VI. Ask questions to which the following sentences could be answers.

1. Fractions like are called proper fractions.

2. In the proper fraction the denominator is greater than the numerator.

3. In the improper fraction the denominator is less than the numerator.

4. A mixed fraction contains an integer and a proper fraction.

5. There also exist equivalent fractions.

6. The little boy was able to multiply and divide fractions.

7. If you write first = and then = it means that you have replaced 3 with n.

8. The words "choose" and "select" mean one and the same thing.

9. If you change a fraction from to you will reduce it to its lowest terms.

10. This text deals with various kinds of fractions.

11. The quantity of material has to be reduced.

Lesson 12

Irrational numbers

For all practical purposes of measuring, the rational numbers are entirely sufficient. Even from a theoretical viewpoint, since the set of rational points covers the line densely, it might seem that all points on the line are rational points. If this were true, then any segment length would be commensurable with the unit. But the situation is by no means so simple. There exist incommensurable segments, for example, the diagonal of the square and its side. In other words, if we assume that to every segment corresponds a number giving its length, then there exist numbers which cannot be rational. We call them irrational numbers. If we choose the side of the given square as the unit of length, then, its diagonal will be . The number is an irrational number.

We shall give a general definition of irrational numbers.

Let us consider any sequence I₁, I_{2, …,} I_n … of intervals on the number axis with rational endpoints. Let us suppose that every interval is contained in the preceding one and that the length of the n-th interval I_n tends to zero as n increases. Such a sequence is called a sequence of «nested intervals". Now we formulate as a basic postulate of geometry: corresponding to each such sequence of nested intervals there is precisely one point on the number axis which is contained in all of them. This point is called by definition a real number. If it is not a rational point it is called an irrational number.

The mathematically important point here is that for these irrational numbers, defined as sequences of nested intervals, the operations of addition, multiplication, etc. and the relations of "less than" and "greater than", are capable of immediate generalization from the field of rational numbers in such a way that all the laws which hold in the rational number field are preserved. For example, the addition of two irrational numbers and can be defined in terms of the two sequences of nested intervals defining and respectively. We construct a third sequence of nested intervals by adding the initial values and the end values of corresponding intervals of the two sequences. The new sequence of nested intervals defines + . Similarly, we may define the product , the difference - , and the quotient . On the basis of these definitions the arithmetical laws can be shown to hold for irrational numbers also.