Data expansion:
The study of quantity begins with numbers, which are familiar natural numbers and integers as well as rational numbers and irrational numbers described in arithmetic. Specifically, for the need of counting, human beings abstract natural numbers from real things, which is the starting point of all "numbers" in mathematics.
Natural number does not close subtraction. For closed subtraction, we extend the number system to integers. In order not to close division, but to close division, we extend the number system to rational numbers; For open root operation, we extend the number system to algebraic number (in fact, algebraic number is a broader concept).
On the other hand, for the limit operation is not closed, we extend the number system to real numbers. Finally, in order to prevent negative numbers from operating to even powers in the real number range, we extend the number system to complex numbers. A complex number is the smallest algebraic closed field containing real numbers. We perform four operations on any complex number, and the simplification results are all complex numbers.
Another concept related to "quantity" is the "potential" of infinite sets, which leads to cardinality and another infinite concept: Alev number, which allows meaningful comparison between the sizes of infinite sets.
The evolution of mathematics can be regarded as the continuous development of abstraction or the extension of subject matter, and the eastern and western cultures have also taken different angles. European civilization developed geometry, and China developed arithmetic.
The first abstract concept is probably number (China's arithmetic), and its cognition that two apples and two oranges have something in common is a great breakthrough in human thought.
Besides knowing how to calculate the number of actual objects, prehistoric humans also knew how to calculate the number of abstract concepts, such as time: day, season and year. Arithmetic (addition, subtraction, multiplication and division) will naturally occur.
In addition, you need writing or other systems that can record numbers, such as Mu Fu or chips used by the Incas. There are many different counting systems in history.
In ancient times, the main principles in mathematics were the study of astronomy, the rational distribution of land and grain, taxation and trade. Mathematics is formed to understand the relationship between numbers, measure land and predict astronomical events. These needs can be simply summarized as the study of quantity, structure, space and time in mathematics.
After the Renaissance in Western Europe from ancient Greece to16th century, elementary mathematics, such as elementary algebra and elementary trigonometry, has been basically complete, but the concept of limit has not yet appeared.