binomial theorem

简明释义

二项式定理

英英释义

例句

1.In my algebra class, we learned how to apply the binomial theorem 二项式定理 to expand expressions like (x + y)^n.

在我的代数课上，我们学习了如何应用二项式定理 binomial theorem 来展开像 (x + y)^n 这样的表达式。

2.The binomial theorem 二项式定理 allows us to express (a + b)^n as a sum of terms involving binomial coefficients.

通过二项式定理 binomial theorem，我们可以将 (a + b)^n 表达为包含二项系数的项的和。

3.Using the binomial theorem 二项式定理, we can find coefficients in the expansion of binomials.

通过使用二项式定理 binomial theorem，我们可以找到二项式展开中的系数。

4.The binomial theorem 二项式定理 is fundamental in combinatorics and is used to calculate probabilities.

在组合数学中，二项式定理 binomial theorem 是基础，并用于计算概率。

5.In probability theory, the binomial theorem 二项式定理 helps in solving problems related to binomial distributions.

在概率论中，二项式定理 binomial theorem 有助于解决与二项分布相关的问题。

作文

The binomial theorem is a fundamental principle in algebra that describes the algebraic expansion of powers of a binomial. A binomial is a polynomial with two terms, typically expressed in the form (a + b). The binomial theorem provides a formula for expanding this expression raised to a positive integer exponent. The theorem states that: (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where n is a non-negative integer, and k ranges from 0 to n. Here, 'n choose k' or C(n, k) represents the binomial coefficient, which is calculated as n! / (k!(n-k)!). This coefficient tells us how many ways we can choose k elements from a set of n elements, and it plays a crucial role in the expansion of the binomial expression.Understanding the binomial theorem is essential not only for students studying mathematics but also for professionals in various fields such as physics, engineering, and economics. The applications of the binomial theorem are vast and varied. For example, in probability theory, it helps in calculating probabilities in binomial distributions, which model scenarios with two possible outcomes, such as success or failure. Furthermore, the binomial theorem simplifies computations in combinatorics and provides insights into polynomial identities.In practical applications, the binomial theorem can be used to approximate complex functions. For instance, when dealing with exponential growth or decay, the theorem allows mathematicians and scientists to express these functions in a more manageable form. This approximation is particularly useful in calculus, where understanding the behavior of functions is crucial.Moreover, the binomial theorem has historical significance. It was first discovered by ancient mathematicians, including the Indian mathematician Bhaskara II and later expanded upon by European mathematicians like Isaac Newton. Newton's work on the binomial theorem laid the foundation for modern algebra and calculus, influencing countless mathematicians and scientists who followed.In conclusion, the binomial theorem is not just a mathematical formula; it is a powerful tool that enables deeper understanding and exploration of various mathematical concepts. Its relevance spans across multiple disciplines, making it a vital topic for anyone interested in mathematics or its applications. By mastering the binomial theorem, one gains essential skills that can be applied in both academic and real-world situations, highlighting the importance of this theorem in our understanding of mathematics and its practical uses.

二项式定理是代数中的一个基本原理，描述了二项式的幂的代数展开。二项式是一个包含两个项的多项式，通常以(a + b)的形式表示。二项式定理提供了一个公式，用于扩展这个表达式升至正整数指数。该定理指出： (a + b)^n = Σ (n 选 k) * a^(n-k) * b^k，其中n是非负整数，k的范围从0到n。在这里，“n 选 k”或C(n, k)代表二项式系数，其计算为n! / (k!(n-k)!)。这个系数告诉我们可以从n个元素中选择k个元素的方式有多少种，这在二项式表达式的展开中起着至关重要的作用。理解二项式定理不仅对学习数学的学生至关重要，而且对各个领域的专业人士也很重要，例如物理、工程和经济学。二项式定理的应用广泛且多样。例如，在概率论中，它帮助计算二项分布中的概率，该分布模拟具有两个可能结果（例如成功或失败）的场景。此外，二项式定理简化了组合数学中的计算，并提供了对多项式恒等式的深入理解。在实际应用中，二项式定理可以用来近似复杂函数。例如，当处理指数增长或衰减时，该定理允许数学家和科学家以更易管理的形式表达这些函数。这种近似在微积分中尤为有用，因为理解函数的行为至关重要。此外，二项式定理具有历史意义。它最早由古代数学家发现，包括印度数学家巴斯卡尔二世，后来被像艾萨克·牛顿这样的欧洲数学家进一步扩展。牛顿对二项式定理的研究奠定了现代代数和微积分的基础，影响了无数后来的数学家和科学家。总之，二项式定理不仅仅是一个数学公式；它是一个强大的工具，使人们能够更深入地理解和探索各种数学概念。它的相关性跨越多个学科，使其成为任何对数学或其应用感兴趣的人必须掌握的重要主题。通过掌握二项式定理，人们获得了可以应用于学术和现实世界情况的重要技能，突显了该定理在我们理解数学及其实际用途中的重要性。