Vocabulary Notes

1. distinct boundary - чітка межа

2. the division being conditional. - розподіл є умовним

3. volumes of solids - об'ємів твердих (геомет¬ричних) тіл

4. consideration of variable quantities - урахування змінних величин

5. called differential calculus - що називається диференціальним численням

6. the close relationship - тісний взаємозв'язок

7. the coordinate method - метод координації

8. graphs are used for representing relations - графіки використовуються для ілюстрації залежності

9. these divisions are represented - ці розділи викладаються

10. the theory of random processes – теорія випадкових процесів

11. the theory of stability – теорія стійкості руху

Read the text and speak on the following topics

1. Higher mathematics vs. elementary mathematics .

2. What made you choose mathematics as the special interest of your life?

Text 2 complex numbers

Make a written translation of the following text.

For many reasons the concept of number has had to be extended even beyond the real number continuum by the introduction of the so-called complex numbers. It was the need for more freedom in formal calculations that brought about the use of negative and rational numbers. The process which first requires the use of complex numbers is that of solving quadratic equations.

We recall the concept of the linear equation ax = b, where the unknown quantity x is to be determined. The solution is simply x = b/a and the requirement that every linear equation with integral coefficients. a ≠ 0 shall have a solution necessitated the introduction of the rational numbers. Equations such as

x² = 2

which have no solution x in the field of rational numbers, led us to construct the wider field of real numbers in which a solution does exist. But even the field of real numbers is not wide enough to provide a complete theory of quadratic equations. A simple equation like

x² = — 1

has no real solution, since the square of any real number is never negative.

Guided by these considerations we begin our systematic exposition by making the following definitions. A symbol of the form a + bi, where a and b are any two real numbers, shall be called a complex number with real part a and imaginary part b. The operations of addition and multiplication shall be performed with these symbols just as though «i» were an ordinary real number , except that i shall always be replaced by — 1. More precisely, we define additional multiplication of complex numbers by the rules

(a + bi) + (c + di) = (a + c) + (b + d) i,

(a + bi) (c + d i) = (ac — bd) + (ad + bc)i.

On the basis of these definitions it is easily verified that the commutative, associative and distributive laws hold for complex numbers. Moreover, not only addition and multiplication, but also subtraction and division of two complex numbers lead again to numbers of the form a + bi, so that the complex numbers form a field. The field of complex numbers includes the field of real numbers as a subfield, for the complex number

a + 0· i is regarded as the same as the real number a.

The geometrical interpretation of complex numbers consists simply in representing the complex number z = x + yi by the point in the plane with rectangular coordinates x, y. Thus the real part of z is its coordinate, and the imaginary part is its y-coordinate. A correspondence is thereby established between the complex numbers and the points in a «number plane», just as a correspondence was established between the real numbers and the points on a line, the number axis.

If z = x + yi is any complex number, we call the complex number z == x + yi the conjugate of z. The point z is represented in the number plane by the reflection of the point z in the x-axis as in a mirror. The real number is called the modulus of z, and written

.

The value p is the distance of z from the origin. The angle between the positive direction of the x-axis and the line Oz is called the angle of z, and is denoted by φ.

An algebraic number is any number x, real or complex, that satisfies some algebraic equation of the form a_nxⁿ+ a_n-1x^n-1 + ··· + a₁x + a₀ = 0 (n > 1, a_{n ≠} 0), where the a₁ are integers. For example, is an algebraic number for it satisfies the equation

x² — 2 = 0.

The concept of algebraic number is a natural generalization of rational number, which constitutes the special case when n = 1. Not every real number is algebraic. This may be shown by the following theorem proved by Cantor.

The totality of all algebraic numbers is denumerable. Since the set of all real numbers is non-denumerable, there must exist real numbers which are non-algebraic.

A method for denumerating the set of algebraic numbers is as follows. To each equation with integral coefficients the positive integer

is assigned as its "height". For any fixed value of h there are only a finite number of equations with height h. Each of these equations can have at most n different roots. Therefore there can be but a finite number of algebraic numbers whose equations are of height h. We can arrange all the algebraic numbers in a sequence by starting with those of height 1, then taking those of height 2, and so on. Real numbers which are not algebraic are called transcendental.