Matrix Theory and Linear Algebra (562420), страница 16

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Binomial

Binomial, algebraic expression that consists of exactly two terms separated by + or -, such as x + y or ab - cd. The binomial theorem asserts that the general expansion of a binomial, such as (x + y), raised to the nth power is given by

The general coefficient of the kth term in the above expression is

and is usually denoted by the symbol (). The expansion of (x + y)ⁿ contains n + 1 terms. Formulated in medieval times, the binomial theorem was developed (about 1676) for fractional exponents by the English scientist Sir Isaac Newton, enabling him to apply his newly discovered methods of calculus to many difficult problems. The binomial theorem is useful in various branches of mathematics, particularly in the theory of probability.

Pi

Pi, Greek letter () used in mathematics as the symbol for the ratio of the circumference of a circle to its diameter. Its value is approximately 22/7; the approximate value of  to five decimal places is 3.14159. The formula for the area of a circle, A = r² (r is the radius), uses the constant. Various approximations of the numerical value of the ratio were used in biblical times and later. In the Bible, the value was taken to be 3; the Greek mathematician Archimedes correctly asserted that the value was between 3 10/70 and 3. With computers, the value has been figured to more than 200 billion decimal places. The ratio is actually an irrational number, so the decimal places go on infinitely without repeating or ending in zeros. The symbol  for the ratio was first used in 1706 by the Welsh mathematician William Jones, but it became popular only after its adoption by the Swiss mathematician Leonhard Euler in 1737. In 1882 the German mathematician Ferdinand Lindemann proved that  is a transcendental number—that is, it is not the root of any polynomial equation with rational coefficients (for example, x³ - 5/7x² - 21x + 17 = 0). Consequently, Lindemann was able to demonstrate that it is impossible to square the circle algebraically or by use of the ruler and compass.

Linear Programming

Linear Programming, mathematical and operations-research technique, used in administrative and economic planning to maximize the linear functions of a large number of variables, subject to certain constraints (see Algebra; Function; Mathematics). The development of high-speed electronic computers and data-processing techniques has brought about many recent advances in linear programming, and the technique is now widely used in industrial and military operations. See Computer.

Linear programming is basically used to find a set of values, chosen from a prescribed set of numbers, that will maximize or minimize a given polynomial form (see Theory of Equations), and this is illustrated by the following example of a particular kind of problem and a method of solution. A manufacturer makes two varieties, V₁ and V₂, of an article having parts that must be cut, assembled, and finished; the manufacturer knows that as many articles as are produced can be sold. Variety V₁ takes 25 min to cut, 60 min to assemble, and 68 min to finish; it yields $30 profit. Variety V₂ takes 75 min to cut, 60 min to assemble, and 34 min to finish, and yields a $40 profit. Not more than 450 min of cutting time, 480 min of assembly time, and 476 min of finishing time are available each day. How many articles of each variety should be manufactured each day to maximize profit?

Let x and y be the numbers of articles of varieties V₁ and V₂, respectively, that should be manufactured each day to maximize profit. Because x and y cannot be negative numbers,

The cutting, assembly, and finishing data determine the following inequalities and equalities:

The profit is given by

The problem is to find the values of x and y, if any, subject to restrictions (1) through (5), that will maximize the linear polynomial or linear form (6).

The equation 25x + 75y = 450 represents a straight line in the Cartesian plane (see Geometry); if point P has coordinates (r, s), P is above the line, on the line, or below the line, as 25r + 75s is greater than, equal to, or less than 450. Therefore, condition (3) is satisfied by the coordinates of any point in the half plane determined by the line 25x + 75y = 450 and all points below it. Similarly, each of the conditions (1), (2), (4), and (5) is satisfied by the coordinates of a point in a certain half plane. To satisfy all five conditions, the point must lie on the boundary or interior of the convex, polygonal region OABCD in Figure 1. The region is convex because if R and S are any two points of the region, so is every point of the line segment RS; it is polygonal because its boundary consists of line segments.

The equation 30x + 40y = p, indicating the profit, also represents a straight line; the equation 30x + 40y = p’ represents a parallel line that is above, coincides with, or is below the first line, as p’ is greater than, equal to, or less than p. The profit will be maximized by choosing the line, of the family of parallel lines, that just touches the region OABCD above, namely, the line through the vertex B(3,5). The manufacturer will earn a maximum profit (of $290) if 3 articles of variety V₁ and 5 of variety V₂ are made per day. Any other quantities of the two varieties, within the constraints of the time limitations, will yield a smaller profit.

Linear programming is applied to many other kinds of problems, and many other methods of solution exist, but the above example is generally illustrative.

Inequality (mathematics)

Inequality (mathematics), mathematical relationship that makes use of the way in which numbers are ordered. Figure 1 shows the symbols used to denote inequality. For example, the inequality 3 < 10 says that the number 3 is less than the number 10. The inequality x²≥ 0 expresses the fact that the square of any real number is always greater than or equal to zero.

Inequalities often arise in describing areas and volumes. For example, if P is any point on the diagonal of the square shown in figure 2, then the area of the two rectangles that are shaded blue is always less than or equal to (≤) the area of the two squares that are shaded red.

The solutions of an inequality such as -2x + 6 > 0 are the values of x for which the expression -2x + 6 is greater than zero. The rules of algebra can be applied to solve this inequality, except that the direction of the inequality must be reversed when multiplying or dividing by negative numbers. So, to solve the inequality -2x + 6 > 0, first subtract 6 from both sides of the inequality to get -2x> -6. Next, divide both sides of -2x> -6 by -2, reversing the direction of the inequality since -2 is negative. This gives x< 3, meaning that any value for x that is less than 3 will be a solution of -2x + 6 > 0.

Root (mathematics)

Root (mathematics), in mathematics, a number that when multiplied by itself a stated number of times yields as a result a second, given number. The index of the root is the number of times that the root appears in the multiplication. For example, the numbers 2 or -2 are the square, or second, roots of 4; 3 or -3 are the square roots of 9; and 4 or -4 are the square roots of 16. A cube, or third, root of 8 is 2, and a cube root of -8 is -2; a cube root of 27 is 3, and of -27 is -3, and so on. The usual symbol for root is √ with the appropriate index, except in the case of the square root, which is written without an index. For example

It is also possible to express roots in the form of fractional exponents; thus

In an algebraic equation, when a quantity inserted in place of the unknown quantity renders the equation a true statement, that quantity is called a root of the equation. See Power.

Fractal

Fractal, in mathematics, a geometric shape that is complex and detailed in structure at any level of magnification. Often fractals are self-similar—that is, they have the property that each small portion of the fractal can be viewed as a reduced-scale replica of the whole. One example of a fractal is the “snowflake” curve constructed by taking an equilateral triangle and repeatedly erecting smaller equilateral triangles on the middle third of the progressively smaller sides. Theoretically, the result would be a figure of finite area but with a perimeter of infinite length, consisting of an infinite number of vertices. In mathematical terms, such a curve cannot be differentiated (see Calculus). Many such self-repeating figures can be constructed, and since they first appeared in the 19th century they have been considered as merely bizarre.

A turning point in the study of fractals came with the discovery of fractal geometry by the Polish-born French mathematician Benoit B. Mandelbrot in the 1970s. Mandelbrot adopted a much more abstract definition of dimension than that used in Euclidean geometry, stating that the dimension of a fractal must be used as an exponent when measuring its size. The result is that a fractal cannot be treated as existing strictly in one, two, or any other whole-number dimensions. Instead, it must be handled mathematically as though it has some fractional dimension. The “snowflake” curve of fractals has a dimension of 1.2618.

Fractal geometry is not simply an abstract development. A coastline, if measured down to its least irregularity, would tend toward infinite length just as does the “snowflake” curve. Mandelbrot has suggested that mountains, clouds, aggregates, galaxy clusters, and other natural phenomena are similarly fractal in nature, and fractal geometry's application in the sciences has become a rapidly expanding field. In addition, the beauty of fractals has made them a key element in computer graphics.

Fractals have also been used to compress still and video images on computers. In 1987, English-born mathematician Dr. Michael F. Barnsley discovered the Fractal Transform^TM which automatically detects fractal codes in real-world images (digitized photographs). The discovery spawned fractal image compression, used in a variety of multimedia and other image-based computer applications.

Construction of a Fractal Snowflake

A Koch snowflake is constructed by making progressive additions to a simple triangle. The additions are made by dividing the equilateral triangle’s sides into thirds, then creating a new triangle on each middle third. Thus, each frame shows more complexity, but every new triangle in the design looks exactly like the initial one. This reflection of the larger design in its smaller details is characteristic of all fractals.

Julia Set

The fractal shown here is the graphical representation of a mathematical function called the Julia set. The set is named after French mathematician Gaston Julia, who worked on the mathematics of fractals early in the 20th century, before the term “fractal” was coined by Polish-born French mathematition Benoit Mandelbrot in 1975. The pattern of the whole shape in a fractal repeats itself on smaller and smaller scales, so that magnifying a fractal produces a shape that is similar to the original.

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