What happened to the rationality

Get to know irrational numbers

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In these explanations you will learn what relationships exist between the sets of rational, irrational and real numbers.

The rational numbers

The set of rational numbers (ℚ) consists of all numbers that can be represented as the quotient of two whole numbers. Since all natural numbers can be represented as, natural and whole numbers are also rational numbers.
The numbers,,, ... are rational numbers.
A decimal number is a rational number if it ...
,, and are rational numbers.

The irrational numbers

Irrational numbers are numbers that cannot be represented as the quotient of whole numbers. Irrational numbers are decimal numbers with an infinite number of places after the decimal point that are not repeated periodically. These include, for example, the roots of natural numbers that are not square numbers. The circle number ... is also an irrational number - it is not a periodic decimal number. The irrational numbers expand the number range erweitert of the rational numbers to the number range ℝ of the real numbers.
is an irrational number.
Not all roots are irrational.
is not an irrational number.
is not an irrational number.

The real numbers

The set of real numbers ℝ consists of the rational numbers and the irrational numbers. The range of real numbers includes the other number ranges known to you: Every natural number is an integer. Every integer is a rational number. Every rational number is a real number.

Proof of irrationality

You cannot use the calculator to decide whether the result of a calculation is an irrational number, as it can only display a limited number of places after the decimal point. The result is rounded. The square root of a natural number is irrational if at least one of the prime factors occurs in its prime factorization in an odd number. In particular, the square root of a prime number is always irrational. The proof is usually given indirectly, here for example for 2.
So there is a contradiction in assuming that there is no shortening! The assumption that it would be rational is therefore wrong. Then it can only be irrational.