 1. ivan_sizov says:

We have infinitely many points here and there. To compare 2 infinities, we must somehow characterize them, mathematicians use the so-called power. Two sets have the same cardinality if a one-to-one correspondence can be established between their elements. Therefore, the sets of points on your line segments are equally powerful. Theory on the topic

2. nekto_v_palto says:

First, what you are asking is called the “cardinality” of a set. For sets with a finite �number of elements, the cardinality is equal to this number. For sets with an infinite number of elements, the case is distinguished when all elements can be “counted” (to establish a one-to-one correspondence with the elements of the set of natural numbers). In this case, mathematicians say that the cardinality of a set is “countable” or simply “a countable set”. Finally, there are sets with an infinite number of elements that cannot be “counted” (there are no one-to-one correspondences between the elements of this set and the elements of the set of natural numbers). In this case, the cardinality of the set is said to be a “continuum”. In some sense, the cardinality of the continuum is superior (is “greater than”) counting power.

Second, what numbers are you asking about? The set of integers in the interval [1;2] consists of two elements (power is equal to two) – one and two, and in the interval [1;3] – of three (one, two and three). Here the cardinalities of the sets are not equal to each other. �

The sets of rational numbers (equal to the ratio of two integers) on both intervals are infinite and countable, that is, their powers are equal. You can prove the equality of these powers by proving the fact that both are countable. Or you can simply choose a one-to-one correspondence between their elements. Such a correspondence will be, for example, Y=3×X/2. It matches any rational X in the interval [1;2] with one and only one Y in [1;3], which is also rational (since 3 and 2 are integers, and rational X is, in turn, the ratio of two integers). Moreover, no rational Y from [1; 3] will be omitted, since there is always a rational X=2×Y/3 from [1;2].

Sets of irrational numbers (cannot be represented as a periodic fraction; irrational, for example, is the number e=2,71828…) on both intervals are infinite and have continuum powers (not countable ones). In particular, we can prove the equality of these powers by using the same maps Y=3×X/2 and X=2×Y/3, where X and Y are �irrational.

Finally, the sets of real numbers (subsets of which are rational and irrational sets) on both intervals are also infinite and have equal cardinalities (continuum).

UPD. As for tags, it's not “algebra”, but “set theory”. “Society” has nothing to do with it either.