Real Number
Real analysis is the branch of mathematics that studies the structure of the real numbers and concepts such as limits and continuity defined on them. WIIS provides materials for systematically learning the foundations of analysis, focusing on topics such as the definition of the real numbers, sequences, limits, continuous functions, series, and sequences of functions. Emphasis is placed on developing mathematical rigor through definitions, theorems, and proofs while carefully understanding concepts such as the completeness of the real numbers and convergence, thereby building a foundation for university-level mathematics, including calculus and topology.
Fields of Real Number
If the real numbers are defined as infinite decimals, every discussion concerning real numbers must ultimately be carried out in terms of infinite decimals, which is cumbersome. As an alternative, we define the real numbers using an axiomatic approach. Here, we explain the axioms that characterize the real numbers.
New Materials
We define the natural numbers using inductive sets and study the successor operation, addition and multiplication on the natural numbers, concrete methods of computation, closure properties of subtraction and division, and the order relation on the natural numbers.
In the axiomatic theory of the real numbers, two binary operations called addition and multiplication are defined on the real number system, and it is postulated that they satisfy the properties of a field. Any property concerning these operations is regarded as valid only after it has been derived from these axioms.
The binary operation called division is defined indirectly in terms of multiplication. Likewise, properties of division are regarded as valid only after they have been proved from the axioms governing the properties of multiplication.
In the axiomatic theory of the real numbers, a binary operation called multiplication is defined on the real number system, and it is postulated to satisfy the properties of a commutative group (abelian group). Any property concerning multiplication is regarded as valid only after it has been derived from these axioms.
The binary operation called subtraction is defined indirectly in terms of addition. Likewise, properties of subtraction are regarded as valid only after they have been proved from the axioms governing the properties of addition.
In the axiomatic theory of the real numbers, a binary operation called addition is defined on the real number system, and it is postulated to satisfy the properties of a commutative group (abelian group). Any property concerning addition is regarded as valid only after it has been derived from these axioms.
Learning Guide
Related Fields
Logic provides the foundation for mathematical proof and reasoning. Through propositional logic and predicate logic, WIIS systematically explains logical formulas, truth values, quantifiers, and the basic ideas behind proof. These materials provide a starting point for studying university-level mathematics and other theoretical disciplines.
Set theory is the common language of modern mathematics. Focusing on sets, functions, relations, cardinality, and order relations, this course systematically explains the fundamental concepts needed to understand a wide range of mathematical fields. It provides a foundation for further study in areas such as real analysis, linear algebra, and topology.