The Concept of Mathematical Proof

Proof is a funny word it can mean different things to different people. For example there is a difference in the concept proof between scientists and mathematicians.  The difference is actually quite subtle but without understanding it then you have little chance of understanding the work of any mathematician.

Classical mathematical proof is to start with a set of statements and axioms  which are true, or at least are self evidently true.  Then by using logic and deduction, you can move step by step to a conclusion.  As long as the underlying axioms are correct, and the logic is flawless then the conclusion should be undeniable.  The conclusion is termed the ’theorem’ by the mathematician.

A mathematical theorem is dependent on the process of logic and so should be true forever.  To the mathematician, their proofs are absolute there is no scale or level to this proof.  When you compare this to the scientists proof you see the underlying difference.  The scientist will put forward a hypothesis to explain a phenomenon  or occurence.  Then it is observed and compared with this hypothesis – the scientist will then gather evidence in favour of the hypothesis or against it.  Eventually evidence will way up either overwhelmingly in favour or against the hypothesis which will decide if it becomes part of scientific law.

The level of proof is not absolute in science, it is never approaching the level of mathematical absolute proof.  The scientists rely on perception and observation both of which can be subjective and are certainly fallible.

To quote Arthur C Clarke

“if an eminent scientist states that something is undoubtedly true, then it is likely to be proved false the next day.” 

Mathematical proof is absolute and devoid of any doubts, whereas scientific proof is frequently rewritten or proved to be false.  Take for instance the wonderful BBC program on Fermat’s Last Theorem and you’ll see some of the difficulties involved in attaining the level of mathematical proof.  It is often available on the BBC Iplayer if you check – if you live outside the UK – try this technique to watch Iplayer abroad.  It works by masking your true IP address and allowing you to use a false one in order to bypass the geotargeting restrictions.   You can even do this by simply using something called Smart DNS where you don’t even change your address –

Playing The Numbers Game

Mathematics in business is usually left to the financial team but numbers are getting ever increasingly important in tracking the development and popularity of a business be it with a new starter company or an established company that is seeking information on its customer base.

One specific way we are seeing this in practice is through social media with most companies having Facebook pages or Twitter accounts. This allows them to stay in contact with their customers as well as gauge responses to certain things. For example following a tesco facebook marketing campaign you would be able to see how liked the specific campaign is as well as read comments to really see what is being said about it.

The number of followers or ‘likes’ on a Facebook page can obviously show how popular a company or brand is but can also be of use by putting information direct to the consumer allowing them to take advantage of deals.   It’s much better to do this sort of analysis first to get an idea about the response.  It’s kind of like of using something like a roulette sim to randomize responses to test your hypothesis.

This can work well for companies, although, as was experienced by Tesco in the Scottish town of Greenock it can also mean information about mistakes can spread like wildfire. On discovery that a deal regarding beer was put through wrong offering the customer a ridiculously low price word spread across social media sites leading to crowds of people attempting to take advantage of the mistake before it was rectified.

Here we can see the power of Social media and that is why more and more companies are trying to harness that power and use it to benefit both themselves and their customers. Keeping a good open communication has proved to be very beneficial to the companies that are already taking advantage of this simple way of getting into peoples homes.

I believe in the future we will see a greater dependence on this sort of interaction.

Additional Reference: British TV for Expats.

Math is an essential tool in the world of finance

Math skills are essential to maintaining financial viability.  In today’s credit-crazy society, it is easy to become lost in the convoluted world of variable interest rates, compounding payment schedules and minimum payment allowances.  Most people have no way to break down the numbers to understand the methodology behind them.  The rules are designed to be hard to understand.  A bankruptcy lawyer is equipped to examine your financial portfolio to determine your best course of action when faced with monetary difficulties.

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The ability to apply math to everyday situations is not something that comes naturally to most people.  Rather, it is necessary to practice a commonsense approach to educate yourself on the necessary topics.  A comprehensive fiscal and mathematical vocabulary is essential for navigation of the complex legal and fiduciary environment that we find ourselves in today.  Without adequate preparation, hard work and dedication can go to waste.  In understanding everything there is to know about financial responsibility, bankruptcy attorneys learn to see all sides of an issue.  They are well-equipped to deal with the harsh realities of fiscal difficulties, yet they understand that at the heart of every distressed business or individual, there are families that are depending on fair disposition of circumstances.

Behind the numbers there are people who need solutions that work so that they can live their lives, not under the looming shadow of impending disaster, but in the light of hope for the future.  By assessing your financial portfolio and making statistically informed decisions, it is possible for even the most seemingly hopeless financial obstacle to be overcome.  Fully understanding the nature of debt requires complex analysis and calculation of debt tables, interest rate formulas, percentages, statistics and much more, along with a complete knowledge of the governmental policies and procedures.  Only bankruptcy attorneys are equipped with the math and other skills necessary to fully assess your financial situation and how to fix it.

Further Reading

How to Watch BBC Abroad

Maths in property

As a mathematics professor I have always found that shopping is the most arduous task in our society today. House closing rarely takes into account basic mathematical precision that would allow for easier agreements to be reached. It’s difficult to display exactly how frequently this occurs but I’ll do my best by telling you about my latest experience with house rental in the great state of Oklahoma.

When I was last searching for  houses to rent I came across a beautiful property and fell in love right away. It had everything I wanted. A Playroom for the kids, a swimming pool for the wife, and a fire pit for me.  All these thing we first read about in Skymall and, of course, wanted for ourselves. I drew the line at the replica sarcophagus coffee table.

We progress through many stages of meetings, after one month we were adjusting closing costs when my broker informs me,  the current owners were adjusting the price bye $5000 for furniture in the home. I made the argument a two part response. Firstly, I said I do not want your furniture. Secondly, it is inconceivable that your bed and sofa were worth $5000. I countered with the offer of individual pricing per item totaling somewhere around $1500. When they scoffed I had no problem walking away from the deal
Another example is when my brother purchased his very first home. It had to do with bills and receipts of previous tenants’ electricity and gas bills. When my brother moved in, he was billed immediately for a year is negligent payments. It took three weeks of e-mails and threatening letters to reach a resolution that did not cost my brother anything but his time. Very luckily for him. If a mathematics-based system could surplant the often arbitrary system we have today we would be much better off.

The Mathematics Behind Compounding Interest

Compound Interest.  It can be your best friend (in your investment or savings account) and your greatest enemy (debt).  The mathematics behind how your money or debt compounds can be beautifully simplistic or complex depending on how you look at it.  For those of us interested in a simplistic, visual approach to understanding how it works, we have a great tool available to us known as a compound interest calculator.

Anyone would like to know how much interest he or she has to pay on a mortgage or a loan. You would also want to keep track of the amount of interest to pay over a given period of time if you are borrowing money. The simplest formula to use when calculating interest rate is P x(1 + r/n)^ Yn where P = principal, r = rate, n = number of times per year for interest compounding and Y = the number of years for the loan. Sometimes, it can be challenging to manually calculate the interest on a loan and you can also make mistakes. Instead of calculating manually, you can use a compound interest calculator to calculate your compound interest.

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With the compound interest calculator, you can know how much you shall have saved after a certain number of years and you can also know how much of your final balance is due to make more interest for you. The calculator is very easy to use. You just need to enter the needed information in the fields specified. You should enter your original deposit in the principal box, your APY in the Rate box and the length of time in the Years box. After inputting the information as required, you can then hit on calculate to get the results. You can either select to calculate your rate using months or years.

The compound interest calculator can be very important to both bankers and those holding savings accounts. Whether you are a lender or a borrower, you will find this tool very useful when it comes to keeping track of how much money you are earning or how much money you would have to pay on a loan.  Obviously there are differences based on your location but also most financial transactions are fairly standardized in the developed world – there are EU rules for example which control transactions in Europe.  It is an easy to use tool that does not need any special skills. You can find the software online. You can do all your calculations just sitting in front of your computer.

Financial management is something that some people take for granted and it can be very crucial when it comes to financial stability. You might be losing a lot of money without actually understanding how if you cannot track your savings properly. Compound calculators will make things very easy even for dummies. They can also be very useful when you need to determine a loan program to get if you are looking for a loan. When shopping around, you can use this calculator to compare the interest rates between the different lending companies before selecting which one to get.

The mathematics of search engine optimization

I have been blogging for some time now and I know for a fact that the internet has definitely become a pervasive medium. As a matter of fact, it has been proven to be very valuable and useful tool in various aspects of man’s life. May it be in the field of entertainment, information, communication, and even commerce, the fact that the internet is of great help will always stand. One of the applications of the internet for business and commerce is through internet marketing. I know that for a fact since there are really, really a lot of advertisements that seem to be all over cyberspace. As a matter of fact, there are a lot of internet marketing companies getting into business nowadays because of the high demand for their services. But how do these companies run seo campaigns?

A recent article published by an seo agency states that social networking, link building and providing excellent content is a sure fire way to increase your rankings but the most critical aspect is Page Rank which determines how much authority and trust a website can confer to another site. A simplified equation for Page Rank is as follows:

PR(A) = (1-d) + d(PR(t1)/C(t1) + … + PR(tn)/C(tn))

This is the original equation but it is obvious that the original equation has been tweaked as the years went on by. From the equation, an important element is the amount of “vote sharing” or outbound links a site has. The more links a site gives out, the more diluted each link can confer Page Rank to another site.

Perhaps you have already known this equation. Essentially, search engine optimization is a strategy used in internet marketing with the aim of making your business or website rank higher in the search results page of search engines. As you may know, ranking higher in search results page will give you great leverage over your competitors. This is mainly because it gives you more and better exposure. Of course, the tendency for consumers is to click or follow the links of the businesses that appear higher in the search results page. Thus, this may eventually lead to more sales and better profit for your business.

Do The Math On Your Real Estate Purchases

Every good real estate investing decision I have made is based on both logic and emotion; both head and heart.  Maybe I’m particularly disciplined or have a head for numbers, but far too often I come across people (particularly “lifestyle buyers” buying real estate in an exotic country) who let their emotions run away with them.  They end up forgetting to run the numbers, ignoring the math and getting completely consumed with the idea of owning a particular property.  They become so consumed with this idea of a new lifestyle, that logic goes out of the window.

Too often this ends badly a few years down the line.  The property does not perform as they hoped and then end up disappointed with their purchase.

Granted if your looking at real estate in an emerging market such as Nicaragua, Belize or Kenya for example, it can be hard to find reliable data on which to base your investing decisions.  Unlike in the US, Canada, or other mature markets, official statistics are not collated centrally and there is often no Multiple Listing Service (see more).  It’s practically impossible to find out exactly what properties sold, for what price and for how long they were on the market before they sold.

So in order to let the numbers drive your investment strategy, you have to collect the data yourself.  Whenever I am purchasing real estate in a country where it’s hard to get my hands on solid market data, I build out my own spreadsheet of asking prices.  I know up front that asking prices are not the same as sales prices, but in a world without official data, they are the next best thing.

There have been many times when collecting the numbers and doing the math have saved me from an expensive mistake.  Just last year when researching Nicaragua real estate, I fell in love with an ocean view villa just outside the town of San Juan del Sur.  I was so captivated by the view that I almost pulled the trigger on the day I first saw the property.  Luckily I managed to restrain myself and spent the next 5 days building a spreadsheet of the asking prices of comparable properties.  With these “comps” in hand I quickly realized that the property was overpriced.

I managed to rein set my head into the decision making and stop my heart from taking over.

Can you do the same?

Math: How to Be Better at It

There’s no arguing the fact that math is the most popular subject in the world. It’s hard to find a career where some basic knowledge or usage of math is not required. Math is also firmly tied to logic. People who are great at math are often able to see logical solutions in complicated solutions.  This might sound unimportant until you consider the fact that problem solvers are exactly the kind of people that end up running companies. Learning math can not only improve your own critical thinking abilities, but can prepare you for many scientific careers and further study, in fields such as engineering, particle physics, nuclear technology, greenhouses gases observational sciences and work in many other scientific research facilities across the world. If you wish to improve your skill and knowledge in Math, the following are some important facts that you need to be aware of.

Math Courses are Cumulative.
You can’t expect to be poor at basic math and then great at handling complex equations. It just doesn’t work that way. If you’re going to succeed at math, then you’ll need to have a firm foundation in the basic courses. Start with the easy courses. Review all your math courses beginning with those from your older classes to your current classes. Focus on the areas that you are uncomfortable with. Math’s complexities mean that you may end up struggling in the beginning. But if you stick at it, you’ll find yourself become a better Math’s student.

You need to be involved.
The only way you’ll ever get better at math is if you pick up a pen and work on a couple of problems on your own. Never ignore your maths homework. Instead, make a habit of working on your assignments every day. The longer you spend at math, the easier it will become. You won’t get far by just listening to your instructor during courses.  Copy out the examples that your instructor uses and try and rework them in your spare time. Math requires your commitment and involvement. It’s the only way it works.

Understand Why.
Some students fly through their math scores by memorizing dates, formulas or certain rules. You may earn an ‘A’ this way but it won’t make you a better student. In order to become a better maths student, you need to understand why certain formulas are used in different situations. Learning more about the way math problems are solved will teach you how to handle problems of your own. Advanced math problems typically involve dozens of different math formulas at varying stages. Unless you understand what these are and how they work, you’ll find it harder solving practical math applications.

Visualize a Problem: Math is so much more than just numbers. You’ll find it easier handling math problems if you try visualize what you’re doing. Sometimes a math problem might require many long steps.  Try and use shortcuts when applicable. This will improve your speed at handling questions and reduce the risk of you getting confused.

Challenge Yourself: So now you’re good at handling certain problems. Don’t stop there. A mathematician’s journey never truly ends. Continue to challenge yourself every day with newer equations. This will increase your speed and improve your confidence in handling advanced questions.  Try using maths in unfamiliar situations to improve your mental dexterity.  For example if you enjoy casino games – you can find a free online roulette game with no download and practice games.  Try out your probability in games of chance like roulette or assess the odds of drawing another card in Blackjack!

Math has many levels and you will need to study them in a certain order to learn it properly. If you have not had any math since high school, then you should begin with basic introductory classes to trigonometry, algebra and calculus. Once you have overcome these subjects, you can start specializing in any other scientific field that you are interested in, such as greenhouse gas research, physics or astronomy.

The Mathematics Of Refrigerator Reviews And Financing

Math skills are highly underrated by some people.  What they fail to recognize is that expertise in mathematics gained in school can translate into improvements in the quality of your life throughout adulthood.  This idea baffles many students, who fail to see the benefits that the math they learn in high school will provide later in life.

Let us look at one typical example as an example – the buying process for a new refrigerator.  Too the poorly informed it may seem like there is not much use for mathematics in this decision making process, but this could not be further from the truth!

There are many people who actually make their living from refrigerators besides those who design, build and sell them.  Just take a look at YouTube and you’ll find a host of individuals making videos about reviews, how to customize and even how to repair them.  You’ll see lots of adverts and sales on major TV channels, try this little app which allows you to watch British TV overseas and see how many mentions there are.

To start with, when looking at refrigerator reviews – or even just reading a list of specifications – shoppers will be taking in data like width, height, depth, volume.  An educated mind is able to tie these together into an underlying whole and understand the mathematics behind them.  When math skills are lacking, these will just be a string of numbers.

With adequate math skills one is able to take these numbers extracted from a refrigerators review and tie them back to the dimensions of your kitchen, or of your proposed installation area, and have a full understanding of any limitation that may prohibit the installation of the model under consideration.

Even once you have analyzed refrigerator reviews and decided which model to purchase; mathematics still has a further role to play.  Will you pay cash, or will you finance your purchase?  If a retailer is offering a great deal on financing so as to move more products, it may make sense to take advantage of low interesting financing.

However, these types of calculations are not simple.  To make the right decision the consumers’ needs to have knowledge and awareness of items like discount rates, opportunity costs, percentages, interest rates, finance charges and the like.

While concepts like these are not difficult, a proper understanding does require a solid mathematical background.  Decisions on financing are very important as they can potentially cost a buyer hundreds of dollars if poorly made.

Hopefully this example of the use of mathematics in reviewing refrigerators for purchase – and exploring financing options – will demonstrate to teens the need to focus their attention on their math studies while at school.


Espresso Machine Engineering

When you’re thinking of buying something nice for your kitchen and you’re going through websites reading things like espresso machine reviews, the last thing on your mind is probably the amount of precision engineering and math that goes into designing and building these complex beasts.

There are more cups of coffee drunk each day in the USA than the leading brand of cola. But these days a lot of people are taking the wise move of buying a home espresso machine to reduce the cost of their daily coffee.

The very first espresso machine was invented by Luigi Bezzera in 1901 but since then, the technology, engineering and math involved in designing an espresso maker has advanced significantly.

Espresso machines work by forcing hot water through compacted coffee at very high pressure. In particular, super-automatic espresso makers are complex systems which deal with every function from the grinding of the beans to the production of the final espresso. So what things does the engineer have to consider when designing one?

  • Managing the temperature
  • Timing
  • The flow of water in the machine
  • The physical portafilter design

As you can imagine a good knowledge of math would be essential in designing a system that does all of these things simultaneously.   It’s certainly a good bet to guess you’re going to need some high level design and mathematical modelling experience. So if you’re interested in going into the world of kitchen appliance design then remember it’s not all about the looks. The engineering of these modern appliances is impressive so stick in at your math class if you want to be the next Luigi Bezzera.