mathematic axiom

简明释义

数学公理

英英释义

例句

1.Understanding the mathematic axiom 数学公理 of addition is crucial for solving algebraic equations.

理解加法的数学公理对解决代数方程至关重要。

2.The mathematic axiom 数学公理 of equality states that if a equals b and b equals c, then a must equal c.

等式的数学公理指出，如果a等于b，且b等于c，那么a必须等于c。

3.The concept of parallel lines never meeting is based on the mathematic axiom 数学公理 of Euclidean geometry.

平行线永不相交的概念基于欧几里得几何的数学公理。

4.In geometry, the statement that through any two points there is exactly one line is a fundamental mathematic axiom 数学公理.

在几何学中，任何两个点之间恰好有一条直线的说法是一个基本的数学公理。

5.One important mathematic axiom 数学公理 in set theory is that for any set, there exists a set that contains only that set.

集合论中的一个重要数学公理是，对于任何集合，都存在一个仅包含该集合的集合。

作文

In the realm of mathematics, certain foundational principles serve as the bedrock upon which more complex theories are built. These fundamental truths are known as mathematic axioms (数学公理). An axiom is a statement that is accepted as true without proof, and it acts as a starting point for further reasoning and arguments. The significance of mathematic axioms (数学公理) lies in their ability to provide a consistent framework within which mathematical discourse can occur.One of the most famous examples of a mathematic axiom (数学公理) is Euclid's fifth postulate, also known as the parallel postulate. This axiom states that if a line segment intersects two straight lines and forms interior angles on the same side that sum to less than two right angles, then the two lines will eventually meet on that side. This seemingly simple statement has profound implications in geometry, leading to the development of different geometrical systems. In fact, the exploration of non-Euclidean geometries arose from questioning the validity of the mathematic axiom (数学公理) regarding parallel lines, showcasing how axioms can shape entire fields of study.The role of mathematic axioms (数学公理) extends beyond just geometry; they are integral in various branches of mathematics, including algebra, calculus, and set theory. For instance, in set theory, one of the commonly accepted axioms is the Axiom of Extensionality, which states that two sets are equal if they have the same elements. This axiom is crucial for understanding the concept of sets and their relationships.Moreover, the use of mathematic axioms (数学公理) allows mathematicians to establish proofs. By building upon these axioms, mathematicians can derive theorems and corollaries that expand our understanding of mathematical concepts. The process of proving a theorem often involves demonstrating that it logically follows from the established mathematic axioms (数学公理) and previously proven theorems. This hierarchical structure ensures that mathematics remains coherent and reliable.However, the acceptance of certain mathematic axioms (数学公理) can be subjective and may vary across different mathematical systems. For example, while Euclidean geometry relies on the parallel postulate, hyperbolic geometry does not accept this axiom, leading to different conclusions about the nature of space and lines. This variability highlights the philosophical aspect of mathematics, where the choice of axioms can influence the direction and scope of mathematical inquiry.In conclusion, mathematic axioms (数学公理) are essential components of mathematical thought. They provide the foundational truths that underpin the vast and intricate world of mathematics. Understanding these axioms not only enriches our comprehension of mathematical principles but also enhances our ability to engage with complex problems and theories. As we delve deeper into the world of mathematics, recognizing the importance of mathematic axioms (数学公理) will empower us to explore new ideas and foster innovation in this ever-evolving discipline.

在数学领域，某些基础原则构成了更复杂理论的基石。这些基本真理被称为数学公理（mathematic axioms）。公理是一个被接受为真的声明，无需证明，它作为进一步推理和论证的起点。数学公理（mathematic axioms）的重要性在于它们能够提供一个一致的框架，在这个框架内，数学讨论得以进行。最著名的数学公理（mathematic axioms）之一是欧几里得的第五公设，也称为平行公设。这个公理指出，如果一条线段与两条直线相交，并在同一侧形成的内角和小于两个直角，那么这两条线最终将在该侧相遇。这个看似简单的陈述在几何学中具有深远的影响，导致了不同几何系统的发展。事实上，非欧几里得几何的探索源于对关于平行线的数学公理（mathematic axioms）有效性的质疑，展示了公理如何塑造整个研究领域。数学公理（mathematic axioms）的作用不仅限于几何；它们在代数、微积分和集合论等多个数学分支中都是不可或缺的。例如，在集合论中，通常接受的一个公理是外延公理，该公理指出，如果两个集合具有相同的元素，则这两个集合是相等的。这个公理对于理解集合及其关系至关重要。此外，使用数学公理（mathematic axioms）使得数学家能够建立证明。通过建立在这些公理之上，数学家可以推导出扩展我们对数学概念理解的定理和推论。证明定理的过程通常涉及证明它在逻辑上是从既定的数学公理（mathematic axioms）和以前证明的定理中得出的。这个层级结构确保了数学保持连贯和可靠。然而，某些数学公理（mathematic axioms）的接受可能是主观的，并且可能因不同的数学体系而异。例如，虽然欧几里得几何依赖于平行公设，但双曲几何并不接受这一公理，从而导致对空间和直线性质的不同结论。这种变异性突显了数学的哲学方面，其中公理的选择可能影响数学探究的方向和范围。总之，数学公理（mathematic axioms）是数学思想的重要组成部分。它们提供了支撑广泛而复杂的数学世界的基础真理。理解这些公理不仅丰富了我们对数学原理的理解，还增强了我们处理复杂问题和理论的能力。当我们深入探索数学世界时，认识到数学公理（mathematic axioms）的重要性将使我们能够探索新思想并推动这一不断发展的学科的创新。