Mathematics Has Always Been More Than a Subject to Me
Mathematics has never been merely a collection of formulas, rules and procedures to me.
Long before I became a professional electrical engineer, mathematics had already become a way of thinking. I was always fascinated by logical structures, numerical patterns, elegant solutions and the ability of mathematics to describe both abstract ideas and real-world phenomena with remarkable precision.
Electrical engineering strengthened this way of thinking even further. As an engineer, I learned that a good solution is rarely only about knowing a formula. It is about understanding a system, recognizing constraints, simplifying complexity, and creating something that works reliably in practice.
That same mindset naturally led me toward programming.
Programming gave me a way to transform mathematical ideas into practical tools. A formula written on paper can explain a concept, but software can make that concept interactive. A mathematical method can remain an abstract procedure, but a program can turn it into a useful application that helps people calculate, test, compare, explore, and learn.
Over time, this combination of mathematics, engineering, and programming led me to create two educational projects:
- NajduzaRec.rs, launched in August 2019
- Infinite Math World, launched in early December 2024
Although these two websites have different identities, they are deeply connected.
Infinite Math World is primarily a place where I write about mathematics, physics, electrical engineering and related scientific topics. It is a platform for detailed explanations, educational articles, mathematical discussions and deeper explorations of scientific ideas.
NajduzaRec.rs, on the other hand, is an interactive educational platform. It is built around web applications that allow users to work directly with mathematical, logical, linguistic and algorithmic problems.
In a sense, Infinite Math World explains ideas, while NajduzaRec.rs allows users to interact with them.
This article is the story of NajduzaRec.rs: how it began, how it evolved, what mathematical and algorithmic ideas stand behind it, and why I believe educational software can play an important role in popularizing mathematics, logic, programming, and problem solving.
From Engineering Thinking to Educational Software
Why Mathematics and Programming Naturally Belong Together
For me, mathematics and programming have always belonged together.
Mathematics gives us structure. Programming gives that structure an executable form. Mathematics helps us define a problem precisely. Programming allows us to test solutions, automate procedures, and make abstract reasoning available to users through software.
This connection is especially powerful in education.
When a student solves a mathematical problem manually, they often focus on the current example. But when the same concept is implemented in software, the student can test many examples, observe patterns, compare results, and develop a broader understanding.
That is one of the main reasons why I became interested in educational software.
A mathematical concept becomes far more accessible when users can interact with it.
Turning Abstract Ideas Into Practical Tools
Many mathematical ideas seem abstract when they are presented only through symbols.
However, when those ideas become part of an application, they become practical.
For example, a quadratic equation is not only an expression written in the form:
ax² + bx + c = 0
It is also a structure whose solutions depend on the discriminant. It is a model that appears in physics, geometry, engineering, and optimization. When implemented as an interactive solver, it allows users to observe how different coefficients change the nature of the roots.
The same is true for prime numbers, logarithms, systems of equations, complex numbers, divisibility rules, and combinatorial problems.
Software does not replace mathematical understanding.
But it can support it.
It can make mathematical ideas easier to test, easier to visualize, and easier to apply.
Why Educational Software Fascinated Me
Educational software fascinated me because it connects three things I value deeply:
- mathematical correctness
- engineering efficiency
- practical usefulness
A good educational application must be accurate. It must produce correct results and respect mathematical logic.
But accuracy alone is not enough.
The application must also be efficient, stable, and understandable. It must respond quickly. It must handle user input properly. It must present results clearly. It must support learning rather than confuse the user.
That combination of mathematical rigor and practical usability is exactly what made the development of NajduzaRec.rs so interesting to me.
The Story Before the Website
Programs That Existed Before NajduzaRec.rs
NajduzaRec.rs officially appeared in August 2019.
However, the website itself was not the beginning of the project.
For years before that, I had been creating various standalone programs related to mathematics, word search, numerical calculations, prime numbers, algebraic equations, and other computational problems.
Some of those programs later became part of NajduzaRec.rs.
Others still exist only as private programs and are not publicly available on the website.
This is an important part of the story because the website was not created from nothing in one month. The public web platform was created in approximately one month, but the ideas, algorithms, logic, and many software solutions behind it had existed much earlier.
In other words, NajduzaRec.rs was not a sudden project.
It was the result of years of earlier work.
Years of Experimentation Before August 2019
Before the website existed, I had already experimented with different kinds of programs.
Some were mathematical. Some were linguistic. Some were algorithmic. Some were designed for testing numerical properties, while others were created to solve more specific problems.
This earlier period was extremely important.
It allowed me to develop and test ideas before they became public web applications. It also allowed me to understand which tools could be useful to other people, not only to me.
That is how the idea of turning those programs into a website gradually became natural.
The question was no longer:
Can I create these programs?
The question became:
Can I make them available online as educational tools?
Building the Website While Working a Full-Time Job
When NajduzaRec.rs was created in August 2019, I was also working a regular full-time job.
That made the development process more demanding.
The website was built in approximately one month, but that month was not isolated from everyday professional responsibilities. It required discipline, organization, and enthusiasm.
This is one of the reasons why I consider NajduzaRec.rs more than a technical project.
It represents a serious personal investment in educational software.
It required not only knowledge of mathematics and programming, but also persistence.
Why the Website Was Created in About One Month
The reason the website could be created relatively quickly is that many of the core ideas already existed.
The logic of several applications had been developed earlier.
The algorithms had been tested.
The mathematical concepts were already familiar.
The challenge was to transform existing ideas and programs into a public web platform.
This required adapting them for online use, organizing the website, designing user input forms, presenting results clearly, and making the applications accessible to visitors.
A desktop program and a web application may solve the same problem, but they are not the same product.
A public web application must be simple enough for users to understand immediately, while still being accurate and reliable behind the scenes.
That transition from private programs to a public educational platform became the foundation of NajduzaRec.rs.
The Original Idea Behind NajduzaRec.rs
Inspiration From the Famous Longest Word Game
The name NajduzaRec.rs comes from the first and most recognizable application on the website: finding the longest possible word from a given set of letters.
The inspiration came from the first game from the Serbian television quiz Slagalica.
The idea is simple and familiar.
A set of letters is given, and the goal is to form the longest valid word.
For a human player, this is a challenge of vocabulary, memory, intuition, and speed.
For a computer program, however, the problem is completely different.
A program must search systematically.
It must check whether a candidate word can be formed from the given letters.
It must respect language rules.
It must deal with many possible words.
It must return the correct result quickly.
This is where a word game becomes an algorithmic problem.
Looking at a Word Game Through an Engineer’s Eyes
As an engineer and programmer, I did not see the “Longest Word” problem only as a game.
I saw it as a system.
The input consists of letters.
The database consists of valid words.
The algorithm must determine which words can be formed from the input letters.
The output must be the longest acceptable result.
If more than one result has the same maximum length, there must be a clear rule for choosing which one is displayed.
This kind of thinking is typical in engineering.
A seemingly simple task is analyzed as a system of rules, constraints, data, and outputs.
That is exactly how the first application of NajduzaRec.rs was developed.
The Challenge of Finding the Longest Possible Word
At first glance, one might assume that the program simply searches through a dictionary.
But the problem is more subtle than that.
A general dictionary is not necessarily suitable for this task. It may contain words of many different lengths, unsuitable forms, duplicates, or entries that do not follow the rules of the intended game.
For the application to work properly, the word database had to be specialized.
The program had to answer several questions:
Word Length
Which word lengths should be included?
For this application, both the Serbian and English databases contain exclusively words from 9 to 12 letters.
This is not an accidental detail.
It is a deliberate design decision based on the purpose of the application.
Letter Frequency
It is not enough to check whether a word contains only letters that appear in the input.
The program must also check frequency.
If a word contains a certain letter twice, then the input letters must also contain that letter at least twice.
For example, if a candidate word requires two occurrences of the same letter, one occurrence in the input is not enough.
This makes the problem more precise.
Duplicate Elimination
A high-quality database must not contain unnecessary duplicates.
Duplicate entries can distort results, slow processing, and reduce the quality of the application.
For that reason, custom programs were used to detect and remove duplicates.
Language-Specific Rules
The Serbian and English versions require different linguistic considerations.
The Serbian version is especially demanding because grammatical forms matter greatly.
Nouns and adjectives must be in the nominative form. Verbs must be in the infinitive. Gender and grammatical form must also be considered where relevant.
This makes the Serbian database much more than a simple list of collected words.
It is a carefully filtered and continuously improved linguistic resource.
Building Specialized Word Databases
More Than 64,000 Serbian Words
The Serbian database currently contains more than 64,000 words.
These words are not included randomly.
They are selected, filtered, organized, and continuously improved.
The goal is not only to have a large database.
The goal is to have a useful and reliable database.
For a word-finding application, quality is more important than raw quantity.
If the database contains invalid forms or unsuitable words, the result may be technically generated but educationally or linguistically wrong.
That is why the Serbian database requires continuous attention.
More Than 73,000 English Words
The English database contains more than 73,000 words.
Like the Serbian database, it contains exclusively words from 9 to 12 letters.
The English version has its own structure and sorting logic according to the English alphabet.
Although English grammar differs from Serbian grammar, the same basic principles apply:
- duplicates must be removed
- unsuitable entries must be filtered
- the database must be optimized for the application
- the words must be organized for fast search
The English version expands the reach of the project beyond Serbian-speaking users and makes the platform more internationally accessible.
Why Both Databases Contain Only Words From 9 to 12 Letters
This point is especially important.
The Serbian and English databases do not mostly contain words from 9 to 12 letters.
They contain exclusively words from 9 to 12 letters.
That distinction matters.
The purpose of the application is to find long words from a given set of letters. Therefore, shorter words are not relevant to the main goal of this specific tool.
Including unnecessary short words would make the database larger without improving the purpose of the application.
This is a good example of problem-specific optimization.
A specialized application should not necessarily use the largest possible dataset.
It should use the most appropriate dataset.
Why These Databases Are Not Ordinary Dictionaries
A dictionary is designed for general reference.
The NajduzaRec.rs word databases are designed for algorithmic search.
That difference changes everything.
An ordinary dictionary may be organized for human reading.
The NajduzaRec.rs databases are organized for fast computation.
An ordinary dictionary may contain many word forms.
The NajduzaRec.rs databases must follow the rules of the quiz.
An ordinary dictionary may include words of all lengths.
The NajduzaRec.rs databases focus exclusively on words from 9 to 12 letters.
The databases were built with a specific purpose: to allow the program to find valid long words quickly and consistently.
The Difference Between Database Size and Database Quality
It is easy to be impressed by large numbers.
More than 64,000 Serbian words and more than 73,000 English words are significant quantities.
But the real value of these databases is not only their size.
The real value is their structure.
A smaller, cleaner, better organized database is often more useful than a larger but inconsistent one.
For NajduzaRec.rs, database quality means:
- correct word length
- no duplicates
- appropriate grammatical form
- meaningful sorting
- compatibility with the search algorithm
- continuous improvement
That is why the database-building process remains one of the most demanding and important parts of the entire project.
Sorting Words for Speed and Accuracy
Longest Words First
The word databases are sorted in a very specific way.
First, words are arranged by decreasing length.
This means that words with 12 letters come first.
Then come words with 11 letters.
After them come words with 10 letters.
Finally, words with 9 letters appear.
This structure directly supports the purpose of the application.
Since the goal is to find the longest possible word, the program should begin with the longest candidates.
If a valid word is found, there is no need to search for shorter words.
The result is already optimal in terms of length.
Alphabetical Ordering Within Each Word-Length Group
Within each group of words having the same length, the words are sorted alphabetically.
This ordering is language-specific.
Serbian Alphabet Ordering
In the Serbian database, words of the same length are sorted according to the Serbian alphabet.
This ensures that if several valid Serbian words have the same maximum length, the displayed result is the first one according to Serbian alphabetical order.
English Alphabet Ordering
In the English database, words of the same length are sorted according to the English alphabet.
This means that the English version follows the same logic, but adapted to English alphabetical order.
A Simple Idea With a Huge Impact
This sorting principle may appear simple, but it has a major impact.
The database is structured in the same order in which the algorithm needs to search.
That is the key.
Instead of searching randomly or checking all possible candidates before deciding, the program starts from the most valuable candidates: the longest words.
The first valid result has special meaning because of the structure of the database.
How the Database Structure Enables Instant Results
Because the words are sorted from longest to shortest, the program can stop as soon as it finds the first valid match.
This is one of the central optimizations of the “Longest Word” application.
The program does not need to continue searching after it has found a valid word if that word belongs to the longest currently searched group and appears first according to the sorting rule.
Returning the Longest Possible Word
The first valid match is not arbitrary.
It is the longest valid word according to the database order.
This allows the application to display the result quickly.
Handling Multiple Words of the Same Length
If several valid words have the same maximum length, the sorting rule determines which one appears first.
For Serbian, the result is the first valid word according to the Serbian alphabet.
For English, the result is the first valid word according to the English alphabet.
This makes the application deterministic, consistent, and easy to understand.
Collecting, Filtering, and Verifying Words
Combining Multiple Sources
Both the Serbian and English databases were built using multiple sources.
However, collecting words is only the beginning.
Raw word lists are rarely ready for serious use in an educational application.
They must be cleaned.
They must be filtered.
They must be checked.
They must be organized.
This process requires both programming and linguistic attention.
Removing Duplicates With Custom Software
To improve the databases, I created the program that check for duplicate words and arrange entries properly.
Duplicate removal is essential because repeated words add no real value to the database.
They can only make processing less efficient and the database less clean.
Custom software helped automate part of this work.
But automation alone was not enough.
Why Automated Filtering Is Not Enough
A program can detect duplicates.
A program can sort words.
But not every linguistic issue can be solved automatically with complete confidence.
This is especially true for Serbian.
The Serbian language has rich grammatical structure, and the correct form of a word matters.
A word may exist, but it may not be suitable for the rules of the quiz.
That is why manual verification is necessary.
The Importance of Manual Verification
The Serbian database requires careful manual checking.
This work is demanding, but it is essential for quality.
Nouns in Nominative Form
Nouns should appear in the nominative form.
This includes both singular and plural forms where appropriate.
Other case forms are not suitable if they do not follow the intended rules of the application.
Adjectives in Nominative Form
Adjectives must also be checked carefully.
They should be included in appropriate nominative forms.
This is important because Serbian adjectives change according to gender, number, and case.
Verbs in Infinitive Form
Verbs should be represented in the infinitive.
This creates consistency and avoids unsuitable conjugated forms.
Gender Classification
For Serbian words, gender can also matter.
Masculine, feminine, and neuter forms may require attention depending on the word type.
This adds another layer of complexity to the verification process.
Continuous Database Improvement
The databases are not static.
They are continuously improved.
New words can be added.
Incorrect entries can be corrected.
Better filtering can be applied.
The database is therefore not only a technical component of the application.
It is an evolving linguistic and algorithmic resource.
Beyond the Public Website: Private Programs and Advanced Versions
Software That Is Not Publicly Available
An interesting part of the NajduzaRec.rs story is that not everything I have developed is publicly available on the website.
Some of my programs are more advanced than the public web versions.
They still exist as private standalone programs.
This is especially true for the “Longest Word” application.
I have private Serbian and English programs that go beyond the functionality currently available on the public website.
Generating All Possible Valid Words
The public website version is focused on quickly displaying the longest valid word.
The private programs can do more.
They can generate all possible valid words from the given set of twelve letters.
These words are then sorted from 12 letters down to 9 letters.
If several words have the same length, they are sorted alphabetically, according to the Serbian alphabet in the Serbian version and according to the English alphabet in the English version.
This is a more complete output.
Instead of only answering:
What is the longest word?
the private program answers:
What are all possible valid words from these letters, sorted by length and alphabetically?
Why These Versions Go Beyond the Website
The public website and the private programs have different purposes.
The website version is designed for speed, simplicity, and easy public use.
The private versions are designed for deeper analysis and completeness.
This distinction is important.
A public web application should usually be clean and simple.
A private analytical tool can be more detailed and more complex.
Both approaches are useful, but they serve different needs.
The Serbian Longest Word Program Demonstration
A presentation of the private Serbian program for finding possible words from given letters is available on YouTube:
Program for finding the longest word from given letters
This program demonstrates that the idea behind NajduzaRec.rs exists beyond the website itself.
The website is the public educational platform, but behind it there is a broader body of software development.
Different Tools for Different Purposes
This is an important software engineering lesson.
The best tool depends on the context.
A public application should be accessible.
A private program can be exhaustive.
A simple interface may be best for most users.
A detailed output may be best for analysis.
NajduzaRec.rs represents the public, educational, user-friendly side of a much broader development process.
Creating Meaningful Letter Challenges
Manual Letter Entry
The “Longest Word” application allows users to enter letters manually.
This gives users complete control.
They can test combinations of their choice.
They can reproduce examples.
They can experiment with specific letters.
Manual entry is important because it makes the application flexible.
Random Letter Generation
The application also provides random letter selection.
However, this random selection is not a naive process.
It is not based simply on choosing twelve random letters independently.
That approach would often produce uninteresting or nearly useless combinations.
Instead, the program uses a smarter method.
Why Pure Randomness Would Be a Bad Idea
If twelve letters were selected purely at random, many combinations would not contain a meaningful long word.
This could make the application frustrating.
A user might receive a set of letters with no good solution.
That would reduce the educational and entertainment value of the tool.
A good random challenge should be random, but not meaningless.
Selecting a Real Word First
The program first selects a word from the database according to a random criterion.
This guarantees that the letter set is based on a real word.
That means a valid solution exists.
Shuffling the Letters
After selecting a word, the program shuffles its letters.
This creates the actual challenge for the user.
The word is hidden inside the shuffled letters.
The user must discover it.
Completing All Twelve Positions
If the selected word contains fewer than twelve letters, the program adds random letters until all twelve positions are filled.
This keeps the format consistent.
The user always receives twelve letters, but at least one valid word is guaranteed to exist.
Ensuring That a Valid Solution Exists
This design creates a balance between randomness and structure.
The challenge feels random, but it is not empty.
It is generated intelligently.
The user receives a meaningful puzzle, and the application can provide a valid result.
This is another example of how careful programming improves user experience.
From One Word Game to an Educational Platform
Why the Project Expanded
Although NajduzaRec.rs started with the “Longest Word” idea, it gradually expanded into a broader educational platform.
This expansion was natural.
My interests were never limited to word games.
They included mathematics, programming, number theory, algebra, complex numbers, logarithms, roots, equations, and combinatorial thinking.
Once the website existed, it became possible to bring more of these ideas together in one place.
Educational Software as a Natural Evolution
The development of additional applications followed the same general philosophy:
- identify a mathematical or logical topic
- create a reliable algorithm or calculation method
- design a useful interface
- allow users to interact with the concept
This is how a single application gradually became a platform.
A Platform for Mathematics, Logic, Programming, and Language
Today, NajduzaRec.rs connects several fields:
- language
- mathematics
- programming
- logic
- algorithms
- education
This makes the website difficult to describe with only one label.
It is not only a word game website.
It is not only a mathematical calculator website.
It is an educational software platform.
Overview of the Thirteen Educational Applications
NajduzaRec.rs contains thirteen web applications, including tools related to:
- finding the longest word
- prime numbers
- systems of two linear equations
- systems of three linear equations
- quadratic equations
- cubic equations
- logarithms
- nth roots
- greatest common divisor and least common multiple
- divisibility of numbers
- multiplication and division of complex numbers
- square roots of complex numbers
- Lotto 7/39
Each application has its own purpose, but all of them belong to the same educational vision: helping users explore mathematical and logical ideas through interaction.
The Mathematics Behind NajduzaRec.rs
Prime Numbers
Prime numbers are one of the most important topics in number theory.
A prime number is a natural number greater than 1 that has exactly two positive divisors: 1 and itself.
This definition is simple, but the mathematical depth of prime numbers is enormous.
Prime numbers are the building blocks of the natural numbers because every integer greater than 1 can be expressed as a product of prime numbers.
Why Prime Numbers Matter
Prime numbers appear in many areas of mathematics and computer science.
They are important in number theory, cryptography, algorithms, divisibility, and many other fields.
Their simplicity makes them accessible to beginners, while their deeper properties continue to fascinate mathematicians.
Generating Primes in Large Numerical Ranges
A prime-number application allows users to find primes in a selected range.
From a programming perspective, this requires checking divisibility efficiently.
The larger the range, the more important efficiency becomes.
Exporting Prime Numbers to Text Files
I also have a private version of a prime-number program that is not publicly available on the website.
This version can export all prime numbers from a selected range into a TXT file and provide the total count of generated prime numbers.
YouTube Demonstration of the Prime Number Program
A presentation of this private prime-number program is available on YouTube:
Program for finding prime numbers in a given range
This program is another example of how the work behind NajduzaRec.rs extends beyond the public web applications.
Systems of Linear Equations
Systems of linear equations are fundamental in mathematics, physics, engineering, economics, and computer science.
NajduzaRec.rs includes applications for solving systems of two and three linear equations.
Solving Two Linear Equations
A system of two linear equations often represents two lines in a plane.
Depending on their relationship, the system may have:
- one solution
- no solution
- infinitely many solutions
This has a clear geometric meaning.
Two lines may intersect, be parallel, or coincide.
Solving Three Linear Equations
A system of three linear equations usually involves three variables.
Geometrically, such equations can represent planes in three-dimensional space.
The relationships between these planes can produce different types of solutions.
An interactive solver helps users test different systems and understand how coefficients influence outcomes.
The Importance of Linear Systems in Engineering
Linear systems are especially important in engineering.
They appear in circuit analysis, structural analysis, signal processing, control systems, numerical methods, and many practical problems.
This makes them a natural part of an educational platform created by an electrical engineer.
Quadratic Equations
Quadratic equations are among the most recognizable topics in algebra.
A general quadratic equation has the form:
ax² + bx + c = 0
where a is not equal to zero.
However, if you set a = 0 in the application, you will get the solution to the corresponding linear equation.
The Discriminant
The discriminant is given by:
D = b² − 4ac
This expression determines the nature of the roots.
If D > 0, the equation has two distinct real roots.
If D = 0, the equation has one repeated real root.
If D < 0, the equation has two complex conjugate roots.
Understanding the Nature of the Roots
The discriminant is not only a calculation.
It provides insight into the structure of the equation.
It tells us whether the solutions are real or complex, distinct or repeated.
Educational Value of Interactive Solvers
An interactive quadratic equation solver helps users test many examples quickly.
Instead of only memorizing formulas, users can observe how changes in coefficients affect the roots.
This helps build deeper understanding.
Cubic Equations
Cubic equations are more complex and mathematically richer than quadratic equations.
A general cubic equation has the form:
ax³ + bx² + cx + d = 0
where a is not equal to zero.
Why Cubic Equations Are More Challenging
Quadratic equations are relatively straightforward because their solution formula is widely known and manageable.
Cubic equations are more subtle.
Their solution methods are more advanced, and their numerical behavior can involve additional difficulties.
Numerical and Computational Challenges
When mathematical formulas are implemented in software, unexpected issues can appear.
A formula that seems simple on paper may require careful handling in a computer program.
This is especially true when roots, powers, negative numbers, real values, and complex values interact.
Odd Roots of Negative Numbers
Odd roots of negative numbers are a particularly interesting topic.
For example, the cube root of a negative number should be negative in the real-number system.
However, computational systems may handle expressions involving fractional exponents in ways that produce unexpected results.
Lessons From the Windows XP Odd Roots Bug
I discussed one such issue in my Infinite Math World article:
This example shows that even apparently simple mathematical operations can become subtle in software.
Why Mathematics and Software Must Work Together
A correct mathematical idea must be implemented carefully.
A program must respect mathematical meaning, not only syntax.
This is one of the reasons why cubic equations and root calculations are so interesting from both a mathematical and programming perspective.
Logarithms and Nth Roots
Logarithms and roots are fundamental mathematical operations.
They appear in algebra, science, engineering, exponential growth, signal processing, and many other areas.
Logarithms as Inverse Operations
A logarithm is the inverse operation of exponentiation.
The statement:
logₐ(x) = y
means:
aʸ = x
This relationship is central to understanding logarithms.
Practical Applications
Logarithms appear in many real-world contexts, including decibel scales, pH values, exponential growth, information theory, and electrical engineering.
Understanding Root Calculations
Nth roots also require careful understanding.
Even roots and odd roots behave differently, especially when negative numbers are involved.
The interactive tool help users explore these differences directly.
Divisibility, GCD, and LCM
Number theory is also represented on NajduzaRec.rs through divisibility, greatest common divisor and least common multiple tools.
Divisibility Rules
Divisibility rules help users determine whether one number is divisible by another.
These rules reveal patterns in the decimal representation of numbers and are useful in arithmetic and number theory.
Greatest Common Divisor
The greatest common divisor is the largest number that divides two or more integers without a remainder.
It is important in fraction simplification, number theory, modular arithmetic, and algorithms.
Least Common Multiple
The least common multiple is the smallest positive number divisible by each of the given numbers.
It is useful in fractions, periodic processes, scheduling problems, and many mathematical applications.
Hidden Number Patterns
These topics may appear elementary, but they reveal important numerical structures.
Interactive tools help users recognize and test these patterns.
Complex Numbers
Complex numbers are especially important to me as an electrical engineer.
In school mathematics, they are often introduced as an extension of the real numbers.
In engineering, however, they are practical and indispensable.
Multiplication and Division
Multiplication and division of complex numbers require understanding both real and imaginary components.
Division usually involves multiplying by the conjugate of the denominator.
This is a procedure that students often need to practice carefully.
Square Roots of Complex Numbers
Finding square roots of complex numbers is more advanced.
It requires deeper algebraic reasoning or the use of polar representation.
An interactive tool can help users understand the structure of these operations.
Why Complex Numbers Matter in Electrical Engineering
Complex numbers are essential in alternating current circuit analysis, impedance, phasors, signal processing, control theory, Fourier analysis, and many areas of applied engineering.
For an electrical engineer, complex numbers are not artificial mathematical objects.
They are practical tools.
From Abstract Mathematics to Real Applications
Complex numbers are a perfect example of how abstract mathematics becomes useful in real systems.
This makes them a natural part of NajduzaRec.rs.
Lotto 7/39 and Combinatorial Thinking
NajduzaRec.rs also includes a Lotto 7/39 application.
Although lottery topics may appear different from algebra or number theory, they are closely related to combinatorics and probability.
Combinations and Probability
In a 7/39 lottery system, the number of possible combinations is:
C(39, 7)
The general formula is:
C(n, k) = n! / (k!(n − k)!)
This calculates the number of ways to choose k elements from n elements when order does not matter.
Understanding Large Numbers
Combinatorial numbers grow very quickly.
This is one reason why many people underestimate how many possible outcomes exist in lottery systems.
Mathematics helps make this clearer.
Mathematical Thinking and Random Events
The goal of such an application is not to guarantee prediction.
The goal is to encourage mathematical thinking about combinations, probability, and randomness.
Why Interactive Learning Matters
Mathematics Should Be Explored, Not Memorized
One of the main ideas behind NajduzaRec.rs is that mathematics should be explored.
Memorization has its place, but it is not enough.
Students and users understand mathematics more deeply when they can test ideas.
They need to see what happens when values change.
They need to compare examples.
They need to make mistakes and correct them.
Interactive tools support that process.
Immediate Feedback Accelerates Understanding
One of the advantages of educational software is immediate feedback.
A user can enter values and instantly see the result.
This makes learning faster and more active.
Instead of waiting to check an answer later, the user can immediately compare expectations with reality.
Making Mathematics Less Intimidating
Many people see mathematics as difficult or intimidating.
Interactive applications can reduce that fear.
They allow users to approach problems gradually, test examples, and build confidence.
Mathematics becomes less distant when users can interact with it directly.
Educational Software as a Bridge Between Theory and Practice
Educational software connects theory with practice.
It shows that formulas are not isolated symbols.
They can become working tools.
This is one of the reasons why I believe projects like NajduzaRec.rs have real educational value.
Serbian and English Versions: Two Languages, One Vision
Supporting Education in Serbian
The Serbian version of NajduzaRec.rs is important because it provides educational tools in my native language.
Mathematics is universal, but language still matters.
Users understand concepts more easily when the interface and explanations are available in a language familiar to them.
Reaching an International Audience
The English version allows the project to reach a wider audience.
Since English is widely used in science, mathematics, technology, and education, the English version gives NajduzaRec.rs an international dimension.
Maintaining Two Separate Databases
The Serbian and English versions require separate word databases.
They are not translations of one another.
Each database has its own linguistic rules, sorting system, and structure.
Similar Goals, Different Linguistic Challenges
The goal is similar in both languages: to find valid long words efficiently.
But the linguistic challenges are different.
This makes the bilingual nature of the project both demanding and valuable.
ASP.NET and the Technical Side of the Project
Why Web Applications Matter
NajduzaRec.rs was developed using ASP.NET technology.
A web application has one major advantage: accessibility.
Users do not need to install a desktop program.
They can access the tools through a browser.
This makes the educational content easier to reach.
Transforming Desktop Programs Into Online Tools
Transforming standalone programs into web applications requires adaptation.
The mathematical logic may remain similar, but the user experience changes.
A web application must be clear, stable, responsive, and easy to use.
User Experience and Accessibility
Good educational software must not only calculate correctly.
It must also guide the user clearly.
Input fields, buttons, output formats, and explanations all influence the usefulness of the application.
Input Validation, Stability, and Reliability
Public applications must handle user input carefully.
They must prevent invalid entries from creating confusion.
They must remain stable.
They must produce meaningful output.
This technical side is often invisible to users, but it is essential.
NajduzaRec.rs and Infinite Math World
Two Projects With a Common Foundation
NajduzaRec.rs and Infinite Math World have different forms, but they share the same foundation.
Both are built around mathematics, education, logic, and scientific curiosity.
NajduzaRec.rs began earlier, in August 2019.
Infinite Math World came later, in early December 2024.
Tools and Explanations
NajduzaRec.rs provides tools.
Infinite Math World provides explanations.
One allows users to interact with mathematical ideas.
The other allows readers to explore those ideas in greater depth.
Why Interactive Software and Educational Articles Complement Each Other
A mathematical tool is useful, but users may also need theoretical context.
An article can explain the idea, while an application can demonstrate it.
Together, they create a stronger educational experience.
Building a Broader Educational Ecosystem
In that sense, the two websites form a broader educational ecosystem.
They are different projects, but they support the same mission: making mathematics, science, programming, and logical thinking more accessible.
What Makes the Project a Long-Term Journey
Continuous Database Improvement
The word databases continue to improve.
This is especially important for the Serbian version, where grammatical verification requires continuous attention.
Database improvement is not a one-time task.
It is an ongoing process.
New Ideas and Future Development
NajduzaRec.rs is not limited to its current form.
There are still programs that are not part of the website.
Some of them may become public in the future.
New ideas may also lead to new tools.
Programs That May Become Public
The existence of private programs shows that the project has a wider background than visitors can currently see.
The website presents one part of the work.
The broader development process continues beyond it.
Why Educational Software Is Never Truly Finished
Educational software can always be improved.
Interfaces can become clearer.
Databases can become better.
Algorithms can be optimized.
Explanations can be expanded.
New applications can be added.
For that reason, I do not see NajduzaRec.rs as a finished product in the absolute sense.
I see it as an evolving educational project.
Lessons Learned While Building NajduzaRec.rs
Invisible Work Behind Visible Results
One of the most important lessons is that users often see only the final result.
They click a button and receive an answer.
But behind that answer there may be years of work, thousands of decisions, database filtering, algorithm design, testing, correction, and refinement.
Good software often hides its complexity.
The Importance of Data Organization
The “Longest Word” application shows how important data organization can be.
Sorting words from longest to shortest and alphabetically within each length group is a simple idea, but it has a major effect on performance and clarity.
Sometimes, a well-organized database is as important as the algorithm itself.
Why Simplicity Often Beats Complexity
A solution does not always need to be more complicated to be better.
Sometimes the best solution is the one that uses a simple structure intelligently.
This is true in mathematics, programming, and engineering.
Mathematics and Programming Strengthen Each Other
Mathematics helps us understand structure.
Programming helps us turn structure into tools.
Each field strengthens the other.
NajduzaRec.rs is a practical result of that relationship.
Enthusiasm as a Long-Term Driving Force
A project like this cannot be built without enthusiasm.
Technical knowledge is necessary, but it is not sufficient.
Long-term projects require curiosity, patience, persistence, and the belief that the work has value.
NajduzaRec.rs represents that kind of long-term dedication.
Final Thoughts
NajduzaRec.rs publicly began in August 2019, but its real story started much earlier.
It grew from years of standalone programs, mathematical curiosity, algorithmic experimentation, database construction, and the desire to create useful educational tools.
What began with the idea of finding the longest possible word from a set of letters gradually developed into a broader educational platform containing thirteen web applications.
The project now connects language, mathematics, logic, programming, number theory, algebra, complex numbers, combinatorics, and interactive learning.
The Serbian and English word databases, containing exclusively words from 9 to 12 letters, were not designed as ordinary dictionaries. They were created as specialized, optimized structures for fast algorithmic search.
The public website shows one side of the project.
Private programs, continuous database improvements, and additional tools show that the broader development process is still active.
For me, NajduzaRec.rs is much more than a website.
It is the result of mathematical thinking, engineering discipline, software development, linguistic attention, and long-term enthusiasm.
It reflects my belief that mathematics should be explored actively, that programming can make abstract ideas practical, and that educational tools can help people approach difficult concepts with more confidence and curiosity.
Infinite Math World and NajduzaRec.rs represent two sides of the same educational vision.
One explains.
The other allows interaction.
Together, they reflect my continuing effort to make mathematics, science, programming, and logical thinking more accessible, more engaging, and more useful to everyone who wants to learn.
That is the real story behind NajduzaRec.rs.
It is a story about mathematics, algorithms, engineering, language, education, and the lasting value of turning ideas into working tools.