Take a line formed from the rational numbers: all rationals may be enumerated in a list based on their ‘height’ (defined as the numberator plus denominator) and thus arranged along a line. A geometric line consists of an infinite number of points arranged in a sequence: $p_1$ may come after $p_2$ or before, but not both after and before. This almost goes without saying, and is implicit in the definitions of rational and irrational numbers and lines. Lines are sequences of points, but the real numbers are non-enumerable Two related paradoxes regarding real numbers are described, which imply a number of interesting properties about dynamical systems. Each gives a closer approximation.Irrational numbers on the real line | Form and Formula Irrational numbers on the real line Home Page Decidability Irrational numbers on the real line The approximation 3.14 for π does not come close enough for the purpose, then 3.142, 3.1416, or 3.14159 can be used. However, one can always come as close to the exact real-number answer as one wishes. When one wants to use an irrational number such as π, √3, or e in a computation, one must replace it with a rational approximation such as 22/7, 1.73205, or 2.718. Pi is one.Įxcept for rare instances such as √2 ÷ √8, computations can be done only with rational numbers. The real numbers also include numbers which are "none of the above." These are the transcendental numbers, and they are uncountable. These include the natural numbers, the integers, the rational numbers, and the algebraic numbers (algebraic numbers are those which can be roots of polynomial equations with integral coefficients). The real numbers have many familiar subsets which are countable. In the case of the irrational numbers, however, there are so many of them that every conceivable listing of them will leave at least one of them out. The set of natural numbers is countable because the ordinary counting process will, if it is continued long enough, bring one to any particular number in the set. An infinite set of numbers is "countable" if there is some way of listing them that allows one to reach any particular one of them by reading far enough down the list. In fact, by using infinite decimals to represent the real numbers, the mathematician Cantor was able to show that the number of real numbers is uncountable. The square root of any prime number is irrational. Point P is not the only irrational point. Point P represents a real number because it is a definite point on the number line, but it does not represent any rational number a/b. It would fall inside a subdivision, not at an end. Even if there were a million, a billion, a billion and one, or any other number of uniform subdivisions, point P would be missed by every one of them. The Pythagoreans were able to show that no matter how finely each unit was subdivided (uniformly), point P would fall somewhere inside one of those subdivisions. To see what this meant, imagine a number line with an isosceles right triangle drawn upon it, as in Figure 1. The Pythagoreans were able to show, however, that the hypotenuse of an isosceles right triangle could not be measured exactly by any scale, no matter how fine, which would exactly measure the legs. Prior to this discovery, people believed that every number could be expressed as the ratio of two natural numbers ( negative numbers had not been discovered yet). It is thought that the first real number to be identified as irrational was discovered by the Pythagoreans in the sixth century B.
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