algebraic number

简明释义

代数数

英英释义

例句

1.The square root of 2 is an example of an algebraic number.

平方根2是一个代数数的例子。

2.An algebraic number is a solution to a polynomial equation with integer coefficients.

一个代数数是具有整数系数的多项式方程的解。

3.Complex numbers can be algebraic numbers if they satisfy a polynomial equation.

如果复数满足多项式方程，它们可以是代数数。

4.Every rational number is also an algebraic number.

每一个有理数也是一个代数数。

5.The set of algebraic numbers includes all integers.

代数数的集合包括所有整数。

作文

In the realm of mathematics, numbers are not just mere symbols; they hold profound meanings and relationships that extend far beyond basic arithmetic. Among the various categories of numbers, the concept of an algebraic number stands out as a fascinating subject of study. An algebraic number is defined as any complex number that is a root of a non-zero polynomial equation with rational coefficients. This definition may sound complex at first, but it opens the door to understanding a wide range of mathematical phenomena.To illustrate this concept, consider the simple equation x² - 2 = 0. The solutions to this equation are the numbers √2 and -√2, both of which are classified as algebraic numbers because they satisfy a polynomial equation with rational coefficients (in this case, the coefficients are 1, 0, and -2). Similarly, integers such as -3, 0, and 5 are also algebraic numbers, as they can be viewed as roots of polynomials like x + 3 = 0, x = 0, and x - 5 = 0, respectively.The importance of algebraic numbers extends into various fields of mathematics, including number theory and algebra. For instance, the set of all algebraic numbers forms a field known as the field of algebraic numbers. This field is significant because it allows mathematicians to explore properties of numbers that arise from polynomial equations, leading to deeper insights into the structure of numbers themselves.Moreover, algebraic numbers can be categorized further into different classes. For example, rational numbers, which can be expressed as the quotient of two integers, are a subset of algebraic numbers. On the other hand, irrational algebraic numbers, such as √2 or √3, cannot be expressed as a simple fraction but still satisfy polynomial equations with rational coefficients. This distinction highlights the rich diversity within the category of algebraic numbers.Another intriguing aspect of algebraic numbers is their relationship with transcendental numbers. While every rational number is an algebraic number, not every real or complex number is algebraic. Transcendental numbers, such as π and e, cannot be the roots of any polynomial with rational coefficients, setting them apart from algebraic numbers. This contrast emphasizes the unique properties of algebraic numbers and their role in the broader landscape of mathematics.In conclusion, the study of algebraic numbers offers a glimpse into the intricate world of mathematics, where numbers are categorized based on their relationships with polynomial equations. Understanding algebraic numbers not only enriches our knowledge of mathematics but also enhances our appreciation for the beauty and complexity inherent in numerical relationships. As we delve deeper into the subject, we uncover the layers of meaning and significance that algebraic numbers embody, ultimately revealing the elegance of mathematical thought.

在数学领域，数字不仅仅是简单的符号；它们蕴含着深刻的意义和关系，远远超出了基本的算术。在各种数字类别中，代数数的概念作为一个引人入胜的研究主题脱颖而出。代数数被定义为任何复杂数字，它是具有有理系数的非零多项式方程的根。这个定义乍一听可能显得复杂，但它为理解广泛的数学现象打开了大门。为了说明这一概念，考虑简单的方程 x² - 2 = 0。该方程的解是数字 √2 和 -√2，这两个数字被归类为代数数，因为它们满足具有有理系数的多项式方程（在这种情况下，系数为 1、0 和 -2）。同样，整数如 -3、0 和 5 也是代数数，因为它们可以视为多项式的根，例如 x + 3 = 0、x = 0 和 x - 5 = 0。代数数的重要性延伸到数学的各个领域，包括数论和代数。例如，所有代数数的集合形成一个称为代数数域的域。这个域之所以重要，是因为它允许数学家探索源自多项式方程的数字性质，从而深入了解数字本身的结构。此外，代数数还可以进一步分类为不同的类别。例如，可以表示为两个整数商的有理数是代数数的一个子集。另一方面，像 √2 或 √3 这样的无理代数数，无法用简单的分数表示，但仍然满足具有有理系数的多项式方程。这种区别突显了代数数类别内的丰富多样性。代数数的另一个引人入胜的方面是它们与超越数的关系。虽然每个有理数都是代数数，但并不是每个实数或复数都是代数数。超越数，如 π 和 e，不能是任何具有有理系数的多项式的根，这使它们与代数数区分开来。这种对比强调了代数数的独特性质及其在更广泛的数学领域中的作用。总之，代数数的研究提供了对数学复杂世界的洞察，在这个世界里，数字根据它们与多项式方程的关系进行分类。理解代数数不仅丰富了我们对数学的知识，还增强了我们对数字关系中固有的美丽和复杂性的欣赏。当我们深入探讨这一主题时，我们揭示了代数数所体现的意义和重要性的层次，最终展现了数学思想的优雅。