1, closed, real number set is closed for the four operations of addition, subtraction, multiplication and division (divisor is not zero), that is, the sum, difference, product and quotient of any two real numbers (divisor is not zero) are still real numbers.
2. Orderliness, the set of real numbers is ordered, that is, any two real numbers A and B must satisfy and only satisfy one of the following three relationships, A.
3, transitivity, real number size is transitive, that is, if a > b and b >；; C has an Archimedes property, and real numbers have Archimedes property, that is, (inverted A)a, b∈R, if A >；; 0, and then a positive integer n, na> B.
The development of real numbers;
Around 500 BC, Greek mathematicians headed by Pythagoras realized that rational numbers could not meet the needs of geometry, but Pythagoras himself did not admit the existence of irrational numbers.
2. Real numbers were not widely accepted in Europe until17th century. 18th century, calculus was developed on the basis of real numbers. 187 1 year, German mathematician Cantor first put forward a strict definition of real numbers.
3. According to daily experience, the set of rational numbers seems to be "dense" on the number axis, so the ancients always thought that rational numbers could meet the actual needs of measurement.
4. Take a square with a side length of 1 cm as an example, how long its diagonal is, and under the specified accuracy (for example, the error is less than 0.00 1 cm), the accurate measurement result can always be expressed by rational numbers (for example, 1.4 14 cm).