TEXT II. RATIONAL AND IRRATIONAL NUMBERS

Quotients of integers a/b (where b¹0) are called rational numbers. For example, 1/2, -7/5, and 6 are rational numbers. The set of rational numbers, which we denote by Q, contains Z as a subset. The students of mathematics should note that all the field axioms and the order axioms are satisfied by Q.

We assume that every student of mathematical department of universities is familiar with certain elementary properties of rational numbers. For example, if a and b are rational numbers, their average (a+b)/2 is also rational and lies between a and b. Therefore between any two rational numbers there are infinitely many rational numbers, which implies that if we are given a certain rational number we cannot speak of the "next largest" rational number.

Real numbers that are not rational are called irrational. For example, e, p, e^π are irrational.

Ordinarily it is not too easy to prove that some particular number is irrational. There is no simple proof, for example, of irrationality of e^π. However, the irrationality of certain numbers such as √2 is not too difficult to establish and, in fact, we can easily prove the following theorem:

If n is a positive integer which is not a perfect square, then √n  is irrational.

Proof. Suppose first that n contains no square factor > 1. We assume that √n is rational and obtain a contradiction. Let √n = a/b, where a and b are integers having no factor in common. Then nb² = a² and, since the left side of this equation is a multiple of n, so too is a². However, if a² is a multiple of n, a itself must be a multiple of n, since n has no square factors > 1. (This is easily seen by examining factorization of a into its prime factors). This means that a = cn, where c is some integer. Then the equation nb² = a² becomes nb² = c²n², or b² = nc². The same argument shows that b must be also a multiple of n. Thus a and b are both multiples of n, which contradicts the fact that they have no factors in common. This completes the proof if n has no square factor > 1.

If n has a square factor, we can write n = m²k, where k > 1 and k has no square factor > 1. Then √n = m√k; and if √n were rational, the number √k would also be rational, contradicting that was just proved.

I. Match the terms from the left column and the definitions from

the right column:

II. Translate into Russian.

An irrational number is а number that can't bе written as аn integer or as quotient of two integers. Thе irrational numbers are infinite, non-repeating decimals. There're two types of irrational numbers. Algebraic irrational numbers are irrational numbers that аrе roots of polynomial equations with rational coefficients. Transcendental numbers аrе irrational numbers that are not roots of polynomial equations with rational coefficients; p and e are transcendental numbers.

III. Give the English equivalents of the following Russian words and word combinations:

отношения целых, множитель, абсолютный квадрат, аксиома порядка, разложение на множители, уравнение, частное, рациональное число, элементарные свойства, определенное рациональное число, квадратный, противоречие, доказательство, среднее (значение).

IV. Translate the following sentences into English and answer to

The questions in pairs .

1. Какие числа называются рациональными?

2. Какие аксиомы используются для множества рациональных чисел?

3. Сколько рациональных чисел может находиться между двумя

любыми рациональными числами?

4. Действительные числа, не являющиеся рациональными, относятся к категории иррациональных чисел, не так ли?

V. Translate the text from Russian into English.

Обычно нелегко доказать, что определенное число является иррациональным. Не существует, например, простого доказательства ирра­циональности числа e^π . Однако, нетрудно установить иррациональность

определенных чисел, таких как √2 , и, фактически, можно легко доказать

следующую теорему: если п является положительным целым числом, которое не относится к абсолютным квадратам, то √ n является иррациональным.

TEXT III. GEOMETRIC REPRESENTATION OF REAL NUNBERS AND COMPLEX NUMBERS

The real numbers are often represented geometrically as points on a line (called the real line or the real axis). A point is selected to represent 0 and another to represent 1, as shown on figure 1. This choice determines the scale. Under an appropriate set of axioms for Euclidean geometry, each point on the real line corresponds to one and only one real number and, conversely, each real number is represented by one and only one point on the line. It is customary to refer to the point x rather than the point representing the real number x.

Figure 1

The order relation has a simple geometric interpretation. If x < y, the point x lies to the left of the point y, as shown in Figure 1. Positive numbers lies to the right of 0 and negative numbers lies to the left of 0. If a < b, a point x satisfies a< x < b if and only if x is between a and b.

Just as real numbers are represented geometrically by points on a line, so complex numbers are represented by points in a plane. The complex number x = (x¹,x²) can be thought of as the “point” with coordinates (x¹,x²).

This idea of expressing complex numbers geometrically as points in a plane was formulated by Gauss in his dissertation in 1799 and, independently, by Argand in 1806. Gauss later coined the somewhat unfortunate phrase “complex number”. Other geometric interpretations of complex numbers are possible. Riemann found the sphere particularly convenient for this purpose. Points of the sphere are projected from the North Pole onto the tangent plane at the South Pole and, thus there corresponds to each point of the plane a definite point of the sphere. With the exception of the North Pole itself, each point of the sphere corresponds to exactly one point of the plane. The correspondence is called a stereographic projection.

I. Match the terms from the left column and the definitions from th e right column: