Year: 2025

Understanding the Power of e number: The Unsung Hero of Mathematics

Post author By Aida Alsina
Post date August 5, 2025

When discussing fundamental mathematical constants, pi often takes the spotlight. However, another number, equally vital yet less celebrated, is Euler’s number e. Known simply as e, this constant appears in countless mathematical contexts and real-world applications. In this article, we will explore the origin, properties, and applications of e, positioning it as an indispensable tool in the worlds of mathematics, science, and finance.

What Is Euler’s Number e?

Euler’s number e is an irrational and transcendental number, approximately equal to 2.71828. It serves as the base of natural logarithms and arises naturally in various growth processes and limit calculations.

The Origin of e: A Practical Example

Consider the following real-life scenario: Imagine a bank offering a 100% annual interest rate on a principal investment of 1€. If interest is compounded once at the end of the year, you would have 2€. However, if the bank compounds interest semiannually at 50% every six months, the calculation changes:

After six months: 1€ × 1.5 = 1.5€
After another six months: 1.5€ × 1.5 = 2.25€

Now, let’s divide the year into three periods of four months each, applying an interest rate of 33.33% every four months:

After four months: 1€ × 1.33 = 1.3333€
After eight months: 1.3333€ × 1.3333 ≈ 1.7777€
After twelve months: 1.7777€ × 1.3333 ≈ 2.3703€

So, if you receive infinite payments, will you have infinite money? Unfortunately, the answer is no. As the frequency increases -from quarterly to monthly, to daily, and beyond- the final amount approaches a limit. The formula to calculate this is:

mathtype formula Image created with Wiris’ MathType

As n approaches infinity, the value converges to e. This simple yet powerful example demonstrates the natural emergence of e in exponential growth, explaining its foundational role in continuous compounding interest models.

Fundamental Properties of e Number

Irrationality: e cannot be expressed as a simple fraction.
Transcendence: e is not a root of any non-zero polynomial equation with rational coefficients.
Infinite Series: e can be represented as the sum of the infinite series:

equation with mathtype

Image created with Wiris’ MathType

Natural Logarithm Base: The function uses e as its base, integral to calculus and natural growth models.
Euler’s Identity: Considered one of the most beautiful equations in mathematics:

eⁱ^π+ 1 = 0

This equation elegantly connects five fundamental mathematical constants: 0, 1, e, i, and π.

Applications Across Disciplines

Euler’s number e is not only a fundamental constant in pure mathematics but also plays a pivotal role in a range of practical applications across various scientific and technological fields. From modeling growth processes to risk management, e is indispensable in understanding and predicting complex phenomena.

In Mathematics

Calculus: Euler’s number is central to calculus, particularly in the context of exponential growth and decay. One of the most important properties of e is that the derivative of the function e^x is e^x itself. This unique property makes it extremely useful in solving differential equations that describe dynamic systems, such as population growth, radioactive decay, and heat transfer. It simplifies the analysis of these systems because the function is self-replicating under differentiation.
Complex numbers: Euler’s formula, e^ix = cos(x) + isin(x), is one of the most profound equations in mathematics, linking exponential functions with trigonometric functions. This relationship is crucial in fields like electrical engineering and signal processing, where it simplifies the analysis of oscillating systems and waveforms. It allows for a more intuitive understanding of rotations and oscillations in the complex plane.

In Physics

Radioactive decay: The process of radioactive decay follows an exponential law, which can be modeled using e. The amount of a radioactive substance remaining after a certain period is given by the equation N(t) = N₀e^λt, where N(t) is the amount of substance at time t, N(t) is the initial amount, and λ is the decay constant. This model is used to predict the behavior of radioactive materials over time, which has applications in medicine (e.g., radiology), archaeology (e.g., carbon dating), and nuclear physics.
Thermodynamics: In thermodynamics, the Boltzmann factor (see image below) is essential for determining the probability of a system being in a particular energy state, where 𝐸 is the energy, 𝑘 is the Boltzmann constant, and 𝑇 is the temperature. This exponential relationship is vital in understanding systems in equilibrium, such as the distribution of particles in gases or the behavior of molecules in biological processes.

In Economics and Finance

Continuous compounding: One of the most common uses of e is in the calculation of continuously compounded interest. If interest is compounded continuously rather than at discrete intervals, the formula for the accumulated value of an investment becomes A = Pe^rt, where 𝑃 is the initial investment, 𝑟 is the interest rate, and 𝑡 is the time in years. This formula highlights how the frequency of compounding approaches a limit, with e representing the limit as compounding becomes continuous. It is fundamental in understanding the long-term growth of investments and savings.
Risk models: Financial models such as the Black-Scholes model for option pricing rely heavily on e to account for the stochastic nature of asset prices. The formula for option pricing involves exponential functions to model how the price of an option evolves over time. e also plays a key role in various risk models, including Value at Risk (VaR) calculations, where it helps quantify the likelihood of extreme losses in investment portfolios.

In Technology

Algorithms: Euler’s number is used in algorithmic analysis, particularly when analyzing the performance of algorithms that exhibit exponential growth. For instance, certain random processes and algorithms, such as the analysis of QuickSort (a widely used sorting algorithm), can involve e. Specifically, in probabilistic algorithms and complexity theory, e appears in the expected running time of algorithms, as they often follow exponential distributions or involve recursive calculations that converge to e.
Machine learning and data science: In machine learning, e is frequently involved in algorithms that model growth processes or in optimization algorithms that use exponential decay to adjust parameters over time. For instance, in gradient descent, which is used to find the minimum of a function, learning rates may decay exponentially based on e to optimize performance and prevent overshooting.

In Biology and Medicine

Population growth: One of the most well-known real-world applications of e is in modeling population growth. When populations grow in ideal conditions (without external limitations), the number of individuals follows an exponential growth curve, which is modeled by the equation, P(t) = P₀e^rtwhere 𝑃(𝑡) is the population at time 𝑡, P₀ is the initial population, and 𝑟 is the growth rate. This principle is not only relevant to ecology but also helps in understanding the spread of diseases in epidemiology.
Pharmacokinetics: The exponential decay model is also used in pharmacokinetics, which studies how drugs are absorbed, distributed, metabolized, and excreted by the body. The concentration of a drug in the bloodstream over time often follows an exponential decay, and the parameter governing the decay is related to e. This helps in determining dosing schedules for medications to ensure therapeutic levels are maintained without causing toxicity.

Visualizing Euler’s Number e with Wiris’ Digital Tools

As digital solutions advance, tools like MathType and CalcMe empower users to work effortlessly with e and other complex mathematical expressions. Whether you’re a researcher, educator, or professional, integrating these Wiris’ technologies into your workflow ensures precision and efficiency.

For instance, to deepen the understanding of e’s behavior, one can visualize the limit expression

equation editor

using graphing tools such as CalcMe, a Computer Assisted System (CAS) for algebraic manipulation that allows complex mathematical calculations to be carried over in a self-contained, fast and intuitive way. As increases, the graph approaches the value of number e, offering an intuitive graphical demonstration of this mathematical limit.

Additionally, MathType, the world’s leading equation editor that allows you to write math notation as easily as you write text, enables seamless insertion of complex mathematical expressions into documents and web content, ensuring clarity and precision.

number e graphic

Graph showing the function (1+1/x)^x in red, approaching the value of e in blue.

Perfect for students, teachers, editors, and technical writers, MathType streamlines the creation of high-quality technical and scientific content across various platforms.

For seamless handling of mathematical expressions and visualizations, consider incorporating both MathType and CalcMe into your digital toolkit.

Write your e number equations here!

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Irreplaceable Formulas: MathType and Mondadori Education’s Editorial Excellence

Post author By Aida Alsina
Post date July 28, 2025

From a Useful Tool to a Strategic Company Reference

When Pietro Curci joined Le Monnier (an Italian publishing house) in the mid-1990s, he was already on the lookout for ways to modernize the production of educational content in his company, especially the most complex part: math formula typesetting. As head of the graphic office, his responsibility was clear: ensure quality, consistency, and efficiency in a publishing environment.

It was during those years, well before Le Monnier was acquired by the Mondadori Education Group (the biggest publishing company in Italy) in 1999, that Curci discovered MathType for Word. The equation editor quickly became indispensable, offering a way to produce clean, editable, and reusable formulas, something that was unheard of at the time. The Mondadori Education Group operates in three main areas: books (publishes fiction, non-fiction, children’s, educational, and academic books); magazines (publishes leading Italian print and digital magazines), and retail (manages a large network of bookstores and online sales). It is also active in education, digital content, and international licensing.

When the transition to Mondadori Education happened, so did the natural evolution of MathType from a useful tool to a strategic standard. Today, as Head of the Graphic Office for the Humanities Area for high schools, Curci oversees a large part of the group’s editorial workflow, and MathType remains at its core. Nearly three decades later, it remains a cornerstone of Mondadori Education’s editorial workflow. “Our solution is used by DeaScuola and Rizzoli Education, which are part of the Mondadori Education Group”, explains Curci.

What started as a practical solution for writing equations in Word has grown into something much bigger. Over time, MathType has become deeply embedded in how Mondadori Education works.

It’s not just used by one team or for one task, but it supports different departments and adapts to various parts of the production process. Whether in editorial or digital, MathType has proven to be flexible and reliable, earning its place as a shared reference point across the company.

The Challenges

In the mid-1990s, Le Monnier -and later Mondadori Education- needed a reliable tool for producing complex mathematical content. Curci recalls the limitations of early typesetting systems, particularly when it came to consistency, reusability, and integration with publishing tools like QuarkXPress.

“We needed something stable that would allow us to reuse content across editions and titles,” Curci explains. “MathType gave us formulas as images, and an image, as I always say, is forever. It was revolutionary“.

But over time, challenges evolved. The rise of digital publishing, colored content, and interactive formats pushed editorial tools beyond what black-and-white workflows could support. Ensuring compatibility with new operating systems and modern layout tools became a strategic necessity.

Mondadori Education and Wiris: Joint Efforts to Improve the Product

Mondadori Education adopted MathType in its early versions and never looked back. The company standardized its use across internal and external collaborators. Layout professionals, editors, and authors could all work with the same equation editor, streamlining collaboration.

“With the plugin installed in our platforms, we can extract formulas without redigitizing them. It saves time and reduces errors, something essential when you’re dealing with 5,000+ formulas in a textbook”, affirms Curci.

When Curci and his team encountered barriers -such as color rendering issues, EPS compatibility, or font glyph problems- they didn’t give up with MathType. Instead, they worked closely with Wiris, contributing practical solutions and insights based on their real-world workflow.

MathType’s flexibility in exporting MathML for digital editions, while maintaining high-quality image-based formulas for print, allowed Mondadori Education to meet evolving demands without rebuilding their content library from scratch. As Curci affirms, “Thanks to the embedded MathML, we can easily export the formulas as code and reuse them in digital formats”.

The Implementation Process

The implementation of MathType at Mondadori Education was the result of a clear vision: to establish a unified, efficient workflow for mathematical content creation. Once MathType proved to be a reliable solution, it was adopted as the official standard across the organization. This meant that all external vendors producing math formula typesetting layouts were also required to work with the same tool, fostering consistency and compatibility across every stage of production.

One of the key advantages of MathType was its ability to generate formulas as images, allowing for the long-term reuse of content. Thanks to this, legacy materials could be updated by simply adjusting style preferences, eliminating the need to recreate thousands of formulas from scratch.

The integration of MathType enabled seamless collaboration. Editorial teams received manuscripts with embedded formulas and passed them on to layout professionals, who could extract and edit the formulas directly, greatly reducing manual intervention and improving version control.

Throughout this process, Pietro Curci played a crucial role in bridging the gap between user needs and technical development. His collaboration with Wiris led to the resolution of various challenges, such as color rendering issues in EPS files, overprint settings in InDesign, and glyph compatibility. These contributions helped shape subsequent versions of the equation editor and ensured its alignment with real-world editorial demands.

Why MathType Is Essential to Mondadori Education’s Editorial Excellence

Over the years, MathType has established itself as a trusted and indispensable tool within Mondadori Education’s editorial ecosystem. Its ease of use, compatibility across platforms, and ability to generate both print and digital outputs have made it a core component of the company’s workflow. The following key benefits highlight why MathType continues to play a strategic role in ensuring quality, efficiency, and long-term content reuse at Mondadori Education.

Preserved editorial legacy. “MathType is irreplaceable,” says Curci. “Without it, we would have lost a huge amount of content or had to reinvest in recreating everything”.
Increased efficiency. Automated extraction of formulas from documents, coupled with standardized workflows, greatly reduced manual effort and error rates across the team.
Ease of use. The platform requires no advanced technical skills, making it accessible to a broad range of professionals. “The fact that it has a simple interface doesn’t mean it’s not good. Otherwise: with just a few clicks, you understand what to do to save and typeset formulas”, affirms Curci.
Flexibility and reusability. Formulas can be adapted to new design preferences with minimal effort, reducing rework.
Support for digital formats. Built-in MathML export facilitates the production of reflowable, accessible digital textbooks.
Collaborative customer service. The collaboration with Wiris has enabled Mondadori Education to resolve technical challenges like spot color management, overprinting, and font compatibility, ensuring consistent output across print and digital.
Cross-department adoption. MathType is used not only by in-house editorial teams but also by all external layout professionals who contribute to Mondadori Education’s math formula typesetting publications. Its status as a shared standard enables a smooth production chain.
Future-proofing. Despite working with older machines to maintain compatibility, Curci and his team are eagerly awaiting the new MathType release. The goal: support newer systems while maintaining continuity.

Looking Ahead Together

We’re entering a new era of innovation, driven by the evolution of our products and the needs of the STEM community. At Wiris, we’re actively working on meaningful improvements of MathType to better support today’s editorial and educational challenges. And we’re not doing it alone, as Mondadori Education and Pietro Curci continue to be by our side, offering their insight, testing new features, and helping us shape what’s next.

For Curci, it’s not just about modernization, it’s about continuity: “I’m not looking for glory” he says. “What I care about is sharing knowledge and ensuring that future generations can build on what we’ve created. Experience is meant to be passed on, not buried”.

Mondadori Education’s relationship with MathType is not just about software: it’s about editorial integrity, long-term thinking, and the power of collaboration. “I’ve worked closely with Wiris for years. I’ve seen how they listen. This isn’t just about Mondadori Education: it’s about keeping a shared legacy alive across the publishing world”. “If one day we had to stop using MathType, it would be a significant loss”.

Interested in MathType? Get in touch.

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Teaching Math Online: Transforming Education with Wiris

Post author By Aida Alsina
Post date July 18, 2025

Teaching math online is redefining the educational landscape. The emergence of advanced digital tools has empowered educators to deliver personalized, flexible learning experiences. Among these innovations, innovative online math teaching methods are crucial in enhancing student engagement, knowledge retention, and academic performance. This article explores these transformative strategies and highlights how Wiris enables educators to implement them effectively and efficiently.

Key Trends in Online Math Teaching: Prioritizing Personalization, Engagement, and Flexibility

In an increasingly digital and dynamic educational landscape, modern online math teaching strategies are evolving to address the diverse needs of today’s learners. The focus is no longer solely on content delivery; it has shifted towards creating learning environments that foster personalization, maximize student engagement, and offer flexibility to accommodate different learning styles and paces.

The most prominent and impactful trends in teaching math online today include:

Flipped classrooms: The flipped classroom model transforms the traditional educational paradigm by moving direct instruction outside of the classroom -typically through videos, readings, or online content- and reserving classroom time for active learning. This approach fosters deeper understanding through discussions, collaborative problem-solving, and interactive exercises in math education. Students engage with material at their own pace before class and apply their knowledge during in-person or synchronous online sessions.
Microlearning: Microlearning involves delivering information in small, focused segments that are easier to digest and retain. Rather than overwhelming students with lengthy, complex lessons, microlearning breaks content into concise, manageable units. In innovative online math teaching, this method proves highly effective, as it reduces cognitive overload and enables students to achieve mastery one concept at a time.
Personalized learning: Personalized learning leverages technology to tailor educational experiences to the unique needs, skills, and interests of each student. In the context of online math teaching, this approach allows educators to adapt instruction dynamically, enabling students to progress at their own pace. It ensures that learners receive the right level of support and challenge, enhancing motivation and academic performance.

Wiris solutions, including WirisQuizzes and LearningLemur, represent online tools for teaching math online that operationalize these methodologies, enabling educators to develop dynamic assessments and tailored learning paths for their students.

Real-World Applications: How These Online Math Teaching Strategies Benefit Students

Adopting personalized learning, microlearning and flipped classroom strategies equips students with competencies critical for real-world success. These methodologies not only boost academic outcomes but also nurture skills crucial for thriving in digital and collaborative professional environments.

Through innovative online math teaching methods, educators are fostering the development of vital skills such as problem-solving, critical thinking, and self-directed learning, which are capabilities increasingly demanded in STEM industries and the future workforce.

Flipped Classrooms: Learning by Doing Instead of Memorizing

The flipped classroom model fosters student autonomy by shifting content delivery outside the classroom and reserving in-person sessions for collaborative, higher-order learning activities. Instead of passively listening to lectures, students engage with preparatory materials —such as videos or readings— at their own pace and use classroom time to apply concepts through problem-solving and discussion.

This method promotes critical thinking, deeper understanding, and active participation. It also supports diverse learning styles and encourages students to take greater responsibility for their learning. By transforming the classroom into a dynamic, interactive environment, the flipped model builds key competencies such as communication, collaboration, and analytical skills.

Benefits of the Flipped Classroom Model

These are the main benefits of the Flipped Classroom Model, a teaching approach that is increasingly gaining traction in modern educational methods:

Assign dynamic, preparatory quizzes to prime students for in-class activities. Assigning dynamic, preparatory quizzes ensures students engage with key concepts before class. This preparation allows for more effective in-class sessions focused on applying knowledge rather than covering basics. Tools like WirisQuizzes and LearningLemur can support you in developing these dynamic preparatory quizzes tailored to your learners’ needs.
Foster interactive, discussion-based classroom environments. With prior exposure to the material, class time can be devoted to discussions and collaborative problem-solving. This shift enhances critical thinking and promotes deeper understanding through active dialogue.
Promote active participation and collaborative learning dynamics. Flipped classrooms encourage students to actively participate and collaborate with peers. Working in teams and engaging in problem-solving activities builds not only academic skills but also essential competencies like communication and teamwork.

girl studing online

The Power of Microlearning: Quick Lessons for Better Retention

Microlearning is gaining traction as an effective strategy in online math teaching. This method decomposes complex concepts into manageable units, promoting deeper understanding and better long-term retention.

With WirisQuizzes and LearningLemur, educators can design brief, targeted quizzes that reinforce core concepts efficiently. The immediate feedback mechanisms facilitate real-time learning corrections. These micro-assessments exemplify innovative online math teaching approaches that reduce cognitive load and improve learning efficacy. This tool streamlines microlearning by enabling the creation of interactive, goal-oriented quizzes.

WirisQuizzes incorporates a feature that enhances microlearning by providing tailored feedback for open-ended math questions. It supports three main types of feedback: suggestive feedback, which guides students toward discovering solutions independently; confirmatory feedback, which reinforces correct answers and boosts confidence; and corrective feedback, which identifies mistakes and provides the correct solution. This immediate, adaptive feedback fosters critical thinking and ensures efficient, focused learning, aligning perfectly with microlearning principles.

feedback with wirisquizzes

Personalized Learning: Delivering Tailored Educational Experiences

Personalized learning is a cornerstone of modern educational methodologies, particularly in teaching math online. It leverages technology to customize educational experiences according to the unique needs, skills, and learning speeds of individual students.

WirisQuizzes and LearningLemur are central to enabling this level of personalization:

WirisQuizzes, oriented to higher education, allows teachers to design sophisticated quizzes incorporating mathematical expressions, graphical representations and dynamic text-based responses. Its advanced randomization and evaluation capabilities generate unique assessments for the students, promoting practice through varied and individualized problem sets.
LearningLemur, oriented to K-12 students and seamlessly integrated with Google Classroom, offers a comprehensive platform for customizable quizzes, automated grading, and personalized feedback. It not only saves educators valuable time but also fosters student engagement and mastery. The platform’s extensive library of adaptable exercises and detailed analytics empowers educators to monitor student progress and intervene when necessary, further enhancing personalized learning. In addition, LearningLemur now incorporates an AI math question generator that understands your needs and generates ready-made quizzes in seconds. It’s designed to address one of the most common pain points for educators: how to start creating math assessments efficiently without sacrificing quality.

learning lemur ai questions

What truly differentiates Wiris’ tools from others is their capacity to mathematically interpret any answer, apply rigorous mathematical logic to personalize feedback, and define random variables using advanced algorithms. These capabilities empower educators to offer more precise and responsive learning experiences, ensuring that each student progresses at their optimal pace and receives support tailored to their specific learning trajectory.

By integrating these solutions, educators can create adaptive learning environments where technology dynamically adjusts to student needs, providing instant feedback and continuously aligning educational content with individual learning goals.

Embracing the Future of Education with Wiris

The digital transformation of education demands agile, student-centric teaching methodologies. Teaching math online requires sophisticated tools, and Wiris stands at the forefront, delivering solutions that enable educators to design personalized, interactive learning experiences.

Enhance your online teaching strategy: start using WirisQuizzes today and experience the benefits of next-generation online tools for teaching math online.

Try WirisQuizzes now

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How Online Math Learning with Wiris Transformed UNED

Post author By Aida Alsina
Post date June 25, 2025

Breaking Barriers in Online Math Learning with Wiris

In the ever-evolving landscape of online education, ensuring that mathematical and scientific content is easy to create, share, and evaluate is a challenge.

Universidad Nacional de Educación a Distancia (UNED), Spain’s leading distance-learning university, recognized this challenge early on. With thousands of students across various disciplines, UNED sought a robust solution to enhance its teaching and evaluation processes, particularly in mathematics and science-related subjects.

To tackle these challenges, UNED turned to online math learning with Wiris, integrating WirisQuizzes and MathType into their educational workflow. Having been a valued client of Wiris for over 15 years of the company’s 25-year journey, UNED has continuously trusted in Wiris’ math solutions to support its evolving digital education needs. We sat down with Mari Carmen García Llamas, a professor in the Economics and Tourism Bachelor’s Degrees, and Raúl Morales Hidalgo, Head of the E-learning Platform at UNED, to understand how these tools transformed their math teaching and assessment methods.

The Challenge: Teaching Math in a Digital Environment

Before implementing Wiris’ solutions, UNED faced significant obstacles in creating and communicating mathematical content effectively in an online setting.

From a technical perspective, their biggest challenge was integrating a math-editing tool into their custom learning platform. Faculty members struggled with writing mathematical formulas in forums, assignments, and quizzes resulting in a poorer learning experience for students.

From a teaching perspective, professors required an intuitive solution to easily insert mathematical expressions into quizzes and ensure smooth interaction with students.

Carmen explains:

“Without a tool to write mathematical symbols, explaining calculations was a big challenge. Whether it was an integral or an algebraic equation, it was always difficult to pinpoint where a student had made an error.”

Additionally, UNED needed an assessment tool that allowed mathematical expressions throughout a quizz, from problem statements to answer options. The absence of such a tool limited quiz and assignment creation.

The Solution: Enhancing Online Math Learning at UNED with Wiris

To overcome these challenges, UNED explored available solutions and found WirisQuizzes and MathType to be the most comprehensive and well-integrated tools for their needs. Their decision to implement Wiris’ solutions was based on several key factors:

Seamless integration with OpenLMS that they use in-house

UNED initially discovered MathType through faculty recommendations and was particularly drawn to its integration with Microsoft Office. However, since they had a custom in-house platform, they needed a solution that could be fully embedded.

Wiris’ technical team collaborated with UNED to integrate MathType and WirisQuizzes into their platform, ensuring a smooth transition and enhanced functionality.

User-friendly for professors and students

With online math learning with Wiris, professors could now write and edit formulas effortlessly across various platforms, including forums, quizzes, and assignments. This eliminated previous barriers in math-based communication between students and teachers.

Carmen highlights the impact:

“We needed a tool that was just as easy to use for the teacher as it was for the student. Wiris makes it possible to create math-rich quizzes where every question is fully customizable.”

Encouraging Active Learning by Breaking Away from Memorization

One of the standout features of WirisQuizzes is its ability to introduce randomized mathematical expressions into quizzes. This ensures that each student receives a unique version of an exercise, discouraging memorization and promoting active learning. This was highlighted by Carmen when she stated:

“WirisQuizzes allows us to create quizzes where every exercise is unique. The level of flexibility it provides is amazing.”

A welcome surprise: the power of WirisQuizzes

While UNED initially focused on MathType, they soon discovered that WirisQuizzes offered additional benefits. As they expanded their digital assessment strategies, WirisQuizzes proved to be an unexpected but valuable tool in enhancing student engagement and practice opportunities.

Raúl shares his excitement:

“When we started making changes to our platform, we were pleasantly surprised by WirisQuizzes. It felt like an unexpected gift, and it has been an incredibly powerful one.”

The Implementation Process: A Collaborative Approach

To successfully integrate online math learning with Wiris into their learning environment, UNED, along with Wiris’ technical team, followed a structured implementation process:

Understanding institutional needs

UNED’s faculty and technical teams worked with Wiris to assess their specific requirements, particularly the need for platform integration and math-based assessments.

Customization and integration

Wiris’ technical experts customized the tools to align with UNED’s platform requirements, ensuring a smooth user experience. Furthermore, the team at Wiris assisted in resolving any doubts as well as providing tips and training on how to use WirisQuizzes efficiently.

Deployment across OpenLMS

The integration was successfully implemented on OpenLMS, allowing professors to seamlessly create and edit formulas within quizzes and discussion forums.

Learning Math Online with Wiris – The Results

Following the implementation of online math learning with Wiris, UNED experienced measurable improvements in teaching efficiency and student engagement:

Greater flexibility for teachers in math-based quiz creation.
Improved student engagement through randomized exercises.
Seamless math communication in forums and assignments.
Quick technical support ensures smooth day-to-day usage.

Carmen notes:

“Wiris allows us to create diverse quizzes with distinct options, making every assessment dynamic and interactive.”

Students also benefit from interactive, dynamic learning experiences, where every exercise is different, allowing them to practice multiple problem variations in a single topic.

Future Plans: Looking Ahead with WirisQuizzes

After experiencing the benefits of WirisQuizzes, UNED is eager to see how the tool evolves and continues to support its online math learning needs. While it is currently used for continuous self-evaluation, faculty members appreciate its capabilities and will continue integrating it into their teaching workflows.

Carmen shares:

“More than one of my colleagues has reached out to me about the potential of Wiris’ solutions. I believe we are only beginning to explore what’s possible.”

Rounding Off: UNED’s Endorsement of Online Math Learning with Wiris

With over 15 years of using Wiris products, UNED strongly endorses WirisQuizzes and MathType for their ease of use, powerful assessment features, and seamless integration.

Raúl emphasizes:

“We have always recommended Wiris. That’s why we’ve been with you for 15 years already.”

Try Wiris for Your Institution

UNED’s success story showcases how online math learning with Wiris can revolutionize digital math education.

If you’re looking to improve math-based assessments in your institution and want a seamless math-writing experience for students and teachers, don’t delay—contact us today.

Request a Demo Today

Uncategorized

AI for Math: The Smart Way to Create Quizzes Faster

Post author By Aida Alsina
Post date June 12, 2025

Creating engaging and varied math quizzes can be a surprisingly time-consuming task. Teachers often face the dreaded blank page, struggling to come up with fresh problems that match different levels, topics, and student needs. But that’s about to change. With AI for math making its way into classrooms, quiz creation just got a powerful upgrade.

Meet LearningLemur: Your AI Question Generator

LearningLemur is an intuitive math platform integrated with Google Classroom that provides customizable quizzes, automatic grading, and personalized feedback, saving educators time and enhancing student learning. Its extensive library of customizable quizzes and exercises fosters engagement and mastery, while detailed analytics help educators track progress and identify areas for improvement. From simplifying assignment creation to delivering instant feedback, LearningLemur transforms classrooms into dynamic and interactive learning spaces, making math more accessible and engaging for all students.

Now, Wiris has introduced a new feature within LearningLemur: a math question generator that understands your needs and generates ready-made quizzes in seconds. It’s designed to address one of the most common pain points for educators: how to start creating math assessments efficiently without sacrificing quality.

Here’s what LearningLemur’s new AI-powered tool offers:

Describe the type of questions you want.
Receive an instant set of customized math problems.
Edit, refine, and assign them directly to your students.

Forget about starting from scratch. This tool helps you work smarter, not harder, all within a platform that is intuitive and easy to navigate, even for teachers with limited tech experience.

How does it work?

Using this tool is simple and straightforward. Teachers input the number of questions, select the question type (e.g., open answer, multiple choice) and finally provide a prompt (e.g., “Make fractions sum problems for students of grade 12. Ask them to be simplified.”).

The AI handles the rest, offering draft content that aligns with your instructions. And because it’s powered by the trusted Wiris correction engine, you maintain full confidence in the mathematical accuracy.

Why This AI Feature Matters for Educators

The LearningLemur AI question generator isn’t just about saving time. It’s designed to help teachers focus on what really matters: pedagogy, clarity, and curriculum alignment. While the AI handles the generation, teachers can shape the output by ensuring pedagogical alignment with curricular goals, adjusting the complexity and style of each question, and refining the language and clarity for the specific group of learners. This tool does not replace your expertise but accelerates your workflow, allowing you to dedicate more time to effective teaching.

Tips for Writing Effective Prompts

Crafting the right prompt is crucial to getting the most out of the LearningLemur AI for math tool. Think of it as briefing a colleague or delegating to a student teacher. Be clear, be specific, and include the following:

Specify the exact math topic you want to assess in your students. For example, indicate whether you want exercises on fractions or polynomials.
Define the educational level or age of your students to tailor the difficulty. Questions should be accessible yet challenging, matching the knowledge and skills of your learners.
Add any constraints. If you have specific requirements, such as simplifying answers or using decimals only, be sure to include them. This ensures the generated questions meet your pedagogical or exam criteria.
Optionally, use word problems to add context and generate relatable scenarios. Adding context or real-life situations makes problems more engaging and easier to understand for students. For instance, framing a problem in the context of shopping, travel or science helps connect math to everyday life and increases motivation.

Example Prompts

“Create 6 word problems on solving linear equations for Grade 9. Include at least one with negative coefficients and require students to simplify x.”

“Create 10 questions on simplifying algebraic expressions for 11-year-old students. Include at least one question involving the distributive property and another with factoring.”

“Create 5 questions for first-year university math students on calculating the rank of a matrix using elementary row operations.”

Use Cases in the Real Classroom

The LearningLemur math question generator adapts across levels and contexts:

For daily practice, generate variations to reinforce a concept.
For formative assessment, tailor difficulty and format.
For homework, instantly provide differentiated sets by level.
For exam prep, create question banks by topic or skill.

And since you can edit anything, it integrates seamlessly into any teaching style.

Diverse Learning Needs: Easily Tailor Content to Students’ Needs

From the teachers’ perspective, LearningLemur offers great flexibility, as educators with limited time or resources in creating math problems can now generate high-quality content more quickly. These advantages are particularly evident for teachers managing large classes, as they can personalize content while reducing preparation time.

Additionally, LearningLemur enables teachers to address the diverse needs of their students by creating customized quizzes and exercises tailored to different skill levels and learning requirements. Teachers can also assign targeted homework tasks, allowing students to strengthen specific competencies where they may need additional support.

Empowering Teachers Through Intelligent Automation

AI for math isn’t about replacing educators: it’s about empowering them. LearningLemur is a clear example of how AI question generators can reduce friction in the creative process while leaving full pedagogical control in teachers’ hands.

Whether you’re battling writer’s block or trying to generate multiple versions of a quiz quickly, this tool gives you a solid head start.

Try the AI of LearningLemur now

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The Role of Technology in Modern Mathematics Education

Post author By Aida Alsina
Post date June 12, 2025

Technology is no longer a guest in the classroom: it’s become a core part of the educational experience. And in the world of mathematics, it’s proving to be a powerful ally. From interactive whiteboards to adaptive AI systems, technology in math education is reshaping the way students learn and how teachers teach.

But what are the tools driving this transformation? And how can enterprises and institutions make the most of them while navigating the inevitable challenges?

In this article, we break down the evolution of digital learning tools, explore their practical impact, and look ahead to where educational technology trends are taking us.

How Technology Has Evolved Math Education

There was a time when learning math meant pencil, paper, and a lot of chalk. Today, that equation includes dynamic software, mobile apps, and real-time collaboration platforms. Over the years, technology has evolved from a classroom supplement to a strategic cornerstone of math education.

What changed? The tools got smarter, more intuitive, and better aligned with pedagogical goals. Technology in math education now means:

Receiving instant feedback and adaptive assessments.
Engaging in problem-solving through interactive scenarios.
Collaborating beyond borders with cloud-based platforms.
Visualizing abstract concepts through simulations.
Helping students to reduce math anxiety.

Digital tools for math teaching are not only enhancing understanding but also making math more approachable, flexible, and inclusive.

Key Technologies Enhancing Math Learning

From immersive 3D platforms to algebra systems that respond in real time, the toolbox for math educators is broader and more innovative than ever. Let’s look at some of the most impactful digital learning tools reshaping classrooms today.

AI and adaptive learning

Artificial intelligence is reshaping math education by enabling more personalized and adaptive learning experiences. By analyzing individual performance, adaptive systems deliver tailored exercises, offer real-time hints, and adjust difficulty levels on the fly. Many platforms now include AI-driven features to support this responsiveness.

A clear example is MathType handwriting, a tool that uses artificial intelligence to convert handwritten expressions into clean, digital equations ready to be used in your documents. These innovations represent a step forward in simplifying the transition from analog input to digital output and supporting more intuitive interaction with mathematical content in digital learning environments.

Graphing calculators and CAS tools

Graphing calculators and Computer Algebra Systems (CAS) allow students to explore equations, functions, and transformations in real time. These solutions shift the focus from mechanical calculation to analytical insight. A prime example is CalcMe, Wiris’ JavaScript-based CAS, which enables users to perform complex calculations and visualize mathematical concepts interactively across various devices.

Interactive software

Visual and interactive platforms offer higher engagement. Programs like WirisQuizzes and Learning Lemur enable learners to build, manipulate, and visualize mathematical models.

With MathType, creating complex equations becomes intuitive, helping students and teachers focus on the math, not the formatting.

LMS integration

Platforms such as Moodle and Canvas have become exponentially more powerful with integrated digital math tools. Embedding the previously mentioned platforms within these systems turns traditional coursework into interactive, dynamic experiences that meet learners where they are.

Online collaborative platforms

Math isn’t always a solo activity. Digital platforms make it easy for students to co-create solutions, share ideas, and solve real-world problems together, whether in the classroom or worldwide. Working in teams, sharing solutions, and learning from peers isn’t just good practice; it’s real-world preparation.

Challenges on the Digital Road

Of course, no transition comes without friction. Schools and universities face several challenges in adopting digital tools for math teaching:

Access inequality: Not every student has reliable internet or a personal device.
Teacher training gaps: Tools are only as effective as those using them.
Overreliance risk: Tech should enhance, not replace, mathematical reasoning.
Privacy concerns: Data protection must be part of the planning.
Maintenance and cost: Edtech isn’t a one-time purchase; it’s an evolving ecosystem.

These issues are real, but not insurmountable. They call for thoughtful planning, continuous support, and strategic investment.

Smart Implementation: Making the Most of Digital Tools

What does it take to ensure technology truly enhances math education? How can enterprises and institutions deploy digital learning tools effectively while avoiding common pitfalls?

Start with clear goals. Tech without purpose is just noise. Define your outcomes first, then choose solutions that help achieve them.
Empower teachers and invest in training. Teachers need to feel confident, not just compliant, when using digital tools for math teaching.
Ensure access, make inclusion a priority. Equitable access to devices and software should be foundational, not optional.
Blend old and new. Combine traditional instruction with digital methods. Balance is key: let tech support, not replace, core learning.
Safeguard data. Choose platforms with robust privacy protocols. Make sure every tool complies with education data standards.
Encourage exploration. Let students use technology to tinker, test, and create. Math becomes more meaningful when they take ownership of their learning.

Looking Ahead: What’s Next in EdTech for Math?

The pace of innovation is only accelerating. These educational technology trends are already reshaping what’s possible:

Smarter AI: Learning systems that adapt in real time based on student engagement.
Blockchain records: Tamper-proof credentials that simplify academic verification.
Game-based curricula: Turning math practice into immersive, story-driven experiences.
Immersive AR/VR: Math will move off the page and into the student’s physical space, making abstract ideas concrete.
Global classrooms: Seamless, multilingual collaboration across borders.
Modular microlearning: Bite-sized lessons for modern attention spans.
Accessibility: Digital tools are increasingly designed to support learners with disabilities or diverse learning needs, including visual impairments, dyslexia, and motor challenges.

How Wiris Is Leading the Change

At Wiris, we believe technology in math education should empower without overwhelming. That’s why we’ve built solutions that integrate seamlessly into classrooms while amplifying learning outcomes.

WirisQuizzes: create auto-graded, math-rich assessments inside your LMS.
MathType: author complex math expressions with ease, across any device.
Learning Lemur: generate math content and adapt the learning process to individual student needs.
CalcMe: perform complex calculations and visualize mathematical concepts interactively across various devices.

Whether you’re teaching derivatives or designing STEM assessments, Wiris helps educators turn digital potential into real progress.

Final Thoughts

The classroom is changing, students are changing, technology is changing, and math education must evolve to meet the moment.

By thoughtfully and strategically integrating digital learning tools, schools can unlock deeper understanding, greater engagement, and lifelong skills. At Wiris, we’re proud to help educators worldwide navigate this journey.

Ready to bring your math instruction into the future? Discover how Wiris can help you lead the way with cutting-edge educational technology.

Learn More About Wiris Solutions

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How to Calculate the Area Between Two Curves?

Post author By Aida Alsina
Post date June 5, 2025

Step-by-step Solution to an Oxford Access Exam Problem.

The Oxford University access exams are famous for their difficulty and for challenging students with mathematical problems of high complexity. In this article, we will focus on a 2020 exam problem that involves calculating the area between two curves. Through a clear and detailed analysis, we will break down each step necessary to approach and solve this mathematical challenge.

If you are interested, feel free to also visit our other article, “How to solve an equation of degree 16”, where we explore another challenging Oxford exam problem.

The problem statement is as follows:

A line is tangent to the parabola y=x² at the point (a,a²), where a>0.

The area of the region bounded by the parabola, the tangent line, and the x-axis equals

(a) a²⁄₃, (b) 2a²⁄₃, (c) a³⁄₁₂, (d) 5a³⁄₆, (a) a⁴⁄₁₀

This is a classic calculus problem that, as we will see, requires the use of both derivatives and integrals. Don’t worry; we’ll break it down into three clear and simple steps. Let’s get to it!

How to Find the Equation of the Tangent Line?

Before starting with the calculations, let’s visualize the problem. Below, you’ll see a graph illustrating the parabola y=x² and the tangent line at the point (a, a²), where a>0. In this graph, we can see how the tangent line touches the curve of the parabola at a single point without crossing it.

Now, to proceed, we first need to find the equation of the tangent line at that specific point. Since the tangent touches the parabola at point (a, a²), we need to determine both its slope and its equation.

This is where the derivatives come into play. The slope of the tangent at any point on the parabola is given by the derivative of the function y=x². We start by calculating the derivative: dy/dx=2x

This means that the slope of the tangent at any point on the parabola is 2x. To find the slope at the tangent point (a, a²), we substitute in the derivative:

Therefore, the slope of the tangent line at point (a, a²) is 2a.

Now that we have the slope, we can write the equation of the tangent line using the formula for the line in its form:

y – y₀ = m (x – x₀)

Where m is the slope of the line, and (x₀ , y₀) is the point through which the line passes. In our case, the point of tangency is (a, a²), and the slope is 2a, so we substitute these values in the formula for the line:

y – a² = 2a (x-a)

y – a² = 2ax – 2a²

y = 2ax – a²

So, the equation of the tangent line at the point (a, a²) is:

y = 2ax – a²

Find the Intersection Points

Now that we have the equation of the tangent, we need to find the points of intersection between this tangent and the x-axis, that is, when y=0. This will allow us to determine the limits of the region whose area we want to calculate.

The area to be calculated is marked in orange in the graph below.

Substituting y=0 in the tangent equation and solving for x we obtain:

0= 2ax – a²

2ax = a²

x = a/2

So the tangent cuts the x-axis at point

P=(a/2, 0 )

Calculate the Area Between the Parabola, the Tangent and the X-axis

Once we have the equation of the tangent and we know the points of intersection, we can proceed to calculate the area of the region bounded by the parabola, the tangent and the x-axis. For this, we use the definite integral, which allows us to find the area between two curves.

The area we want to calculate is obtained by subtracting two regions. In the first graph, you can see the total region we are interested in. However, part of this area is specifically delimited in the second graph, which we must subtract to be left with only the part we are looking for.

Let us calculate both integrals:

a) Integral of graph 1:

definite integral of x squared from 0 to a. It equals the integral from 0 to a of x squared dx, which is equal to x cubed over 3 evaluated from 0 to a, resulting in a cubed over 3.

b) Integral of graph 2:

Now that we have solved all the integrals, we subtract the results to obtain the total area:

In conclusion, the area of the region bounded by the parabola y = x², the tangent y = 2ax – a² and the x-axis, is:

a sup 3 / 12

Therefore, option c is the correct answer

And so we come to the end of this fascinating problem! If you enjoyed this analysis or found it useful to better understand the solving process, feel free to share it with other math enthusiasts!

Now we challenge you: do you dare to calculate the area formed by the curve y = x³ and its tangent at the point (1,1)? Share your solution or your ideas in the comments. We’d love to hear your approach!

Try WirisQuizzes to generate this exercise or others!

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Engaged Students, Better Results: La Salle’s Math Learning Platform With WirisQuizzes

Post author By Aida Alsina
Post date May 30, 2025

From Textbooks to a Personalized Digital Math Learning Platform

When La Salle San Ildefonso set out to replace printed math textbooks with digital tools, they needed more than just online worksheets—they needed a complete math learning platform. With WirisQuizzes integrated into Moodle, their teachers built a collaborative system that supports independent learning, personalized feedback, and scalable assessment.

What started as one teacher’s initiative has become a model adopted across multiple La Salle campuses, reshaping how students engage with math.

The Challenge

When La Salle San Ildefonso first decided to phase out physical math textbooks, they faced a critical question: How do we offer high-quality math learning in a wholly digital format without sacrificing feedback, rigor, or accessibility?

As teacher Javier Melchior explained, early digital tools offered little beyond binary right-or-wrong grading

“The only feedback we could give was whether the answer was right or wrong. That wasn’t enough”, Javier explained. ”What students needed was guidance – not just the solution, but the how behind the solution”.

This insight became a turning point for La Salle’s approach to digital education. It wasn’t just about digitizing content but about ensuring that students could learn actively and reflectively within a new system.

At the same time, the shift to digital demanded a scalable infrastructure for:

Creating large volumes of exercises quickly.
Providing real-time, personalized feedback.
Supporting teachers across varying levels of technical proficiency.
Giving students the tools to practice independently and meaningfully.

Other platforms like Google Classroom, while helpful for document sharing, lacked the sophistication for in-depth math evaluation and formative feedback.

The Solution

That’s when WirisQuizzes came into focus—not just as a plugin but as the core engine behind La Salle’s custom-built math learning platform.

Led by Javier and math teacher Guillermo Pérez González, the department began building a shared bank of exercises directly within Moodle using WirisQuizzes. These weren’t just questions—they were intelligent, dynamic exercises that could offer step-by-step feedback, track partial understanding, and regenerate with different data for unlimited practice.

“We discovered the true power of WirisQuizzes when we realized we could break down a problem into parts,” said Javier. “It’s not just about getting the final answer. If a student does the first 20% right, the platform can recognize and score that.”

WirisQuizzes became more than an assignment tool for students—it was their personal study assistant. “Now they ask us, ‘Is there a quiz for this topic?” Guillermo shared. “It’s become a natural part of how they study and prepare.”

The Implementation Process

La Salle’s implementation of WirisQuizzes was a grassroots initiative that grew into an institutional innovation. Here’s how it evolved.

Assessment

The shift began nearly a decade ago when La Salle removed printed textbooks for some math levels. With a digital-first model on the horizon, Javier started exploring alternatives. “I found WirisQuizzes online by chance,” he said. “And after testing its capabilities, I pitched it to my colleagues. From there, it just took off.”

Customization

Using Moodle’s system-level question bank, the teachers created hundreds of questions organized by topic: polynomials, functions, geometry, probability, and more. Every question included custom feedback. “Almost all our questions show the solution process,” said Javier. “This makes it easier for students to learn independently—even their parents can follow along.”

Importantly, this work was collaborative. Teachers built and reused each other’s content. “We never assigned categories formally,” said Guillermo. “If someone had free time, they’d jump in and add to the bank. It’s a shared resource for everyone.”

Deployment

WirisQuizzes was deployed through Moodle across the school. Teachers could:

Select questions from a shared bank.
Build quizzes quickly.
Offer time-limited assessments or practice exercises.
Use quizzes for classwork, homework, or even during substitute hours.

The school also exported question banks to other La Salle campuses in Tenerife, Madrid, and Gran Canaria. Adoption varied depending on each site’s IT support and teacher training, but the system was designed for easy export/import.

Training

While many core users were math or STEM teachers with programming experience, others found the “programming” aspect intimidating. Guillermo acknowledged this gap. “Some teachers hear ‘programming’ and think of C++,” he joked. “But it’s not like that. They’d see how simple it is if they just took one step.”

To address this, La Salle and Wiris discussed tailored training sessions that start simple, focusing on quiz-building basics before moving into more advanced logic and scripting.

The Results

La Salle’s math learning platform, powered by WirisQuizzes, delivered transformative results across several dimensions.

Improved Student Autonomy & Motivation

Students began voluntarily requesting quizzes to practice new topics. “It’s like they have a personal tutor,” said Javier. In one memorable example, he logged in on a weekend to find nine students had already completed a practice quiz and scored a perfect 10. “They were competing healthily. It became fun.”

Collaborative Efficiency Among Teachers

The math department created a bank of over 600–700 questions, reusable across grade levels and subjects. This significantly reduced prep time while increasing question quality and consistency.

Parental Engagement and Transparency

Because quizzes included feedback, parents could follow along, even if they weren’t strong in math. “Some told us they were thrilled to see their child working independently,” Guillermo shared.

Formative Assessment at Scale

Teachers used quizzes to diagnose student progress week by week. “In Bachillerato, I might run three quizzes weekly,” said Guillermo. “And they all count toward the grade. It keeps everyone working consistently.”

“WirisQuizzes isn’t just a quiz engine,” Javier added. “It’s a full feedback system. It’s the backbone of our math learning platform.”

The Future of WirisQuizzes at La Salle

La Salle continues to expand the use of WirisQuizzes beyond math:

They’ve developed questions for economics, technology, and even geography.
Some teachers are exploring how AI-based feedback could provide personalized remediation paths.
A growing number of educators across La Salle’s network are joining upcoming training sessions to learn how to build questions and customize feedback.

They’ve also expressed interest in exploring remote proctoring, integrating with other LMS platforms like Google Classroom, and co-developing content with other schools.

Rounding Off

La Salle’s transformation is a shining example of how a school can build its own collaborative, intelligent, and scalable math learning platform. By integrating WirisQuizzes into every level of instruction—from the classroom to home study—teachers empowered students with autonomy, parents with transparency, and the school with data-driven teaching strategies.

What started as one teacher’s experiment has become a system-wide initiative that is changing how La Salle teaches and how students learn.

👉 Want to build your own feedback-driven math platform?

Get in touch today

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Towers of Hanoi: A Math and Programming Challenge

Post author By Aida Alsina
Post date May 21, 2025

Towers of Hanoi: A Math and Programming Challenge

The Towers of Hanoi is one of the most iconic puzzles in the field of mathematics and computer science. With its simple structure and transparent rules, this problem has captivated mathematicians and amateurs alike, making it an intriguing challenge since its creation.

According to the legend, the Towers of Hanoi were conceived by a group of monks in an ancient temple in India. In the center of this temple, three golden pillars stood, upon which rested 64 disks of various sizes, arranged in the first pillar in an orderly fashion: the largest disk at the base, with progressively smaller disks stacked on top, until reaching the smallest disk, placed at the top.

The monks’ objective was to move all the disks from the first pillar to the third, using the central pillar as an auxiliary. To accomplish this task, they had to follow a strict set of rules:

Only one disk can be moved at a time.
A disc can never be placed on top of a smaller disc.

Legend says that when all the disks were correctly transferred to the last pillar, the world would come to an end. However, how close is the end of the world, really? How long would it take to complete this challenge?

This challenge was officially introduced by the French mathematician Édouard Lucas in 1883 as part of his research in number theory and quickly became a popular problem.

Next, we will analyze the solution of this problem from a mathematical perspective, breaking down its key principles and explore how it transforms into an interesting programming challenge.

Recursion as a Solution to the Puzzle

The challenge of the Towers of Hanoi is not only in moving the disks from one pillar to another following the strict rules but also in calculating how many moves are needed to solve the problem of moving n disks from the origin pillar to the destination pillar.

The most efficient way to approach this problem is through recursion, a fundamental technique in programming. Recursion allows the problem to be divided into smaller subproblems, where the solution of each subproblem leads to the solution of the original problem.

In the case of the Towers of Hanoi, recursion applies naturally, and we can divide the problem of moving n disks into three steps:

Move the first n – 1 disks from the origin pillar to the auxiliary pillar, using the destination pillar as the auxiliary.
Move the largest disc (the disc n) from the origin pillar to the destination pillar.
Move the n – 1 disks from the auxiliary pillar to the destination pillar, using the origin pillar as an auxiliary.

We can repeat this pattern recursively, reducing the number of discs with every step until we reach a base case where only one disk remains, which we just directly move.

In mathematical terms, the minimum number of moves to solve the problem with n disks equals the number of moves needed to solve it for n – 1 disks, plus one to move the largest disk, plus the number of moves needed to solve it for n – 1 disks again. This formula yields the result:

TH( n ) = TH( n – 1 ) + 1 + TH( n – 1 ) = 2TH( n – 1 ) + 1

Where TH( n ) represents the number of moves needed to solve the problem with n disks.

From the above formula, we will substitute each TH( i ) by its corresponding expression until we reach TH( 1 ) . Then, we will replace TH( 1 ) by 1, since, with only one disc, it is possible to move it directly to the destination pillar, which justifies that the minimum number of moves in this case is 1.

Let’s see the expression developed:

We first substitute TH( n – 1 ), and we obtain:

TH( n ) = 2TH( n – 1 ) + 1

TH( n ) = 2( 2TH( n – 2 ) + 1 ) + 1 = 2²TH( n – 2 ) + 3

Now we substitute TH( n – 2 ):

TH( n ) = 2²TH( n – 2 ) + 3

TH( n ) = 2²( 2TH( n – 3 ) + 1 ) + 3 = 2³TH( n – 3 ) + 7

If we continue substituting in this way, we will arrive at a general formula:

TH( n ) = 2^k TH( n – k ) + (2^k – 1)

Where k is the number of steps backward in the recursion.

For k = n – 1 we arrive at TH( 1 ), the base case:

TH( n ) = 2²TH( n – 2 ) + 3

We know that TH( 1 ) = 1, then we obtain:

TH( n ) = 2^{n – 1} · 1 + ( 2^{n – 1} – 1 ) = 2ⁿ – 1

We have shown that the number of moves needed to solve the Towers of Hanoi with n disks is:

TH( n ) = 2ⁿ – 1

Solving the Puzzle Using Mathematical Induction

Another way to approach the solution to the Towers of Hanoi puzzle is to use mathematical induction, a technique used to prove that a statement is true for all natural numbers, based on two fundamental steps: the base case and the inductive step.

In this case, the goal is to show that to move disks from the first pillar to the third pillar, 2ⁿ – 1 moves are required. Let’s break this process down with a demonstration by induction.

Base case:

Suppose we have only one disk. In this scenario, it is clear that only one move is needed to move the disk from the source abutment to the target abutment. That is, for n = 1, the number of moves required is 2¹ – 1 = 1, which is true.

Inductive step:

Now, suppose that the statement is true for n = k, i.e., that to move k disks requires 2^k – 1 moves. What we must prove is that the statement is also true for n = k + 1.

To move k + 1 disks, we first need to move the upper disks from the origin pillar to the auxiliary pillar, using the destination pillar as an auxiliary. According to our inductive hypothesis, this will take 2^k – 1 moves.

Next, we move the larger disk (the disk k + 1) from the origin pillar to the destination pillar, which requires just 1 move.

Finally, we move the k disks from the auxiliary pillar to the destination pillar, using the origin pillar as an auxiliary, which will also take 2^k – 1 moves, according to our inductive hypothesis.

Therefore, the total number of moves required to move k + 1 disks is:

(2^k – 1) + 1 + (2^k – 1) = 2^k+1 – 1

Thus, we have shown that if the statement is true for n = k, it is also true for n = k + 1.

As we have observed, it is true for 1. By mathematical induction, we can conclude that for n disks, the minimum number of necessary moves is 2ⁿ – 1.

Going back to the legend, if the monks were extremely fast and could move a disk in just one second, we could calculate how long it would take to complete the challenge with 64 disks. According to the formula, the total number of moves would be 2⁶⁴ – 1, approximately 18.4 quintillion moves. With each of these taking just 1 second, this gives us 18.4 quintillion seconds. Converting these into years, we get that the total time would be approximately 581.4 billion years. So, if the legend turned out to be true, the end of the world would still be a long way away!

CalcMe formula We note that the function describing the number of moves needed to solve the Towers of Hanoi is exponential, meaning that it grows rapidly as we increase the number of disks. In the following graph, we can visually see how this function spikes, clearly illustrating the exponential growth.

graphic formula

This visual support allows us to confirm, once again, that completing the challenge with 64 discs would take an extraordinary amount of time.

If you liked this mathematical challenge and you were surprised by its solution, don’t hesitate to share this article! And if you dare, test your skills by solving the Towers of Hanoi puzzle: will you manage to move the disks in the exact number of steps? Let us know your experience!

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AI in Mathematics: The Revolution in Scientific Writing

Post author By Aida Alsina
Post date May 6, 2025

Artificial Intelligence (AI) is a field of computer science that seeks to develop systems able to perform tasks that traditionally require human intelligence, such as logical reasoning, problem solving, and learning. Based on advanced algorithms and machine learning models, AI can analyze large volumes of data, identify complex patterns and adapt autonomously to new contexts.

In mathematics, these capabilities are transforming the way knowledge is generated, verified, and communicated. From automating demonstrations to generating detailed explanations, AI is facilitating the work of researchers, teachers, and students. In addition, its ability to process natural language is improving the writing of mathematical papers, making texts more accessible, accurate and structured.

In this article, we will explore how AI in math is revolutionizing mathematical writing and how Wiris’ tools are part of this advancement.

The Role of AI in math and the Creation and Editing of Mathematical Content

Writing scientific papers in mathematics has always been a challenge due to the complexity of their notations and symbols. Traditionally, mathematicians have relied on handwriting or the use of complex typesetting languages. However, with the advent of AI, the process of creating and editing scientific documents has evolved exponentially.

Artificial intelligence has brought multiple benefits to the writing of mathematical papers, facilitating both the writing and the editing and publication of academic papers. Below are some of the main ways in which AI is impacting this area.

Handwriting recognition

One of the most significant advances has been the development of systems capable of interpreting and digitizing handwritten notes. Traditionally, mathematicians wrote their equations and proofs on paper, which was time-consuming if they needed to transcribe them into a digital format. With AI-based handwriting recognition tools, such as those built into MathType, it is now possible to directly handwrite equations into our touch devices and automatically convert them into editable digital text.

This not only saves time but also reduces transcription errors and allows you to work more efficiently. In addition, this type of technology is especially useful for students and teachers, as it facilitates the conversion of notes into organized documents without the need for manual typing.

Formula automation

In the creation of mathematical articles, writing formulas accurately is essential. Traditionally, this required specialized tools that, while powerful, could be complex and demanded a deep understanding of their syntax and structure, such as LaTex.

Advancements in technology have enabled the development of tools like MathType, which simplify equation writing without the need for coding. Instead of memorizing commands and code structures, users can simply enter their formulas through an intuitive graphical interface. This democratizes access to mathematical writing, allowing more people, regardless of their technical expertise, to create high-quality mathematical documents.

Another key aspect is the automatic correction of errors in equations. AI in math can identify inconsistencies in formula writing and suggest real-time corrections, preventing mistakes that could compromise the validity of an academic paper.

Optimization of editing

Another benefit that AI has brought is the optimization of the editing of mathematical documents. In the past, mathematicians and scientists had to manually check their documents for errors in equations, notation and text structure. Today, there are AI algorithms embedded in editing programs that can analyze the consistency of mathematical expressions, suggest improvements in writing and detect inconsistencies.

AI-based writing assistants, such as those integrated into advanced text processors, can help improve the clarity and readability of mathematical articles. In addition, they provide stylistic recommendations to give the paper a logical and easy-to-follow structure, which is crucial in academic paper writing.

The integration of AI into mathematical writing is not just an evolution but a revolution that is transforming the way we create, edit, and share scientific knowledge. Tools like Wiris, with their powerful capabilities, are making mathematical writing more accessible, efficient, and error-free. As these technologies continue to advance, they will further bridge the gap between human intuition and machine accuracy, making the work of researchers, teachers, and students alike easier. If you found this exploration interesting, share it with your peers and join the conversation about the future of AI in math.

If you’d like to find out more about our products, please feel free to get in touch with our Sales team at sales@wiris.com for more information.