Introductory lecture
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Dear students and all participants. I'll read you a short introductory lecture on mathematics. This course is designed for students of all science and engineering specialties. It contains the necessary minimum of theoretical knowledge and practical skills for successful mastery of your profession. The course is divided into 3 modules: mathematical analysis, linear algebra and analytic geometry and also the theory of probability and mathematical statistics.
The word "mathematics" means "learning through reflection" in the ancient Greek language. And indeed, to understand mathematics is necessary logic and intuition, that is a strict logical thinking and intuitive representation of familiar associations, geometric images, spatial imagination.
The object of the mathematical study is quantitative relations and spatial forms of the world around us, that is the reality, and its generalizations. Mathematicians explore a variety of particular and abstract mathematical structures. Mathematics is widely used in other sciences and applications, due to hypothetic-deductive method, mathematical modeling and, in modern conditions, various computer technologies.
Mathematics, first developed as pre-science, that is a set of rules and regulations "do this", "do so" when counting, calculating, measuring, necessary already at the time inthe exchange, trade, agriculture, the observation of the starry sky, shipbuilding, navigation, production of tools and household items.
Further, in the VI century BC in ancient Greek philosophical schools of Thales Miletsky and Pythagoras of Samos mathematics arose as a science, as a deductive science and evidence-based science. The first mathematical knowledge of two and a half centuries were summarized in Euclid's famous work "Elements". So mathematics, as well as all European science, can be assumed to be two and a half thousand years, even two thousand five hundred and fifty years.
The quantitative ratio, the category of number is manifested in mathematics in the concept of number. The development of the concept runs through the history of mathematics, beginning with the first natural numbers (1, 2, ... a lot, they said) through integers, rational numbers, real numbers to complex numbers, quaternions, cardinal and ordinal numbers. Algebra also studied number-like objects: matrix, polynomials, different rows, their sets
And thus the development of the concept of number through arithmetic, the number theory traditional elementary algebra to modern abstract algebra. The basis, of course, is a system of real numbers. You know from the school that real numbers are represented by points of the number line, thereal number are divided into rational numbers, these are fractions of integers of form m/n, for example, 15/7 and other numbers that are not represented in the form of fractions – they are irrational numbers.
The first of these was a square root of 2 as the diagonal of the unit square found in the Pythagorean school. Also, real numbers are divided into algebraic numbers (the roots of polynomials with integer coefficients) and other numbers - transcendental numbers, among which you know numbers e and pi. E - is the base of natural logarithm, pi is defined as the ratio of the circumference to circle diameter.
It should be noted that using parallel programming pi is calculated to 43 trillion signs that, of course, this was not possible previously.
The first module is mathematical analysis. Real number system - is the foundation of classical mathematical analysis, the object of which are numerical functions, i.e. functions f: RR, building graphs and their different properties. The most important possible properties of functions are properties of continuity, differentiability, integrability and measurability of the function. They are all defined in terms of the concept of function limit at the point.
The fundamental idea of passage to the limit permeates the entire mathematical analysis. The course studies the basics of differential and integral calculus. Central concepts here are the derivative and the differential, the uncertain integral, as a set of primitives, which differ from each other by a constant, and the definite integral.
Differentiability of numerical function y = f (x) at x0 means the existence of this function derivative at the point x0, i.e. f '(x0). Visually, it looks as a smooth curve or the function graph or the curve y = f (x) near point x0, without any projections and breaks there. With predetermined accuracy a piece of the curve in point x0 can be replaced by a tangent drawn at this point to the graph of y = f (x).
You know from school that the derivative of function y = f (x) at point x0 is tangent of the slope angle of this line – the slope to x-axis. In the figure, it looks like this. The tangent equation is y = kx + b, where k is the derivative of (x) is positive and continuous on the interval [a, b], then there exists a definite integral of this function in the range from a to b, which according to Newton-Leibniz formula is the difference F (b) - F (a), where function f (x) at x0.
If function y = f F (a) and F (b) are primitives of function f on the interval [a, b], i.e. F '(x) = f (x). It is equal to the area of a figure enclosed between the straight line (Ox-axis, direct lines x = a, x = b) and the curve y = f (x), that you can clearly see in the picture - this is the area of the curvilinear trapezoid ABCD.
The first module also addresses the differential calculus of functions with two or more variables, double integral, curvilinear integrals. Mathematical analysis is well suited to describe various kinds of motion and change, in particular the description of natural, various technological production processes, it is effective in applications.
The second module starts with analytic geometry. Here we consider the category of shape and spatial forms embodied in mathematics in the concept of "geometric figure". Geometric figure is a set of points of some space, say, Euclidean plane P.
If plane P is given by a rectangular Cartesian coordinate system xOy, point A is associated with the ordered pair of coordinates (x, y), x is the projection of point a on x-axis, y is the projection of this point on y-axis. In particular, the origin will be associated with the pair (0, 0), the plane itself – with a plurality of all pairs of real numbers, i.e. it can be R * R, and Figure F is assigned a set of pairs of real numbers corresponding to the points of this figure. That is, we can assume that F is a set of pairs of real numbers.
The set of solutions of the linear equation ax + by = c with real coefficients set a figure on plane P. If a is 0, b is not zero, and C is not equal to zero, there is no solutions of this equation, therefore, the figure is an empty set. If all the numbers a, b, c are equal to zero, we get the whole plane, since any pair of numbers satisfies this equation. If one of the coefficients before the unknown a or b is not zero, then we get a straight line.
The set of solutions of the square equation, say x ^ 2 + y ^ 2 = 4 sets on the plane a circle with radius 2 centered at the origin - point O. In the picture it looks like this. That line with coefficient a = 1/2, if y is moved to the right side, its factor is -1, if c is moved it will be equal to 1, if we write this down, in the form ax + by = c, we see a circle, with the center at point O with radius 2.
The idea coordinatization of geometric space, its points and shapes underlies analytic geometry which started in 16th -17th centuries by French mathematician and philosopher Rene Descartes. Parallel to him, other mathematics work on the creation of analytic geometry, in particular Fermat. In the three-dimensional Euclidean space (this is a cube) R * R * R planes are set by linear equations with three unknowns Ax + By + D = 0, and the lines – by systems of two linear equations, as intersection of two planes.
Second order curves are also considered, these are ellipses, parabolas, and hyperbolas, if viewed in three dimensions, we getn surfaces - ellipsoids, paraboloids and hyperboloids. Connection between figures and numbers already clearly manifested itself in the antiquity in trigonometry. "Trigon" is a triangle, and each triangle can be associated with lengths of its sides or angles, the perimeter and area, the radius of the inscribed or circumscribed circle.
Linear algebra is research theory of methods for solution of linear equation system, with numerical coefficients, which are real. A common method of solving equations systems is the Gauss method - the method of successive elimination of unknowns. Studying the properties of matrixes and determinants is also applied to linear algebra.
We can consider an example of three equations with three unknowns.
Here we denote them x1, x2, x3, instead of x, y, z. Coefficient, for example, a23, which means the second equation coefficient before the unknown x3, b1, b2, b3 are free members. But it is more convenient to work not with the systems, writing the unknowns, but with matrixes. Here is the matrix - this is the main matrix of this system, consisting of coefficients before the unknowns. It is assigned a number, its determinant, which plays an important role in linear algebra. In general, we consider n equations with n unknowns.
Here, let's see geometrically, which cases are possible here. Each of these equations, if it is not trivial, defines a plane, and we have three planes in an ordinary three-dimensional Euclidean space, and the system is their intersection. How may they be located relative to each other?
If the first plane is parallel to, say, the second plane, but they do not match, then there are no solutions, they do not intersect. If they match, then we consider the second and third equations, i.e. two planes, and what situation is possible there? Again, they are parallel – there are no decisions , they coincide – there are infinitely many solutions and they intersect in a straight line - there is also an infinite number of solutions.
But if the two, the first and the second, planes intersect, they are not parallel, they intersect in a straight line, and we already have a straight located relatively the plane defined by the third equation. If, again, the straight line is parallel to the plane, is not included in it, then there is no solution. If it is included there are infinite number of solutions. If the straight line intersects the plane, we get a unique solution.
Thus we see that the solution of a linear equations system, as in this example of three equations with three unknowns, can be 0, there are no solutions, an inconsistent system. One unique solution gives a certain, consistent system. With infinitely many solutions the system is called uncertain consistent. There is a close connection with determinants: it says that the determinant is not equal to 0 if and only if the system has a unique solution.
Apparatus and algorithms of linear algebra are used шт many areas of mathematics: geometry, theory of differential equations, computational mathematics, in a variety of applications. Linear algebra mathematically formalizes fundamental idea of linearity, which we discussed when considering the function derivative at the point.
We turn to the third section - the theory of probability and mathematical statistics. Here we study mathematically expressed concepts of probability and randomness. The different events in the space of events are considered, and each event is assigned a real number from the interval [0, 1], that is non-negative, less than or equal to one. Often, the probability is expressed as a percentage, then it is necessary to multiply it by 100 percent.
Events are divided into possible and impossible, reliable ... Impossible event is an event with 0 probability, it will never happen. Possible event happens for sure, it has a probability of one. Consider the simplest problem as an illustration. A dice is tossed - it's a cube with the number of points on one of six faces from the 1 to 6. What is the probability of event A, meaning that the number drawn is a composite number?
But we know that all the outcomes of such tests number 6 – falling out 1, 2, 3, 4, 5, 6 points. All of them are equally possible (if frauds haven’t made the faces opposite number 6 heavier, so that they will often have to fall 6, then they will always win). These numbers from one to six are divided into three groups: the primes (have exactly two positive dividers), these are 2, 3, 5, compound numbers having more than two dividers or folding into a product of smaller numbers (these are 4 = 2 * 2, 6 = 2 * 3), and number 1 is aloof, it has exactly one divider.
According to the classic definition of probability, the probability of an event A is the number of favorable outcomes divided by the number of equally possible outcomes. In this case, there are 2 composite numbers, total number is 6, 2 divided by 6, the result is 1/3.
In this module we study the properties and formulas of probabilities, discrete and continuous random variables, their most important numerical characteristics such as mean and variance, standard deviation. And the basics of mathematical statistics. The basis of the theory of probability is combinatorics, it is already seen in the problem of tossing two dices.
The simplest combinatorial things occur depending on the sort of outcomes, most likely sums of the numbers. Methods of mathematical statistics are widely used in sociology, during various public opinion polls, in the scientific research analysis.
I must say that mathematics is a developing science, it continues to evolve rapidly. There are new sections in it. At preset we can name, perhaps dozens, if not hundreds of mathematical disciplines. We start with arithmetic, elementary algebra, elementary geometry, and then we have higher mathematics, higher algebra, geometry, mathematical analysis.
Now you know that there is discrete mathematics, game theory, probability theory and mathematical statistics, there is mathematical logic and algorithm theory, differential geometry, etc.
Some problems of antiquity remain unsolved, such as the problem of the twins. The twns are two prime numbers that differ by 2, for example, 3 and 5, 5 and 7, 11 and 13 and so on. It is still not known whether there are infinitely many such pairs, or t some place they are not found in the natural sequence of 1, and so on ad infinitum.
But the problems that mathematicians and the humanity faced for a long time have been resolved recently, in the last quarter of the 20th century Fermat's Last Theorem was proved by Andrew Wiles and finally resolved in 1995, 25 years ago. Fermat's theorem is whether there is a solution in natural numbers of equation x ^ n + y ^ n = z ^ n, where n> 2. If n = 2, this is so-called Pythagorean triples, we already know that 3 ^ 2 + 4 ^ 2 = 5 ^ 2 by the Pythagorean theorem.
For higher numbers there is no solution, and this was proven in 1995. Another problem is Poincare hypothesis, which existed about a hundred years, and was solved by Russian mathematician Grigori Perelman in 2003. So problems can be solved. But many problems are not resolved yet.
One of such important, one might say philosophical, problems to be solved and, of course, with mathematical methods is how far the development of artificial intelligence is possible, where its borders lie, if it is consistent with the human mental power, if it can cover the human intellect, that is to create robots indistinguishable from humans.
I think that this is impossible, but the approach to this in the development of modern computers, which, I repeat, have already counted 43 trillion real digits of Pi, is quite possible, and we can expect the development of artificial intelligence, robotics, but, nevertheless, the human mind is still something divine, no program can program it. Thanks for attention.