 3 terms, this is a trinomial. Since it is the highest degree of all the terms, Make sure that you don’t fall into the trap of thinking it is always the degree of the first term. It is the greatest degree of its terms.

Finding out how to factor polynomials doesn’t have to be hard. The theory of polynomials is quite an important and intriguing portion of mathematics. Given 2 people’s ages, there are numerous mathematical relations you are able to calculate utilizing both of these numbers. Similarly the past two terms have a standard aspect. It’s important to not forget to multiply the terms along with the numbers.

Introducing Differential Equations

A topic that some students of maths can find difficult at first simply because of the number of terms involved and the fact that some of these terms will involve derivatives of one of the variables. The dependent variable is usually designated as ‘y’ and will be derived from another
independent variable – ‘x’

Here’s a quick introduction to differential equations –

As you can see the solutions of a differential equation look different from algebraic equations because they don’t just include numbers but will normally have functions in them too. If you’re having difficulty with the concept remember that the differential equation represents the relationship between a quantity which varies and it’s rate of change. This concept is essential in many forms of scientific investigation.

You will come across differential equations in most areas of science, especially in physics, chemistry and engineering. However they are not limited to these areas, ask any economist and they will have used these equations in all sorts of economic projections and research.

Other definitions you might come across are those of ‘ordinary differential equation’ which refers to the function of a single variable only and some of it’s linked derivatives.

With all these equations, the order is important and is stated with regards the highest derivative, the simplest form is the first order differential equation.

You can even express that laws of nature in the form of differential equations, if you learn the techniques and processes they are invaluable in so many areas of science. Fortunately the internet has lots of courses and explanations. Some of the best are on various academic sites and there’s useful introductions on places like the BBC website too. If you are having problems accessing then see some of the useful links below for bypassing filters and region locks which can affect some people depending on the location of their IP address.

Further Reading:

Accessing Residential VPNs –

Surfing the Internet through a Ninja Proxy –

The Laws of Sines and Cosines

After the 3rd side is figured, the Law of Sines may be used to calculate either of both of the other angles. Make Use of the Law of Cosines to decide on the length of the diagonal of the parallelogram. Make Use of the Law of Cosines to specify the length throughout the lake. In cases like this, utilize The Law of Sines first to seek out either one of both of the other angles, then utilize Angles of the Triangle to get the third angle, then The Law of Sines again to locate the last side.


The Law of Sines is among the tools that makes it possible for us to figure out the triangle. The law of attraction could be successfully implemented into each region of your everyday living.
Then, there is but one law of cosines, though it might be written in several guises.
In both instances, you should already know a side an angle which are opposite of one another. There isn’t any indication of the ideal angle. Thus we’ve shown the law of cosines holds true once the angle is really an ideal angle. As a way to locate the angles, since we now have all 3 sides and one particular angle, we can put on the law of sines to locate one angle then utilize the sum of the angles of the triangle to discover the third.

Triangles are composed of three line segments. This issue involves all 3 sides but just one angle of the triangle. The Law of Sines (or Sine Rule) provides an easy way to prepare proportions to acquire other parts of the triangle that is not necessarily an appropriate triangle. The Law of Cosines (or Cosine Rule) again provides yet another a simple method to work out the measurements of the unknown parts of the triangle which doesn’t have to be an equilateral triangle.

These laws can often sound complicated written down but if you search the internet you’ll find many videos demonstrating calculations.  In fact there is a mathematical series originally released by Channel 4 which has a whole series of these, it’s now on Netflix – try this video – best VPN for Netflix for an illustration.

The gray line symbolizes the third aspect of the triangle that is not known. As a way to make use of the law of sines to solve a triangle, we have to know a minimum of one pair comprising a side along with the angle opposite it. In the event the triangle is just a suitable triangle, you may use simple trigonometric ratios to obtain the missing parts. An oblique triangle is simply any triangle that is not a proper triangle.

Other proofs are available elsewhere. The usage of the Pythagorean theorem as well as the tangent secant theorem could be replaced by one application of the power of the point theorem. For each and every trigonometry function including cos, there’s an inverse function which works in reverse. The cosine function, together with sine and tangent, is among the three most popular trigonometric functions.

James Herriot

Mathematical and Technology Blogger

How to Change IP Address to USA for Netflix

Understanding the Infinitude of Prime Numbers

Now we have only to show that a great number of reals between 0 and 1 is just like the amount of every one of the reals. It’s also simple to develop simple examples of subsets where the number of primes is finite. The group of integers is actually a particularly important series of numbers. The pack of irrationals as well as the group of reals aren’t countable.

Therefore, merely a countable subset of the actual numbers might be named. Mersenne primes, nonetheless, are still crucial in today’s study of numbers. Every positive number may be expressed in only one way for a product of prime numbers. As an example, an infinite variety of numbers are even, however an infinite variety of numbers aren’t even, as they’re odd.


That shows there’s a prime that’s not on our list, which really is a contradiction. This is the way mathematicians utilize these terms beyond formal mathematical logic. The conjecture hasn’t been proved. Assuming that he is true, let’s see that the Cauchy-Schwartz inequality must also be true.

Cubic quantities of dots might be arranged to earn a cube. Hence, the factors of the ideal squares must likewise be growing by 2. Suppose the triangle is really not a suitable triangle. The first triangle is currently in bold lines.

That secondary argument, then, presumes an actual infinite cannot exist. The idea of prime number is so crucial it has been generalized in various ways in a variety of branches of mathematics. It follows that there are various forms of infinity.

Earth-Prime was, basically, our world. They believed the universe could possibly be explained by numbers. Craig uses the claim a true infinite is impossible, consequently, to support an essential premise of his Kalam argument that the universe started to exist.  It forms the base of many well used algorithms that are currently used on the internet, fire a proxy for Netflix and admire how the web site will reflect your ratings and viewing preferences based on your viewing habits.

This is a typical wellspring of confusion as the proof is frequently presented as a proof by contradiction, even though it can be written as an immediate proof utilizing the exact same ideas. Like Euler’s proof, this proof demonstrates that there should be positive integers that are really not of the necessary form. His proof of the theorem is among those cases. Sadly, there’s no elementary proof of that particular actuality.

There are lots of different proofs, needless to say. Thus far, none have been found, and nobody has shown that odd perfect numbers usually do not exist. Unquestionably, Dirichlet’s Theorem is among the big landmarks within the history of number theory. In the history of the issue, many mathematicians believed that there should really be a door, however they couldn’t discover it.

When the prime factorisation of the number was obtained, all its factors can very quickly be written down. The prime factorization of the number is the expression of that particular number as the item of its own prime factors. Consequently, there are a finite variety of factors. Consequently, there are really an infinite variety of multiples.

That’s what an organic number is. These numbers are rather scarce among whole numbers generally. There are several familiar divisibility tests that may be used to learn if a number is divisible by a lot of the one digit numbers. Multiply all of the numbers together.

Further Reading:

Mathematical Models of Netflix VPN Services: , Simpson, Balaton Press

Using Triangles in Trigonometry

There are just three primary functions in trigonometry and they’re called sin, cos, and tan. In regards to trigonometry, you’ll have to know how to solve trigonometric equations. Think about trigonometry for a toolbox. Now for both of the other trig functions.   The tangent could be the last of the 3 principal trigonometric functions. The height of the triangle are available via an application of trigonometry. This is accomplished by using basic trigonometry and of course using triangles.


In this composition, we’ll be continue discussing triangles. This definitely is an absolute spoiler, however, since it will provide you with the true WGS84 co-ordinates for both caches. Many troubles that handle right triangles yield decimal answers.

Thus the sides which make the equal angles will soon be proportional. Sometimes, a may be the identical length as b. All ideal triangles have a lengthiest side that is certainly directly across from the best angle. With time, however, trigonometry was adapted so the angles don’t necessarily represent angles in a triangle. An angle significantly less than one-forth of the circle 8.

We are almost prepared to explain what SOHCAHTOA actually represents, but there’s one point I need to stress that’s missed by the majority of Geometry students. Geometry is, in addition, near trigonometry along with the areas you must focus on include problems involving circles. Games and activities to allow you to learn trigonometry. You need to arrive in the drawing and also the formula shown here.  There are some useful programmes online which explain the more advanced functions of trigonometry, unfortunately some of these are region locked so you may have to change your IP address using a residential VPN service.

So, if you discover a basic Pythagorean Triplet, you can multiply all 3 sides by exactly the same number, and you’ll find another proper angled triangle with 3 entire number sides, along with the exact 3 internal angles as before. The real key to solving the of the correct angled triangle, would be to do as I’ve done here, which is to ensure the proper angle is at among the bottom corners. The most suitable angle triangles may also be of two sorts. An angle with its vertex in the center of the circle 14.

Insert within the diagram every one of the things you’re given. Listed below are the values shown within the diagram in addition to another frequent group of values for this particular triangle. The initial step is always to draw a diagram. Draw this suitable angle into the diagram.

Cache 2 was also put in a hole in the base of the tree. Speed, distance and time could be calculated utilizing a magic triangle. This box just includes a Zip-Loc bag which then contains quite a few envelopes. Place a stick figure in the angle for a point of reference.

The fundamental right triangle rule may be the Pythagorean theorem. In Euclidean geometry, any 3 non-collinear points determine a distinctive triangle plus an exceptional plane. This value may be found utilizing the Pythagorean theorem.

The 3 medians intersect within a point, the triangle’s centroid. The 2 pots aren’t the exact same distance from both reference points but they’re very close, just a few meters away. The 3 altitudes intersect in one point, known as the orthocenter of the triangle. This last example has a Cos value supplied, but it’s for the very best angle within the Triangle.

There are still formulae for finding the 3 entire number sides of the appropriate angled triangle. Hipparchus developed what’s known as the very first trigonometric table. Each one of the above combinations represent the 3 lengths of the proper triangle.

Further Reading

John Harris, Online IP changer, Haber Press:2015

Business Intelligence NAV – Takes Control You Business

If you are looking for the best and most popular business tools one should consider looking into business intelligence. We have heard a lot about business intelligence as an ideal business technique for businesses. Business intelligence for nav is not just a simple business tool but a heavy weapon to use for controlling your business. To keep track the records and important details that’s happening in your entire business then this kind of business tool should you use. Business intelligence is known as BI, it is known as one of the best business tool. Reports from the entire business are all important. It is also essential that you make sure on the kind of business technique you are going to use. Indeed, this business technique should give you at ease in time of gathering the data for the analyzing process.  Businesses are determined when it comes on looking on the right business tool. They want to make sure that everything in their business will be alright. Making sure on the safety of the business not being involved in certain problems like going bankrupt is truly an alarmed. Once a business proven that this bankruptcy is near on your business then thinks about on the best weapon to fight for this problem. 

You want to use a business tool that would assure the efficacy of it. Thus, business intelligence is known to be as an effective business technique. Business intelligence nav turns to be one of the most helpful tools for the entire business. It is especially designed for midsized organization; it is an appropriate tool such as business intelligence nav for the needs of distribution and manufacturing companies. So, businesses don’t have to worry because the business intelligence nav is right now working for the better of any business. Acquiring this tool for your business would take control on the attendance, performance and the reports of your employees. For example, if you track down the days of your employee on when he/she was absent for the payroll, and then this tool would not give you any headache from locating it. Businesses involved money and this is used for the business so that it will grow and earn more money with these business tools. So, it is very important that anything that is helpful for the business should be acquired.

Business intelligence nav is one of the most business techniques that have been used. In time that you handled business, think about on the useful things like on its improvement. Of course, you built the business to improve it and it is vital that the entire business should be controlled well. BI turns to be the first choice of most business managers. They find out that this business tool have helped a lot of businesses. Business intelligence nav is gaining a name in the business industry. It has used by many businesses since they have find out that this is effective. Your businesses would be more effective and efficient if you have used a good and better business technique. Business tool should be the right business technique to make it competitive. Business Intelligence NAV gives you the power to take control of you business and are one of the best business tools you can buy.

Patterns, TV and Teaching

Math is getting a better reputation by the day don’t you think?  One thing that I noticed the other day, was the show Numb3rs. The show is pretty simple to explain, two brothers one of which is a math whiz and local University professor, another is a FBI agent.  The math whiz brother helps solve cases and is actually the main character of the show, probably the first math whiz main character who seems like a regular guy in the history of tv.

A friend who owns a gift basket business loves the show and thinks it should be required watching for anyone looking into the math’s or science’s as a career, because it can show exactly how good and intense it can be.

Do you agree?

Mathematics of Dominoes

Many mathematicians have been fascinated by the chain reaction of a falling domino.  After all even the simple fact that a small domino is able to knock over a much larger one suggests there’s something quite intersting with the chain reaction that is occuring.  It’s obviously due to  the momentum of the first domino but how can we work out the mathematics involved in this process.   After all just look at the momentum produced in this famous domino toppling scene – there’s another one on the BBC Iplayer, check this post and how to access it from the States –


So how does  this chain reaction work and how much bigger can each domino get to maintain the reaction.  There have been lots of guesses and arguments and commonly people tend to believe that each domino can topple one 1.5 times the size.  This is of course also dependent on spacing as well as the momentum generated.

In a physics context it’s quite simple – when you lay a domino on it’s side it stores a certain amount of potential energy.  This energy is released when it’s pushed over.  The force required to knock the domino over is less than that generated when it falls over.  Therefore there is an amplification of the force which means a little domino can knock over a bigger one.

Well a mathematician from the Netherlands has simplified the issue for us all in a mathematical analysis.  His name is Van Leeuwen and makes a number of simplifications.  Things like the friction involved is infinite so there can be no sliding.  No elasticity in the contact between dominos and that they always stay in contact with each other.

Van Leeuwen then demonstrates that when  the dominos are spaced optimally that the maximum growth size of each domino is a factor of two.  So my wife was wrong, I don’t waste too much of my time browsing on the internet – I’ve now discovered this evening how to change to a fake ip address from here and also figured out how many dominos I’d need before I could harness enough energy to start knocking over tower blocks using this chain reaction !!

Further reading