What is the Bunyakovsky inequality? When do students in Vietnam study the Bunyakovsky inequality?

What is the Bunyakovsky inequality?

- The Bunyakovsky inequality is an important inequality in mathematics, particularly useful for solving problems related to vectors, real numbers, and proving other inequalities. It is named after the Russian mathematician Viktor Bunyakovsky.

*Bunyakovsky Theorem

For two sequences of real numbers ((a_1, a_2, ..., a_n)) and ((b_1, b_2, ..., b_n)), we have the following inequality:

((a₁b₁ + a₂b₂ + ... + a_nb_n)² ≤ (a₁² + a₂² + ... + a_n²) (b₁² + b₂² + ... + b_n²))

Equality occurs if and only if: there exists a real number (k) such that (a_i = k b_i) for all (i) from 1 to (n).

Vector form:

The Bunyakovsky inequality can also be written in vector form:

For two vectors (a = (a_1, a_2, ..., a_n)) and (b = (b_1, b_2, ..., b_n)), we have:

(|\vec{a} \cdot \vec{b}| ≤ ||\vec{a}|| ||\vec{b}||)

*Where:

(|a \cdot b|) is the absolute value of the dot product of two vectors

(|a|) and (|b|) are the magnitudes of vectors (a) and (b) respectively

**Example illustrating the Bunyakovsky inequality

Problem: Prove that for all real numbers (a, b, c), we have the inequality:

((a + b + c)² ≤ 3(a² + b² + c²))

Solution:

Applying the Bunyakovsky inequality to the two sets of numbers ((a, b, c)) and ((1, 1, 1)), we get:

((a \cdot 1 + b \cdot 1 + c \cdot 1)² ≤ (a² + b² + c²)(1² + 1² + 1²))

Or:

((a + b + c)² ≤ 3(a² + b² + c²))

Equality occurs if and only if: (a = b = c).

*Note: The information is intended for reference./.

What is the Bunyakovsky inequality? When do students in Vietnam study the Bunyakovsky inequality?​ (Image from the Internet)

When do students in Vietnam study the Bunyakovsky inequality?​

Based on Section 5 of the Mathematics Curriculum issued with Circular 32/2018/TT-BGDDT, the content to be achieved in grade 9 mathematics includes:

*First-degree inequality in one variable

- Inequalities. First-degree inequality in one variable:

+ Recognize the order on the set of real numbers.

+ Recognize inequalities and describe some basic properties of inequalities (transitive property; relationship between order and operations of addition, multiplication).

+ Recognize the concept of a first-degree inequality in one variable, solutions of a first-degree inequality in one variable.

+ Solve the first-degree inequality in one variable.

Therefore, the inequality is taught in grade 9 mathematics.

What are 4 perspectives for constructing the grade 9 mathematics curriculum in Vietnam?

According to Section 2 of the Mathematics Curriculum issued with Circular 32/2018/TT-BGDDT, the 4 perspectives for constructing the grade 9 mathematics curriculum in Vietnam are as follows:

- The mathematics curriculum systematically applies the basic principles in the General Curriculum; inherits and promotes the advantages of the current curriculum and previous programs, selectively incorporates experiences from curriculum development in advanced countries, and approaches the achievements of educational science, considering the economic and social conditions in Vietnam.

Simultaneously, the mathematics curriculum emphasizes several perspectives as follows:

[1] Ensure simplicity, practicality, and modernity

The mathematics curriculum ensures simplicity, practicality, and modernity by reflecting the necessary contents to be addressed in public schools, meeting the need to understand the world as well as the interests and preferences of learners, aligning with the global approach today.

The program systematically maintains the spirit of "mathematics for everyone", where everyone can learn mathematics but can approach it in a way that suits their interests and individual abilities.

The mathematics curriculum places emphasis on applications and connections with practical reality or other educational subjects and activities, especially those intended to implement STEM education, tied with modern developments in economics, sciences, socio-life, and urgent global issues (such as climate change, sustainable development, financial education,...).

This is also manifest in practical activities and experience in mathematics education through various forms such as implementing mathematical study topics and projects, particularly those concerning the application of mathematics in practice; organizing math games, mathematics clubs, forums, seminars, competitions, etc., providing opportunities for students to creatively apply their knowledge, skills, and experience to real-life situations.

[2] Ensure consistency, coherence, and continuous development

The mathematics curriculum ensures consistency, coherence, and continuous development (from grade 1 to grade 12), consisting of two closely linked branches, one describing the development of core content knowledge threads and another describing the development of student competencies and qualities.

Simultaneously, the mathematics curriculum ensures continuity with the early childhood education program, laying the groundwork for vocational education and higher education.

[3] Ensure integration and differentiation

The mathematics curriculum achieves internal integration around three knowledge threads: Numbers, Algebra and Some Elements of Analysis; Geometry and Measurement; Statistics and Probability; achieving interdisciplinary integration through related contents, themes, or mathematical knowledge explored and utilized in other subjects such as Physics, Chemistry, Biology, Geography, Informatics, Technology, History, Arts; achieving both internal and interdisciplinary integration through practical activities and experiences in mathematics education.

Simultaneously, the mathematics curriculum ensures the requirement of differentiation. For all educational levels, mathematics education adheres to the spirit of individualizing learning on the basis that the majority of students (across all regions of the country) meet the required program objectives; at the same time, attention is paid to specialized subjects (gifted students, students with disabilities, students in difficult circumstances,…).

For high school level, the mathematics curriculum contains a system of in-depth learning modules and learning content that helps students enhance knowledge, practical skills, and apply resolutions to problems linked with reality.

[4] Ensure openness

The mathematics curriculum ensures a unified orientation and essential core mathematical education content mandatory for students nationwide, while granting autonomy and responsibility to localities and schools in selecting, supplementing some mathematical educational content and implementing an educational plan suitable to the local context and conditions of educational institutions.

The mathematics curriculum only prescribes general principles, directions for required student competencies and qualities, content education, educational methods, and assessment of educational results, without overly detailed rules, to create conditions for textbook authors and teachers to productively, creatively implement the program.

The program ensures stability and the potential for development during its implementation to match scientific-technological progress and real-world demands.