As a basic concept in mathematics, real numbers are the set of all points on the number axis, including rational numbers and irrational numbers.
A rational number is a number that can be expressed as the ratio of two integers. Specifically, rational numbers can be integers, finite decimals or infinite cyclic decimals. For example, 1, 3. 14 and -2/3 are rational numbers. Rational number is very useful in mathematics, especially in numerical calculation, which provides an accurate and reliable numerical representation method.
Irrational numbers cannot be expressed as the ratio of two integers, they are infinite cyclic decimals. Irrational number is also very important in mathematics, because it appears in many natural phenomena and practical problems, such as pi and the base e of natural logarithm. The existence of irrational numbers also solves some long-standing problems in mathematics, such as the continuum hypothesis.
The range of real numbers has important mathematical properties, such as completeness and order. Completeness means that all the properties in the real number set can find numbers that satisfy this property in the real number set; Orderliness means that the set of real numbers can be sorted according to the size relationship. These properties make real numbers widely used in mathematical analysis, physics, engineering and other fields.
In addition to rational and irrational numbers, real numbers also include some special numbers, such as infinity and infinitesimal. These special numbers also have important significance and application in mathematics. For example, infinity can be used to describe the behavior of some functions at infinity, and infinitesimal can be used to describe the limit behavior of derivatives or partial derivatives of functions at a certain point.
In a word, real number is a very basic and important concept in mathematics, which includes rational number and irrational number. It has important mathematical properties such as completeness and order, and is widely used in mathematical analysis, physics, engineering and other fields. It is very important for students studying mathematics and physics to understand and master the concept and properties of real numbers.