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My Knowledge of Mathematics
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My Knowledge of Mathematics I had my first exposure to mathematics when I was 6 years old in first grade in the autumn of 1965. I learned the 10 Hindu/Arabic numerals, 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 which were introduced into Europe by Leonardo Fibonacci in the 12th century. The curriculum started with simple addition, that is, 1 + 1= 2, then 2 + 2= 4, etc. Then subtraction was introduced, that is 5 - 3= 2, etc. Back in the 1960s, electronic calculators and desktop computers did not even exist then. I remember that I had great difficulty in learning the multiplication tables ranging from 1 x 1= 1, 2 x 3= 6 all the way to 10 x 10= 100, when I was 10 years old attending 5th grade in elementary school in Middle Village, Queens, NY in 1970. After learning multiplication, the much more difficult reverse division was introduced which I recall used to be extremely difficult to figure out by pencil and paper before electronic calculators were introduced around 1976 when I was in high school. Then I learned other concepts like fractions, or ratios of whole numbers and the concept of irrational numbers, starting with pi= 3.141592654.... the ratio of a circle's diameter to its circumference and the the square roots of integers like the square root of 2 = 1.41421233562. . .Actually I first learned the concept when a friend told me that he learned them in school and not in class. In 2nd grade, I learned how to tell time by reading a clock face. I also learned about the Sieve of Eratosthenes which is a method of figured out all the prime integers up to a certain point, say up to 100, by writing them down on a grid and systematically crossing out all the multiples of the numbers up to 10. I much later learned out as an adult, that there is in fact no end to the progression of prime numbers by a method explained in "Euclid's Elements" written back in the 3rd century B.C.E. In Alexandria, Egypt. Later in middle school, I learned a lot about set theory, the use of a number and the concept of logarithms which were developed in the 16th century as a convenient form of multiplication. The outdated slide rule works by the addition and subtraction of logarithms for multiplication, division and extracting roots. My paternal grandfather gave me a slide rule as a which I still have and is now technically an antique as no one used them anymore having been replaced by electronic calculators and other modern computing devices. In the autumn of 1972, I took elementary algebra in middle school and at first it was very difficult to understand. One new concept I learned in algebra was the concept of negative numbers or numbers of a value lower than zero as a solution to algebraic equations. Then when I took more advanced algebra next year, I learned about the concept of imaginary numbers based on the square root of -1 which technically can not have a solution in real numbers. I learned about the quadratic formula as a solution for second degree algebraic equations which I saw the teacher derive on a blackboard in front of the class. Now, when I was a in the mid 1970s, they never taught the class about the solution to higher degree equations, starting with the third degree cubic, the fourth degree quartic. Most especially they did not explain at all that the fifth degree quintic and all others of higher degrees are completely insolvable by the use of algebraic formulas as discovered by the European mathematicians, Galois and Abel with the idea of group theory, early in the 19th century. I only found that out much later as an adult. Back in the spring of 1974, I recall having to laboriously use paper and graph paper to plot out and graph, algebraic equations of two variables, x and y, which was often very difficult. Now since 1994, we now have advanced electronic graphing calculators which can easily do that and are available to any middle and high school student. It was also an ordeal to have to look up logarithms and trigonometric ratios in a table in the back of math textbooks and now we have electronic calculators to easily look them with ease by just punching in the numbers. I think the mathematics course I enjoyed the most was when I took high school geometry in the autumn of 1974 and the spring of 1975. As the teacher explained, the concept of geometry is very different from the more logical approach of arithmetic and algebra. The whole concept of geometric proof is based on logical deduction, that is making a proof by the process of elimination. The whole basis of modern high school geometry is based on the concepts laid out in "Euclid's Elements" composed back over 23 centuries ago. Actually, it was the second biggest bestseller in the world next to the Bible, throughout the middle ages and into the modern era. In fact, it was written before the more modern concepts of trigonometry and algebra were developed as it contains no concepts like measuring angles in triangles by degrees, etc. When I learned geometry, it starts with a progression of dimensions, starting with specifying a position by a point, then a one dimensional line, ray, line segment, etc., then to a two dimensional plane, with the basic shapes, of triangle, quadrilateral, more sided polygons, then to circles, ellipses, parabolas, hyperbolas, etc. I then learned the more complicated solid geometry with basic three dimensional shapes like a cube, pyramid, sphere, etc. In the third book of Euclid's Elements, he lays and explains the properties of the five Platonic Solids, that is solid geometric figures where all the sides and angles are equal, the one most familiar to the layman is the cube. I find it to be rather amazing that he figured out that stuff way back then , three centuries before Christ without the aid of any modern computing devices. In high school in 1976, I took trigonometry which I found is really just an extension of the concepts of geometry based on the trigonometric ratios of the sides and angles of right triangles, that is triangles where one angle is 90 degrees. Of course, figuring out problems in trigonometry is very much easier today with the use of modern graphing calculators. Those calculators can also do complex problems in geometry. When I attended S.U.N.Y. at Stony Brook in the autumn of 1977, I first learned calculus, that is using to concept of computing very small limits as in differential and integral calculus to do problems in the trajectories of moving bodies like artillery projectiles and space probes around the solar system. Actually our modern space program would be completely impossible without calculus. In the few centuries before Christ, the ancient Greeks were beginning to stumble upon the fundamental concepts of calculus, like Zeno's Paradox of an arrow in motion, or the race between the tortoise and the hare, etc. But then the middle ages arrived and no one was interested in mathematics. In the 17th century, Newton and Liebnitz figured out the basic concepts of calculus. It's interesting that we now use Leibniz's notation in calculus and not Newton's today. Calculus is used extensively in engineering and architecture to figure out things like areas under curved lines and surfaces. In the spring of 1994 while in my final semester in S.U.N.Y. at Farmingdale taking Technical Speech, I became obsessed with solving the original 3 x 3 Rubik's cube puzzle which first came out in the Christmas season of 1979/80. Then, I went to the campus<b> library </font></b>and took out several puzzle books on how to solve it. When reading them, I discovered the whole principle of being able to solve it depends on a branch of mathematics called group theory which was first developed by mathematicians in the 19th century. I was able to solve the cube even without the sight of the manual. But then, I became very curious and went to the local bookstores on Long Island and bought more books on the subject. Actually group theory is really an elaborate form of algebra and it can used too solve problems in physics and chemistry. I also tried to learn abstract algebra and Galois theory. Of course, I started to buy several books on topology. In the 18th century, the Swiss mathematician Leonhard Euler tried to figure out the problem of the Seven Bridge of Konigberg, where modern Poland is now. That is actually a problem in graph theory. In studying topology, you find that it is a study of spatial relationships of a very different nature than geometry. In geometry, it is a study of specific measurements and shapes, like the length of line segments, angle measurements, etc, and specific geometric shapes, etc. In topology, it is more of a study of just how two and three dimensional figures are connected together. Actually, most people tend to think more topologically than geometrically in doing things like reading a map, walking around the house or walking to familiar destinations in the local neighborhood. One concept that most people are really not aware of is the four color map problem, that in fact, in drawing up the political divisions within any two-dimensional map, it actually only takes 4 colors to separate any region from any other. There is an elaborate computer proof and I've read a book about it and began somewhat to understand it. In topology, a coffee cup and a donut are very similar because they are both three-dimensional figures with a single hole in them. If you squeeze the hole out of it, the figure becomes a solid three dimensional figure and is thus very different topologically. When you think about the basic geometric ratio, pi= 3.141592654... it was figured out by the ancient Greeks by dividing a circle into ever smaller triangles. The use of the Greek letter pi to designate it wasn't until the 17th century. Now, even though it has probably been computed by modern computers extending into billions of decimal places, there is in fact, no actual circle that has ever been found that is accurate to the full figure of pi than at most five decimal places. I first learned the fundamentals of set theory when I was elementary school around 1971. I have since learned the mathematician Georg Cantor used it in the late 19th century to prove that there is in fact, no actual correspondence between the irrational numbers which have decimal components stretching into infinity and the much more familiar rational numbers , like whole numbers, integers and the ratios of integers or fractions, like 1/2. 3/4, 7/8, etc. In fact, the vast majority of all real numbers are in fact, irrational numbers and the rational numbers are actually extremely rare intervals among them. Back in the stone age, actually most hunter/gatherer people only thought in terms of the natural numbers, that is 1, 2, 3, 4, 5,. . . . The concept of 0 as a numerical place holder signifying nothing wasn't until the 6th century in India. Virtually no one ever thought in terms of fractions or ratios of whole numbers. The use of the term a million to specify a thousand thousands or 1,000 x 1,000= 1,000,000 did not arise until the 15th century in Italy, when bookkeepers and merchants would come upon numbers that big in their computations. Then the term billion equals a thousand millions and trillion which means a thousand billions did not come about until more recently. The notation the ancient Romans used were Roman numerals which are very clumsy and completely useless for numerical computation. The way a number of n objects can be arranged differently is n! or n factorial or the specific in integer being multiple by all the integers smaller than it to 1. A prime number is one which has no factors other than itself and 1. All other numbers are composite numbers which are all the product of a specific combination of prime integers and no other. Modern electronic calculators can be used to compute and figure out all these basic mathematics concepts. In the summer of 1994, the first electronic graphing calculators came out and I bought my first one, the Texas Instruments TI-85 which hitherto to the earlier calculators could do advanced problems in algebra and calculus. I spent most of my 3 days off from employment at nights working out computer programs on the calculator involving problems in those mathematical disciplines. I found it amazing that the calculator could plot two variable algebraic equations. Then in 1996, the more advanced Texas Instruments TI-92 which can do problems in geometry and 3-dimensional graphing along with a lot of other interesting applications. That would have been very handy to use when I took geometry in high school, 20 years earlier. They've since come out with more advanced calculators with color screens. I have also studied a lot about number theory which is about the properties of the whole numbers or integers but actually that is a misconception since a significant portion of number theory is about studying the aspect of numbers with fractional components being irrational that technically are not integers. One interesting concept is of continued fractions and there are degrees to how irrational a number can be as to whether the continued fraction that computes will have an entirely random pattern or that it ever forms a repeating pattern. One interesting concept is that there in fact are no uninteresting numbers as if any particular number is always interesting for some reason and in fact if it were perceived to be uninteresting, it would in fact be interesting for that reason. An other thing is the concept of very large numbers. Although we human beings have among the longest life spans of any animal and we measure our lifespan in years so at the most we are very lucky to reach the age of 100 or a little above it. My paternal grandfather lived to be 98 years old. If we measure our lifespan in seconds, a person has to reach the age of nearly 32 years old to be a billion seconds old and if we're lucky we may live to be over 3 billion seconds old. But that is actually less measured in seconds than the Earth and the rest of the solar system are old in earth years, as the estimated age of the Earth is about 4.54 billion years. The age of the universe since the Big Bang event is estimated to be something like 13.8 billion years. When you study chemistry and physics you come across extremely large numbers like Avogadro's Number equals 6.02214179 x 10^23 which is the number of atoms or molecules of an element or compound respectively as measured in grams. The number of possible combinations of 52 deck of playing card is 52!= 8.065817517. . x 10^67. Now, that number is extremely huge, being by degrees immensely greater that the age of the Earth and the universe as measured in seconds. That obviously explains as to why there is an immense variety to all the different ways, most games of playing cards can be played. The same is true for a lot of other games like chess as when playing a game of chess there are in fact, about 4 billion combinations of moves that the first 4 moves can be played. When I took high school geometry in 1974/75, of course I learned the Pythagorean Theorem, that is for a right triangle, a triangle where one angle is 90 degrees, the square of the hypotenuse of longest side is equal to the sum of the squares of the two smaller sides or legs of the triangle and this can be expressed in integer or whole number values as Pythagorean triples as a^2 + b^2= h^2. This is in fact an example of a Diophantine equation, that is an algebraic equation which only has solutions in integer values. Actually the ancient Greek Pythagoras wasn't the first mathematician to discover the Pythagorean theorem in the 6th century B.C.E. as it had been known to the ancient Babylonians, several thousand years earlier towards the beginning of history. They have found ancient writings of written numerical figures that showed that they were able to compute very high integer values for Pythagorean triples, of integers in the thousand number range that could have not possibly come about by mere chance. In the 17th century, the French mathematician Pierre Fermat claimed to have come up with a proof called Fermat's Last Theorem as to why the Diophantine Equation, a^n + b^n= h^n in fact only has integer solutions for n= 2 and no higher exponential power above it, which he actually didn't. Then in the late 20th century, the mathematician Andrew Wiles did come up with an actual proof to Fermat's Last Theorem. Now I have taken out several<b> library </font></b>books on the subject and looked it up on Wikipedia and I still cannot understand exactly how his proof works and I do have a lot of experience in understanding most mathematical concepts.           |
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