Mathematics

Subject	Mathematics
Semester	1st semester (autumn)
Type	Obvezni
ECTS	9 ECTS
Study programme:	Business Sciences (UNI)
Primary language:	Slovene

Prerequisites
Successful participation in the course requires an active readiness to learn, responsibility for one's own study process and a willingness to work regularly. Prior knowledge of secondary-school mathematics (basic algebraic and analytical content) is desirable, but the course also includes systematic consolidation of key concepts. The key condition for success is not perfect prior knowledge but continuity of work, understanding and active participation. Students can successfully complete the course in two ways: through regular participation in the organised learning process (lectures, ongoing work, seminar paper), or through greater independent engagement, in which they take responsibility for planning their learning and fulfilling all obligations on time. Attendance at the introductory lecture is compulsory; it presents in detail the objectives of the course, the methods of work, the assessment criteria and the possibilities of adapted forms of participation (e.g. in the event of special circumstances). The introductory lecture is the basis for establishing effective cooperation between the student and the course lecturer. In the event of absence from the introductory lecture, the student is obliged to contact the lecturer independently and in good time and to become acquainted with all obligations and the method of work. Active management of one's own study process is an integral part of the competences developed by the course. The use of modern learning tools (e.g. digital tools, software, artificial intelligence) is encouraged, but the student must ensure independent understanding and critical use of the acquired knowledge.

Content (Syllabus outline) 1. Mathematics as a language for describing reality
Sets, relations, mappings. Mathematical formalisation of data and structures.
2. Mathematics as a language for formalising relationships
Logic, numbers, functions. Foundations of mathematical reasoning and modelling of dependencies.
3. Mathematics as a language for formalising processes
Vectors and matrices as carriers of data and transformations (introduction to data structures and linear models).
4. Linear algebra and systems of equations
Matrix calculus, Euclidean spaces, linear dependence, rank of a matrix. Systems of linear equations and the Gaussian algorithm. (link: input-output models, simple economic systems)
5. Optimisation and linear programming
Problem formulation, graphical solving, interpretation of solutions. (link: allocation of resources, costs, profit, decision-making)
6. Sequences, series and financial mathematics
Sequences, limit of a sequence, series. Compound interest. (link: time value of money, investments)
7. Functions and continuity
Properties of functions, continuity, overview of elementary functions. (link: economic functions - costs, revenues, demand)
8. The derivative and the differential in the analysis of change
The derivative as a measure of change. Extrema and optimisation of functions. Analysis of a function. (link: marginal quantities, optimisation of decisions)
9. The integral and aggregation
Indefinite and definite integrals and the relationship between them. Areas, mean values, the improper integral. (link: accumulation, aggregates, averages in economics)
10. Foundations of data thinking and algorithms:
Interpretation of mathematical models in the context of artificial intelligence and decision-making
11. Mathematics in practice: models, data and presentations
Presentation of seminar papers. Use of mathematical models, digital tools and interpretation of results in a business context.
Objectives and competences
1. To consolidate and upgrade secondary-school mathematics in terms of understanding and applying mathematical concepts in modelling business processes, financial instruments and other phenomena in economics. 2. To develop logical, analytical and quantitative thinking and the ability to reason in a structured way in abstract mathematical space and in solving real problems. 3. To connect mathematical knowledge with the use of modern digital tools and environments (e.g. Excel, R, WolframAlpha, open-source analytical environments, artificial intelligence tools - LLMs) and to develop the ability to use them sensibly, critically and effectively in mathematical modelling. 4. To develop the ability to translate simpler professional and business problems into mathematical language (algebraic, analytical, numerical models), to solve them and to interpret the results professionally. 5. To encourage independent research, problem solving and continuous learning through seminar work and independent exercises (seminars), which are linked in content to the lectures and enable an in-depth understanding of the concepts covered. 6. To produce a seminar paper that comprehensively demonstrates an understanding of mathematical concepts, the ability to apply them in an economic context and the use of appropriate analytical tools.

Intended learning outcomes
Upon successful completion of the course, the student develops the following competences: a. Mathematical and analytical competences Understanding of fundamental algebraic and analytical concepts (vectors, matrices, functions, limits, derivatives, integrals) and their interconnections. The ability to think abstractly, to formalise problems and to apply mathematical methods to solving quantitative tasks. Understanding of mathematical models as a tool for describing economic and business phenomena. b. Modelling and problem solving The ability to translate real problems into mathematical models (e.g. optimisation, linear programming, analysis of functions). The ability to select appropriate mathematical methods and to interpret the results in a business context. Development of a systematic and structured approach to problem solving. c. Digital and technological competences Use of spreadsheet tools (e.g. Excel or similar) for organising data and basic modelling. Use of statistical and analytical environments (e.g. R or similar) for basic simulations, optimisation tasks and data analysis. Use of tools for symbolic and numerical computation (e.g. WolframAlpha and related environments). Sensible and critical use of artificial intelligence tools (LLMs) in learning, problem analysis and interpretation of results. d. Information and research literacy The ability to find, select and critically use online, digital and library resources to deepen mathematical and applied knowledge. Understanding of the limitations of the methods and tools used and assessment of the reliability of results. e. Communication and applied competences Clear presentation of mathematical procedures and results in written and oral form (seminar papers, presentations). The ability to link mathematical results to business interpretations and decisions. Development of independence, responsibility and reflection in learning and problem solving.

Learning and teaching methods

Forms of work

• A combination of frontal teaching and interactive work involving students
• Student seminars and presentations of mathematical models
• Guided and independent exercises linked in content to the lectures
• Independent and individual student work (problem solving, project work)
• Work in small groups (discussion, solving more complex tasks)
• Digitally supported learning (use of software tools, online resources and artificial intelligence)

Methods of work

• Explanation and conceptual interpretation of mathematical content
• Conversation, guided discussion and argumentation of solutions
• Case studies (linking mathematical concepts to economic and business problems)
• Problem-based learning and inquiry-based learning
• Solving tasks (from basic to applied and open problems)
• Project and seminar work (development of mathematical models)
• Use of digital tools (e.g. Excel, R, WolframAlpha, LLMs) in analysis and modelling
• Reading, analysing and writing professional texts using mathematical language
• Ongoing checking of understanding and reflection on one's own learning

Assessment

• Assessment is based on a combination of ongoing work, applied project work and a final examination.
• The final grade consists of three components:
• written exam: 50 % (50 points)
• seminar paper: 25 % (25 points)
• seminars (weekly independent exercises): 25 % (25 points)
• a. Written exam (50 %)
• The written exam tests the understanding of fundamental mathematical concepts and the ability to apply them to solving tasks.
• The exam usually comprises 5 tasks. To pass, the student must achieve at least 50 %.
• Exam dates are scheduled in accordance with the academic calendar (usually in the winter, summer and autumn examination periods).
• b. Seminar paper (25 %)
• The seminar paper is the student's independent applied work, in which the student:
• identifies a problem in the field of business sciences,
• formalises it appropriately in mathematical terms,
• uses suitable mathematical and/or digital tools,
• interprets the results in the substantive context.
• The assessment takes into account: understanding of the problem, adequacy of the model, correctness of the procedures, interpretation of the results and the quality of the presentation.
• c. Seminars - weekly independent exercises (25 %)
• Throughout the semester, students complete regular independent tasks (exercises) directly linked to the lectures.
• The purpose of the seminars is:
• ongoing consolidation of the material,
• development of problem-solving skills,
• preparation for the seminar paper and the written exam.
• The assessment takes into account the regularity of submissions, the correctness of solutions, understanding of the procedures and progress during the semester.

Readings

• Starček, Simon. 2026. Matematika. Ljubljana, Fakulteta za pravo in ekonomijo.
• Starček, Simon. 2025. Matematična logika. Nekoč, danes in naprej. Ljubljana, Fakulteta za pravo in ekonomijo.
• Žerovnik, Janez. 2020. Matematika 1. Ljubljana, Univerza v Ljubljani, Fakulteta za strojništvo.