Post type: MathType

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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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.

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Wiris in the Classroom: Learn and Practice with Unlimited Exercises

Post author By Aida Alsina
Post date April 16, 2025

Learning mathematics requires more than just understanding theories: the key to mastering complex concepts and honing skills lies in continuous practice. However, traditional methods often limit the learning process: once the student has solved a problem, the challenge disappears, and exams often repeat the same questions, thus losing their value as an assessment tool. Imagine having the possibility to generate unlimited exercises with random variables, adapted to the needs of each student. With Wiris, this is now possible.

In addition, it has an automatic correction functionality which means that while the exercise is being generated, the answer is calculated simultaneously. This allows instant feedback, making it easier to learn and correct mistakes immediately.
For teachers, this tool offers a key advantage: the possibility of generating an infinite number of exercises, with each one being unique. In this way, teachers can create varied and personalized learning experiences, ensuring that students face new challenges every time they practice.

For their part, students enjoy constant practice which allows them to reinforce their skills without the worry of repeating the same exercises. In addition, this ability to generate unlimited exercises gives them excellent preparation for exams.

Practical example: Solving a system of linear equations

To illustrate how this tool works and how it can be used in the classroom, we will focus on solving systems of linear equations. We chose these types of problems because they represent a common mathematical challenge faced by all students and offer the possibility to explore different problem-solving methods. In addition, systems of equations can have various characteristics, such as being compatible, incompatible, or indeterminate, allowing students to work with a wide range of situations.

Let us imagine that we have the following system of linear equations with three unknowns:

We are going to solve it using Gauss’s Method. The first step is to write the system in the form of an augmented matrix:

Now, we will apply elementary operations to reduce the matrix to its row echelon form.

Therefore, we obtain:

From equation 3 of the system, we find the variable z:

Unlimited generation of systems of linear equations with random variables

Now that we have solved this system, for further practice, we would like to be able to generate new systems of equations of the same form but with random values.

These systems would take the following form:

Below, we will show you a code example to implement this functionality and generate given random systems of equations. With Wiris, it is totally possible!

r() := random(-4,4)

sol = [r() with i in 1..3]

[a,b,c] = sol

repeat

A = [[r() with i in 1..3] with j in 1..3]

until determinant (A) != 0

b = A * sol

ec = {}

for i in 1..3 do

ecaux = A.i * [x,y,z]

ec = append(ec, ecaux = b.i)

end

{ec1, ec2, ec3} = ec

r() := random(-4,4): This function generates a random number between -4 and 4.

sol = [r() with i in 1..3]: Here, three random values are generated that will represent the solutions of the system of equations.

[a,b,c] = sol: The generated values are assigned to the unknowns of the system.

repeat…until determinant(A) != 0: This block ensures that the coefficient matrix A is invertible (i.e., its determinant is not zero), which guarantees the system has a unique solution.

b = A * sol: The result vector b is calculated by multiplying the coefficient matrix A by the solutions sol

for i in 1..3 do: In this cycle, the three equations are generated, using each of the rows of the matrix A and the solutions sol.

{ec1, ec2, ec3} = ec: Finally, the three generated equations are stored in the variables ec1, ec2, and ec3, ready to be used in new exercises.

Using WirisQuizzes and implementing this code as a question, we can obtain the following results:

Reloading another system with the symbol “=” generates a new set of equations, as shown below:

For more details, you can consult the complete product documentation here.

If you have found this article useful, we would love you to share it with other colleagues, teachers or students. We would also be delighted to hear your opinion and any suggestions you may have.

Check Out WirisQuizzes

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What is the Collatz Conjecture?

Post author By Aida Alsina
Post date February 28, 2025

What is the Collatz Conjecture?

Mathematical conjectures are riddles that challenge the human mind: seemingly simple problems that, despite having been verified in millions of cases, still lack a proof. They are seemingly intuitive questions that hide unexpected complexity.

The fascinating thing about conjectures is that, in many cases, they can be empirically verified i.e., verified by an enormous number of numerical examples. However, this verification does not satisfy mathematicians, who seek a theoretical proof, a solid proof that validates the statement in its entirety. Without such a proof, the conjectures remain open challenges.

The Collatz conjecture is a perfect example of this type of mathematical mystery. It was proposed in 1937 by Lothar Collatz and its statement is as follows:

Choosing any positive integer we will apply the following steps:

➔ If the number is even, we will divide it by 2.

➔ If the number is odd, multiply it by 3 and then add 1.

➔ Repeat the process with the new number obtained.

The conjecture says that, no matter what number you start with, you will always get to number 1. Once you get to 1, the process repeats indefinitely: 1 →4 →2 →1

Formally, this can be written as the function f=N →N defined as

Two illustrative examples of the Collatz Conjecture

Let’s look at a couple of simple examples.

Let’s take the number n=6 and apply the steps of the Collatz conjecture:

6 is even so we divide by 2: 6/2=3
3 is odd so we multiply by 3 and add 1: 3·3+1=10
10 is even so we divide by 2: 10/2=5
5 is odd so we multiply by 3 and add 1: 3·5+1=16
16 is even so we divide by 2: 16/2=8
8 is even so we divide by 2: 8/2=4
4 is even so we divide by 2: 4/2=2
2 is even so we divide by 2: 2/2=1

Once we reach 1, the process is repeated: 1 →4 →2 →1.

So we observe that starting with n=6 the conjecture is satisfied.

Let us now take the number n=21 and apply the steps of the Collatz conjecture:

21 is odd so we multiply by 3 and add 1: 21·3+1=64
64 is even so we divide by 2: 64/2=32
32 is even so we divide by 2: 32/2=16
16 is even so we divide by 2: 16/2=8
8 is even so we divide by 2: 8/2=4
4 is even so we divide by 2: 4/2=2
2 is even so we divide by 2: 2/2=1

Again, we arrive at number 1.

Both examples have reached the number 1, but in a different number of steps. In fact, although 21 is a larger number, it reached 1 in fewer steps than 6. This conjecture has been tested for an incredible number of numbers, up to more than 2ˆ60 = 1.152.921.504.606.846.976 cases, without finding any counterexample. However, it still remains a mystery whether there is a number that does not satisfy the conjecture.

Graphical representations of the Collatz conjecture

The directed graph of orbits is a visual representation that facilitates the understanding of the behavior of numbers under the rules of the Collatz conjecture. In this graph, each number is represented as a node, and the connections between them show the steps followed by the conjecture process. When following the sequence of a number, the nodes are connected by arrows that indicate how the number is transformed at each step. Although even numbers are generally omitted from the representation for simplicity, the graph illustrates how the numbers “orbit” around certain cycles, such as 1 →4 →2 →1.

In the image above, we can see an example of the directed graph of orbits, where we have highlighted in black the numbers 3 and 21. The number 3 refers to the sequence of 6, since in this graph the even numbers are omitted to simplify the visualization.

We note that 21 quickly reaches 1, since it only passes through even numbers on its way, as we saw in the previous example. On the other hand, the number 6 first becomes 3, then pass through the number 5, which will finally take it to 1 after several passes through even numbers.

It is also interesting to note that the number 9, although relatively small, follows a longer sequence: it goes through the numbers 7, 11, 17, 17, 13 and 5, and finally reaches 1, after a total of 19 steps.

Below is a graph in which the X-axis shows the different initial integer values, while the Y-axis represents the number of iterations required for each number to reach the value 1.

In summary, the Collatz conjecture is an apparently simple problem that, to this day, still has no formal proof. Although it may seem a statement of little relevance or practical utility, this conjecture has applications in various fields, such as number theory, cryptography, algorithm analysis and artificial intelligence. In these fields, the study of complex sequences and their behavior under specific rules can offer valuable insights for solving larger problems and understanding mathematical patterns.

Have you been fascinated by the simplicity and mystery of the Collatz conjecture? Now it’s your turn: take the number 27 and start the sequence. How many steps will you need to get to 1?

If this challenge got you, don’t keep it to yourself! Share it with others who are curious about mathematical riddles and find out together who can solve the challenge the fastest.

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How to solve an equation of degree 16

Post author By Aida Alsina
Post date February 6, 2025

Step-by-step solution of an Oxford access exam problem

Every year, thousands of students face the challenge of the admission exams to enter the prestigious University of Oxford, a process that tests not only their knowledge, but also their ability to solve under pressure. In particular, the math exam is known for its complexity, posing problems of all kinds, from solving equations to questions of logic, advanced algebra, calculus and number theory. In this article, we will explore one of the problems presented in October 2023.

The problem poses the following question: how many real solutions does the following equation have?

Do you dare to solve the equation before reading the complete solution in the blog?

At first glance, the statement seems simple: an equation with a single variable, where all the numbers involved are integers between 1 and 4. However, the real difficulty lies in correctly undoing the squared parentheses, making sure to consider all possible cases.

A key detail in this type of equations is that, when squaring a number, the same result is obtained for both its positive and negative values. For example:

This occurs because squaring a number eliminates the negative sign. Therefore, when solving a quadratic equation we must take into account the two possible options: the positive and the negative number that can generate that result.

One last observation: if we were to fully develop the parentheses, we would obtain an equation of the form xˆ16+…, which indicates that the equation is of degree 16 and, at most, could have 16 real solutions for x. Therefore, we cannot rule out any of the options presented to us as possible answers.

Since for equations of degree 16 there is no systematic formula like Ruffini’s method for polynomials of degree 3, we will solve this equation by working the parentheses progressively, from the outermost to the innermost.

Specific example of how to solve an equation of degree 16

Let’s get on it!

Let’s start with the outermost parenthesis. If we define t as ((x²-1)²-2)²-3 , then we get t²=4. This gives us two possible values for t: t=2 or t=-2.

Substituting t by its original value in each case, we obtain the following results:

1. If t=2 then: ((x²-1)²-2)²-3 =2 ⇒ ((x²-1)²-2)²=5 and again, substituting r= (x²-1)²-2 we obtain that r²=5 . Let’s look at the two new cases that arise:

1.1 r=√5: substituting r for its original value we obtain (x²-1)²-2 =√5 ⇒ (x²-1)²=2+√5 and again, one last time, we substitute s=x²-1, we obtain that s²=2+√5 and observe the two results again:

When trying to calculate the square root to obtain the value of x, we would be taking the root of a negative number. As a result, the two values obtained will be imaginary numbers.

1.2 r=-√5 substituting r for its original value we obtain (x²-1)²-2=-√5 ⇒ (x²-1)²=2-√5 . Following the same reasoning as in the previous section, given that √4=2, it follows that √5>2 . This implies that 2-√5<0, and when trying to calculate its square root, we would obtain an imaginary number. Therefore, when we continue developing to obtain the value of x, this would also be an imaginary number.

2. If t=-2 then: ((x²-1)²-2)²-3 =-2 ⇒ ((x²-1)²-2)²=1 and again, substituting r= (x²-1)²-2 we obtain that r²=1 . Let’s look at the two cases:

2.1 r=1: substituting r for its original value we obtain (x²-1)²-2 =1 ⇒ (x²-1)²=3 and again, one last time, we substitute s=x²-1, we obtain that s²=3 and observe the two results again:

2.1.1 s=√3 and substituting s for its value, we finally obtain: x²-1=√3 ⇒ x²=1+√3 and therefore

2.1.2 s=-√3 substituting s for its original value we obtain x²-1=-√3 ⇒ x²=1-√3 . This subtraction is negative, since √2≈1,41 and therefore, √3>1,41 with which we deduce that 1-√3<0 , and when trying to calculate its square root, we would obtain an imaginary number.

2.2 r=-1 substituting r for its original value we obtain (x²-1)²-2 = -1 ⇒ (x²-1)² = 1 and again, one last time, we substitute s=x²-1 and we obtain that s²=1 and observe the two results again:

2.2.1 s=1 and substituting s for its value, we finally obtain: x²-1 = 1 ⇒ x²=2 and therefore

2.2.1.1 x=√2 which is a real solution.

2.2.1.2 x=-√2 which is a real solution.

2.2.2 s=-1 and substituting s for its value, we finally obtain: x²-1 = -1 ⇒ x²=0 and therefore the only possible solution is x=0 .

Let’s review all the possible real values we have obtained:

This gives us a total of 7 real solutions for x, so the correct answer is option (c).

Did you get your answer right?

Drawing this equation with our graphing tool is a simple and effective way to check the number of real solutions. Looking at the graph, you can clearly see how the curve interacts with the x-axis, confirming the previous result. Below, we show you the graph to see for yourself:

Done with the WirisQuizzes Assessment tool

Will you dare to take on more challenges in the future?

If you found this content useful or inspiring, please share it with your friends and fellow number lovers!