In 1963 a hilarious comedy by the name of “It’s a Mad, Mad, Mad, Mad World” appeared on the silver screen. The cast included many of the top comedians and actors at the time. As much as laughter is an integral part of our society and daily lives, math in my opinion and experience is even more so. There is not one area of activity that I can think of that does not include some form of math, and it is therefore omnipresent. Mathematics has been named as the “Queen of all Sciences” and I will add to this “The Mother of all Sciences” as well. Yet math is poorly taught in schools, and poorly promoted in our Western society. The Occidental world is steadily losing its vanguard position as developers of new science and technology. NOTE: do not worry there won’t be any dazzling or mesmerizing formulas in this article. Guess what? We are all mathematicians, which is inherent to our intuitive, judgmental, reasoning and cognitive capabilities. I will try and explain myself later in the article.
When I was a child my dad encouraged me, and without any form of coercion, to explore math and use it as a visionary way to better understand and make use of the wonders around us and marvel at the possibilities the universe can provide. As I mentioned in The Big Picture article, I believe the Infinite Creator created all the laws of nature, however mankind has developed math as a way to understand and harmonizing with these laws that GOD has created. I have had my fair share of successes with math in my lifetime, but I fear that math in the last few decades and even today is being taught more as a discipline and tool to simply arrive at calculated results. I truly believe this is a certainly a wrong approach, and it’s no wonder why so many youngsters are steering away from math. Teachers, tutors and educators of math should instead inspire and motivate their students to the fascinations involved with math, and use its astonishing tools to better understand and make use of the splendors of the world around us. And better yet, these tools should enable us to extract greater benefits from the laws of nature and accelerate progression that will serve to benefit all of mankind.
In my last article I mentioned that people are not able to love inanimate things, yet significant discoveries were made by prominent people throughout history that we continue to benefit from. I truly believe these people were fascinated and had great passion as well as a keen interest in their respective fields of endeavour and the projects they were involved with. Any motivation they obtained was most likely inspired by people they were associated with. For instance, I am fascinated and certainly have a keen interest in math, however I am not in love with the discipline. Any motivation that I obtain is from the human interaction and the satisfaction that such achievements will serve to benefit myself and everyone else that I associate with and anyone else who wishes to listen.
Earlier I said that everyone is a mathematician. Here are some examples. When crossing a street on foot, assuming it is not at a corner with street lights or stop signs, we watch traffic in either direction and mentally judge the distance and possible speed on the vehicles coming in our direction either way, then make a decision to either walk across the street, run across or wait until the path is clear. Of course, our judgement takes into account the possible speed of the vehicles, distance apart and other factors. This judgment of course implies CALCULUS, and without the use of numbers or scientific calculator in hand. Now, how about sports, music, medicine, business, arts etc. involving zillions of calculations that are being performed with every step along the way? I refer to all of this as “Math without Numbers”. All that is needed is numbers to formulize the calculations involved. The other aspect is “e” which can be interpreted as natural number base e or irrational number 2.718281828…In my book “e” also means “equations”. I can imagine some grins or sneer on people’s faces. Every day of our lives we make countless equations, most of them unconsciously, and many with analytical thoughts like, “if I want to do this (=) then I should get this” and many more decisions that involve “logic” and “intuitive thinking”. Again, these equations do not necessarily have numbers involved and I refer to them as “qualitative”, whereas when numbers are added to the equation process this becomes “quantitative”. The difference between “qualitative and “quantitative” process is that the latter process helps to rationalize and ensure that both sides of the equation have the same weight or balance value.
Now I’m going to switch gears and get involved with numerical areas. The first one is Prime Numbers. I can imagine people’s faces saying, ah ah! As we know, prime numbers are those whole numbers that are divisible by themselves or 1 and nothing else. In the first 100 numbers there are 25 prime numbers, the first one is 2, which, by convention, is both an even and odd prime number, then 3, 5, 7, 11, 13, 17, 19 etc. Of course, there is an infinite number of prime numbers. I developed a simple Excel spreadsheet that enables identifying whether a number is a prime number or not. Google has a huge list of sites addressing prime numbers; YouTube also has great videos, so I won’t go into this since they do already provide valuable information. Prime numbers provide valuable security and safety encryption benefit features for credit cards etc.
While prime numbers appear to be solitaires and do not seem to be too appealing in the overall list of numbers, yet they do provide discreet support for divisible numbers in the background. As such, divisible numbers can be expressed as prime numbers multiplied together. Example: 128 =2⁵X2. How about 2014 = 2X19X53? As I already mentioned 2 is both an even number and an odd prime number. Please try it and you will see that divisible numbers have prime numbers that multiply together to equal them.
Here is a discovery I made years ago. Each digit between 1 and 9 has an inverse negative number. The list is 1=-8, 2=-7,3=-6, 4=-5, 5=-4, 6=-3, 7=-2,8=-1,9=0. You will probably identify a pattern here. If you are interested in knowing more about this, please reply to the article and I will be delighted to explain this.
I mentioned that equations are a constant and appear everywhere in daily thinking process for everything. When it comes to math equations children, adults and just about everyone has some difficulty at one time or other to resolve equations. Here is a simple acronym to remember and apply: BEMDAS. Brackets, Exponents, Multiplication, Division, Addition and Subtraction to be applied n that order to solve equations. Note: begin be calculating within the most inner bracket numbers first, then progressively move outward to the last outward bracket and keep track of the results, then move on to sequence I have just indicated. Here is a simple equation that without brackets we would not have proper direction to solve: 2+3X5 = ?, but by adding brackets this gives us a clear direction to solve it: (2+3)5 = 25 or 2+(3X5) =17. Same numbers, yet different results obtained.
How about new developments in math like fractals, chaos theory and computer graphics at incredible speeds just to name a few areas evolving. Yes math continues to evolve and there are many opportunities for future generations.
While math is most definitely of practical value everywhere in our daily activities, it can also be fun, mentally stimulating and a good way to stay young and creative at any age. Enjoy! Btw, ask someone, “Have you read a fascinating math book recently?” and watch for words like “What did you ask me?” along with the facial reaction to the question.
In the last few weeks the Globe and Mail, one of Canada’s top newspaper has issued several articles about math, parents, educators and the overall interest in the field. This weekend topped it off with OECD’s Programme for International Student Assessment (PISA) with Canada’s student representation in the test slipping from 10th position (score 532) in 2003 to 15th position (score 518) in 2012. During the same period students from U.S.A. slipped from 38 (score 483) to 45 (481). The U.K. figure for 2003 is not shown but 2012 showed a ranking of 33 (score 494). France slipped from 25th position (score 511) to 32nd position (score 495). During the same period several of the Oriental countries have moved to the top of the list.
These figures represent an emergency situation that needs addressing and reversing in the short term. It has been indicated that teaching basic math skills will help to redress the declining ratings. While I do support more emphasis on teaching basic, intermediate and more advanced math skills, I do firmly believe that children from an early age on should be inspired and create interest to “mathink” concepts, principles and develop their own “qualitative” and “quantitative” equations. The tools and rules of math should then follow to reinforce and apply the analytical thinking process put in place. This can be compared to electricians, plumbers, mechanics, carpenters, professionals etc. who first learn their respective trades, then are given the tools to construct, develop or resolve issues. In summary, first things first to succeed.
***My Little Puzzle or Conundrum***
As site visitors and repliers will have probably observed by now I have a fascination and passion for math. For many years I have played with numbers and divided them with divisible and prime numbers. It’s not so much the whole number or integer portion of the quotient but the decimal portion that extends indefinitely that I am interested in. Quite often, the decimals will demonstrate a sequence or string of digits that repeat over and over. This is referred to as a rational number since the pattern can be identified. However, there are numbers such as pi 3.14159…., the golden ratio 1.618… and natural logarithm 2.7182818…. that are described as irrational numbers since the decimals sequence do not repeat or denote a set recurring pattern.
To demonstrate this let’s take a large number like 123456789 and divide it by, say a prime number like 77 = 1603334.9220779220779220779220779. Now divide the same number a divisible number like 124 = 1603334.9220779220779220779220779. In both instances, the recurring decimal sequence does appear or gets revealed through a close look.
Now for more fascination is when the numbers are divided by prime numbers 37 or its inverse 73. 123456789 divided by 37 = 1603334.9220779220779220779220779 and 123456789 divided by 73 = 1603334.9220779220779220779220779. Reversing the dividend we then obtain 987654321 and dividing by 37 = 26693360.027027027027027027027027 and dividing by 73 = 13529511.246575342465753424657534. These divisions demonstrate sequences of recurring decimals. Now, I invite readers that are interested to take the Windows calculator and divide any number they chose by either 37 or 73 and look closely at the sequence of repeating decimals, assuming of course there is a string of decimals and not something like 551/73 =7. As such, both 37 and its inverse 73 have some special characteristics.
I suspect a number of site visitors and repliers will think that this guy has lots of time on his hands. Yes, but the information I have indicated above was developed when I was in high school and was about 15 years old and completed in preparation for the special math contest I was chosen to participate in the following year. In those days there were no sophisticated calculators where divisions extended with many decimals like the one available in Windows today that I used to demonstrate the above in a few seconds.
***Fermat's Last Theorem - Excel Spreadsheet ****
I would like to add that I drafted an Excel spreadsheet that clearly demonstrates that succeeding numbers to the power of 2 beyond the one described in Fermat's Last Theorem do indicate the widening variances as the equations become larger. The spreadsheet also allows for selecting any power or exponent beyond 2 with the same widening variances result.
To access the spreadsheet, access the below link:
/uploads/2/4/0/7/24074143/fermats_last_theorem.xlsm
***Update 2/15/2016******
EQUATIONS
I believe that all living organisms use a concept for their decision-making process referred to as EQUATIONS. This concept can be as simple as “if I do this, then this will hopefully happen”. Animal species make survival decisions all the time, whether they are based on instinct or experience. We humans are no different in this respect. However, what differentiates us I believe is that unlike other animal species, we make qualitative decisions based on our cognitive reasoning and assessment of situations, and we do strive to rationalize our decisions by adding quantitative elements either to support, change or modify our decisions, or even reject them . An important quantitative element used by our species consists in using the numbering system and the more sophisticated discipline of mathematics with its analytical processes, formulas, symbols and equations.
In this post I would like to demonstrate the use of QUADRATIC EQUATIONS and extend its principle to solve a problem. I will not explain the workings of the process but only indicate that there are many sources which clearly explain the functionality of quadratic equations like this article described in Wikipedia: https://en.wikipedia.org/wiki/Quadratic_equation. Quadratic equations were probably developed some 4,000 years ago by the Babylonians of Mesopotamia as a way to measure the square dimensions of land etc.
We often use the formula ax²+ bx+ c =0 to express the notation of the Quadratic Equation. I would like to apply this formula to resolve a problem that is described in the “the Mathematics Calendar 2014” which has been developed and written by Theoni Pappas, author of The Joy of Mathematics. The problem is x³=12x²+85x. In mathematics, solving problems can usually be reduced by breaking them down into smaller components, and this problem also fits the pattern. Btw, mathematics is a discipline that seeks to identify patterns in everything that needs to be rationalized and resolved. So, we see that each component of the problem includes an unknown factor “x”. Since the equation is supposedly equal on each side, then let’s divide both sides of the equation by the same quantity of “x”. By doing this, we then obtain x²=12x+85. Crossing the factors on the right side of the equation to the left side we then obtain x²-12x-85 = 0. Now this new equation factors into (x-17) X ((x+5), which when multiplied together gives x²-12x-85. In conclusion, the positive root of this equation is 17, while the negative root is -5 following the workings of the Quadratic Equation principle. We have now solved this problem. However here is a more interesting extension aspect to the Quadratic Equation principle which I would like to demonstrate below.
Let’s assume the problem had been a lot more involved such as “x^13-12x^12-85x^11 = 0”. This appears to be a Herculean task to resolve. However, we can again reduce or break the problem down to a simple basic task the fits like hand-in-glove within the QUADRATIC EQUATION principle. Like we did with the initial problem, let’s divide each side of the equation this time by “x^11”, which is the highest divisor to reduce the equation to its basic notation. As such, we now obtain the same start-up equation of “x²-12x-85 = 0”. It should be noted that when you divide exponents with the same base the resulting value is the same base with the difference of the exponents. Since we have reduced the problem to its basic display, the same positive root of 17 and the negative root of -5 will continue to apply. To test this, you may want to apply these roots to this huge equation and you will see that the resulting answer is also 0.
I would like to conclude this exercise with something that I have demonstrated above, yet it is not usually addressed or displayed in the description of the QUADRATIC EQUATION formula. Actually, the description is a subtle or discreet addition. The formula ax²+ bx+ c =0, and this could be more appropriately or visually be described as “ax²+bxᶦ+cx°= 0”. Since xᶦ =x and x°=1, this notation process helps to understand that the exponents are in fact reducing by 1 from the left to the right in the equation. I believe this process should be more clearly explained in the description of the QUADRATIC EQUATION formula. This will also assist in performing more elaborate problems such as I have demonstrated above. Applying this process to the above problem equation we then obtain: “x²-12xᶦ-85x° = 0”. Isn’t math fun and interesting?