The Importance of 1. Real Numbers in Application
Real numbers are very important in mathematics and practical application. In the field of mathematics, real numbers are widely used in numerical calculation, algebraic operation, analytic geometry, mathematical analysis and so on. In physics, engineering, economics and other practical applications, real numbers are used to describe and quantify various metrics, data and variables in the real world. The range and properties of real numbers provide a basis for us to establish mathematical models, conduct scientific research and solve practical problems.
2. The range of rational numbers
Rational number is a number that can be expressed as the ratio of two integers, including positive integer, negative integer, positive fraction and negative fraction. Rational numbers range from negative infinity to positive infinity, including 0. Rational numbers can be expressed in fractional form or decimal form, such as 1/2, -3/4, 0.25, etc. Rational numbers are closed in algebraic operations, that is, the sum, difference, product and quotient of two rational numbers are still rational numbers.
3. The range of irrational numbers
An irrational number is a number that cannot be expressed as the ratio of two integers, and its decimal part will not be repeated, and it is infinitely acyclic. Common irrational numbers are π π, e (the base of natural logarithm), √ 2 (the square root of 2) and so on. Irrational numbers also range from negative infinity to positive infinity, but unlike rational numbers, there is no order of magnitude between irrational numbers. Irrational numbers can be expressed as approximations or infinite cyclic decimals.
4. Completeness of real numbers
The range of real numbers includes rational numbers and irrational numbers, which are isomorphic to a complete number field. Completeness means that every real number in the set of real numbers can be accurately represented, and for any real number, there are other real numbers that can be infinitely close to it. This property plays an important role in analytic geometry, calculus and other mathematical fields, which can ensure the accuracy and continuity of mathematical reasoning.
5. Properties of real numbers
Real numbers have many important properties. First of all, the set of real numbers is closed, that is, the sum, difference, product and quotient of two real numbers are still real numbers. Secondly, real numbers are transitive, symmetrical and reflexive. In addition, the real number set also satisfies Archimedes property, that is, there is an integer between any two positive real numbers. These properties make real numbers the basis of mathematical research and application.
6. Conclusion
Real numbers are the set of all numbers including rational numbers and irrational numbers. Rational numbers include positive integer, negative integer, positive fraction and negative fraction; Irrational numbers include π, e, radical number 2, etc. The closeness, completeness and many important properties of real number sets make them have extensive application value in mathematics and practical applications. Knowing the range and properties of real numbers is helpful for us to better understand and apply mathematical knowledge and make accurate numerical calculation and analysis in practical problems.