What is the real number of 0


It seems that every point on the number line corresponds to a rational number.
Already in classical Greece it was found - to the great amazement - that there are lines in geometry that cannot be measured with rational numbers. According to legends that have been told over and over again, the Pythagoreans are said to have perceived this knowledge as sacrilege, as it contradicted their conviction: 'Everything is (rational) number' and they are said to have been relieved that the messenger of this knowledge, Hippasus of Metapontus, was swallowed up by the sea has been.
It is comparatively easy to show that, for example, the diagonal in a square with side length 1, i.e. the segment $ \ sqrt {2} $, cannot be determined as a fraction of whole numbers and therefore not as a rational number.
Such numbers are called irrational numbers. The best-known irrational numbers are $ \ pi $ and Euler's number $ e $, but many roots of integers such as $ \ sqrt {3} $ or $ \ sqrt {6} $ also belong to this set. Will the rational numbers be the set of irrational numbers added, one obtains the amount of real numbers $ \ mathbb {R} $. Real numbers generally have a finite or infinitely periodic or infinitely non-periodic decimal representation.
The visual appearance of the number line does not change compared to that of the rational numbers. It should be noted, however, that between the infinitely many rational numbers there are still infinitely many irrational numbers.

Number line of real numbers
It can be shown that when the non-rational numbers are inserted into the number line, it is now full, i.e. that every real number corresponds to a point on the straight line and every point to a real number. In the economic context, the assumption is usually made that the variables considered, such as prices or quantities of goods, must not be negative.
In the event that only positive real numbers are considered, the following notation is valid: \ begin {eqnarray *} \ mathbb {R} ^ {+} &: = & \ mbox {{all positive real numbers}}. \ end {eqnarray *}
In the event that only positive real numbers including zero are considered, the following notation is used: \ begin {eqnarray *} \ mathbb {R} ^ {+} _ {0} &: = & \ mbox {{all non-negative real numbers}} \ &: = & \ mbox {{all positive real numbers and the zero}}. \ end {eqnarray *}