commutatively

简明释义

英[kəˈmjuːtətɪvli]美[kəˈmjuːtəˌtively]

交换地

可互换地

具有可换性地

英英释义

单词用法

同义词

反义词

例句

1.This paper realizes object location recognition commutatively. The paper realizes road and some area objects recognition semi-automatically or automatically.

实现了军事目标的交互式识别以及道路和部分面状目标的半自动或自动识别。

2.This paper realizes object location recognition commutatively. The paper realizes road and some area objects recognition semi-automatically or automatically.

实现了军事目标的交互式识别以及道路和部分面状目标的半自动或自动识别。

3.In mathematics, addition is often described as being done commutatively 交换地, meaning that a + b = b + a.

在数学中，加法通常被描述为 交换地，这意味着 a + b = b + a。

4.When working with vectors, the dot product can be calculated commutatively 交换地 as A · B = B · A.

在处理向量时，点积可以 交换地 计算，A · B = B · A。

5.The operations in this algebraic structure are defined to be commutatively 交换地, allowing for flexibility in calculations.

在这个代数结构中，运算被定义为 交换地，这使得计算灵活。

6.In the context of set theory, union of sets is performed commutatively 交换地; A ∪ B = B ∪ A.

在集合论的背景下，集合的并集是 交换地 进行的；A ∪ B = B ∪ A。

7.The rules of arithmetic state that multiplication is commutatively 交换地 applied, so 3 × 4 = 4 × 3.

算术规则说明乘法是 交换地 应用的，因此 3 × 4 = 4 × 3。

作文

In mathematics, the concept of operations being performed commutatively is fundamental to understanding how different mathematical processes interact. To say that an operation is commutative means that the order in which two elements are combined does not affect the final result. For example, when we add two numbers, say 3 and 5, it does not matter whether we calculate 3 + 5 or 5 + 3; both yield the same result of 8. This property can be observed in various arithmetic operations, particularly addition and multiplication, which are both commutative. However, it is important to note that not all operations share this property. Subtraction and division, for instance, are not commutative because changing the order of the numbers will generally produce different outcomes. Understanding the commutative nature of these operations can greatly simplify calculations and help in solving more complex mathematical problems. Moreover, the commutative property has implications beyond basic arithmetic. In algebra, when dealing with variables, the commutative property allows us to rearrange terms freely. For example, in the expression a + b, we can switch the order to b + a without changing the value of the expression. This flexibility is incredibly useful when simplifying equations or combining like terms. Similarly, in multiplication, the expression ab can be rearranged to ba, again without affecting the outcome. The significance of commutative operations extends into other areas of mathematics, such as set theory and matrix algebra. In set theory, the union of sets is commutative; that is, A ∪ B is the same as B ∪ A. This property helps in understanding relationships between different sets and in performing operations involving multiple sets efficiently. In matrix algebra, however, we find that while addition is commutative, multiplication is not. The product of two matrices A and B is not necessarily equal to the product of B and A, which highlights the unique characteristics of different mathematical structures. In real-world applications, the commutative property simplifies tasks in various fields, including computer science, physics, and economics. For instance, when programming algorithms, understanding which operations are commutative can lead to optimizations in code execution. In physics, the laws of motion often rely on commutative properties to predict outcomes based on different initial conditions. Economists also utilize commutative principles when analyzing supply and demand equations, allowing them to interchange variables without altering the overall economic model. In conclusion, the concept of operations being performed commutatively is a cornerstone of mathematics that facilitates a deeper understanding of how numbers and variables interact. Recognizing which operations are commutative can simplify calculations, enhance problem-solving skills, and provide insights across numerous disciplines. As we continue to explore the vast world of mathematics, the commutative property remains a vital tool, aiding us in our quest to make sense of numerical relationships and their applications in everyday life.

在数学中，运算以交换律进行的概念是理解不同数学过程如何相互作用的基础。说一个运算是交换的意味着两个元素结合的顺序不会影响最终结果。例如，当我们加两个数字，比如3和5时，无论我们计算3 + 5还是5 + 3，结果都是8。这一特性可以在各种算术运算中观察到，特别是加法和乘法，它们都是交换的。然而，重要的是要注意，并非所有运算都具有这一特性。减法和除法，例如，并不是交换的，因为改变数字的顺序通常会产生不同的结果。理解这些运算的交换性质可以大大简化计算并帮助解决更复杂的数学问题。 此外，交换性质的意义超越了基本算术。在代数中，当处理变量时，交换性质使我们可以自由地重新排列项。例如，在表达式a + b中，我们可以将顺序切换为b + a，而不会改变表达式的值。这种灵活性在简化方程或合并同类项时极为有用。同样，在乘法中，表达式ab可以重新排列为ba，结果也不受影响。 交换运算的重要性延伸到数学的其他领域，如集合论和矩阵代数。在集合论中，集合的并集是交换的；即A ∪ B与B ∪ A是相同的。这一特性有助于理解不同集合之间的关系，并有效地执行涉及多个集合的运算。然而，在矩阵代数中，我们发现虽然加法是交换的，但乘法却不是。两个矩阵A和B的乘积不一定等于B和A的乘积，这突显了不同数学结构的独特特性。 在现实应用中，交换性质简化了计算机科学、物理学和经济学等各个领域的任务。例如，在编写算法时，理解哪些运算是交换的可以导致代码执行的优化。在物理学中，运动定律通常依赖于交换特性来预测基于不同初始条件的结果。经济学家在分析供需方程时也利用交换原则，使他们能够在不改变整体经济模型的情况下互换变量。 总之，运算以交换律进行的概念是数学的基石，它促进了对数字和变量如何相互作用的更深刻理解。认识到哪些运算是交换的可以简化计算、增强解决问题的能力，并为多个学科提供见解。随着我们继续探索广阔的数学世界，交换性质仍然是一个重要工具，帮助我们理解数值关系及其在日常生活中的应用。