The Hidden Wonders of Mathematics: Power, Discovery, and Application

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Welcome to Hattennoki's blog/article series! I am Jaspreet Sethi, the author and founder of Hattennoki, and I’m excited to share my thoughts on brain science, learning, and development with all of you.

Say you're tasked with opening a locked door given the clue: "The code is the largest prime factor of 48,803."

At first glance, you might think, "I’ll never need this in real life. Besides, prime factors seem like strange things to care about. So, why spend time on this problem?"

Here’s Why: The skills you use here – breaking it down, finding patterns, using logic, and applying a systematic approach – are universal. These skills, honed through mathematics, can secure much more than doors; they form the foundation of modern cryptography, which keeps our digital world safe.

Prime numbers and their properties lie at the heart of cryptography, the science of protecting information. Every time we make a secure online purchase, send an encrypted email, or log into our bank account, prime numbers and the math behind them are working in the background to keep our data safe.

So, while you might never directly use prime factors in your daily life, understanding how they work unlocks the ability to comprehend the very technology that powers our digital world.

Math educators often hear the perennial question: "When will I ever use this?" Some learners are genuinely curious, while many others derail the class and set it off on tangents. Over the years, I have refined and developed a simple answer: “Math is not just about finding answers. It’s about learning how to ask better questions.”

The beauty of pure math, like number theory, is that it offers a deeper, often more exciting connection to the way the world works – even if those connections aren't always immediately obvious.

1. Real-World Examples Don’t Always Hook Learners

With the internet at our disposal, teachers are increasingly working hard to find applied examples to help learners understand the relevance of math. Here are a few common ones and why they might not always work:

Statistics and Probability: Many teachers explain how statistics can help you understand grades or sports outcomes, or how probability applies to gambling. But unless the learner plans to become a grader, is a huge sports fan, or has experience with gambling, they might tune out. Additionally, learners may view statistics as something useful only in specific situations, rather than as a universally applicable skill.

Algebra and Budgeting: "You’ll need algebra to calculate your taxi fare or phone bill!" While money is important, learners might not connect solving for x in 10 + 5x = 30 to a real-life situation involving fixed and variable costs. The abstract equation may feel disconnected from their current life experiences.

Geometry and Architecture: "Understanding angles will help you in architecture and designing buildings!" True, but how many high schoolers are thinking about designing buildings right now? Geometry, too, can seem like an abstract concept with little personal relevance.

These examples, though based on real-world applications, can miss the mark because they don’t resonate with the lived experiences of many learners. For some, math feels more like a disconnected exercise in rote memorization of formulas and theorems, rather than something that connects to the world they know and live in.

My foremost advice to educators is that every topic does have relevance, but let’s keep it short and simple to begin with, rather than introducing new jargon to explain existing jargon.

• Learning about compound interest might seem irrelevant until you realize that it’s how your savings grow over time – something that impacts your financial future.

• Studying quadratic equations might feel abstract, but they model real-world phenomena like the path of a basketball or the trajectory of a rocket.

• The Pythagorean theorem might not come up every day, but it's behind the geometry of an unimaginable life without computer screens, camera lenses, or bridges.

• Understanding probability can feel like a distant concept, but it underpins every lottery, weather forecast, and risk assessment you encounter daily.

• While learning about statistics might feel like just another calculation, it's the key to understanding everything from poll results to health trends.

• The idea of angles and geometry might seem irrelevant, but they determine the design of everything from video games to movie sets to skyscrapers.

• Algebra can feel like an exercise in isolation, but it's actually the foundation of all computer programming and encryption technologies, keeping our digital world secure.

• Learning about functions may seem abstract, but they’re what power the algorithms behind social media feeds, online shopping recommendations, and traffic routing.

• Math’s role in optimization may feel disconnected from daily life, but it’s at the heart of everything from delivery routes to website loading times.

• Although geometry can feel theoretical, it’s the reason why we can use GPS navigation systems to get from one place to another with pinpoint accuracy.

2. Give Access to Pure Math and Activate the Joy of Number Theory and Abstract Thinking

Number theory is an area of mathematics that is beautifully abstract, built around problems that seem simple at first but become deep and profound as you dive in. This is where the "real" joy of math lies, and it’s also where you can show learners that math isn't just about applications but about cultivating a certain way of thinking.

Here is an example of number theory: A great example of pure math is the study of prime numbers. They might seem like simple, standalone numbers, but there are a lot of connections.

Prime Numbers: These are numbers greater than 1 that have no divisors other than 1 and themselves. The first few are 2, 3, 5, 7, 11, and so on. Prime numbers are the building blocks of all numbers. Every whole number is either a prime or can be factored into primes. This is known as the Fundamental Theorem of Arithmetic. It’s like math’s version of DNA. All numbers are made from primes, and yet primes themselves are indivisible. When you study number theory, you aren’t just studying isolated problems – you’re studying the very structure of numbers, how numbers interact, and how even the simplest of number patterns can reveal the deepest truths of mathematics.

3. How Does This Connect with Kids?

If learners are questioning the utility of all this, how do we show them that this kind of abstract thinking is still valuable, even if it’s not directly "useful" in their day-to-day lives?

Math as a Way of Thinking: Number theory and other branches of pure math teach logical thinking and problem-solving skills that transfer into every area of life, even if the connection isn’t immediate. It’s about building a mindset that’s comfortable with tackling complex problems, breaking them down into manageable steps, and finding patterns that might not be obvious at first glance.

Indirect Usefulness: Sure, you might not use prime numbers directly to plan your finances or design a building. But what if I told you that cryptography (the math behind secure online communication) is based on number theory? If you use a credit card, send an email, or do anything online that involves privacy, you’re using cryptography, which relies heavily on prime numbers. That’s a tangible indirect connection to something abstract. If you want a glimpse of the math, here goes:

First, we will have to understand the basics of RSA encryption. RSA stands for Rivest-Shamir-Adleman, the last names of the three inventors of the algorithm - Ron Rivest, Adi Shamir, and Leonard Adleman. RSA is a widely used asymmetric encryption algorithm that secures data transmitted over the internet, such as in email encryption, online banking, and secure communications.

RSA is one of the most popular encryption methods used for securing online communications, such as credit card transactions, emails, and private messages. It involves two keys: a public key (which everyone can see) and a private key (which only the recipient knows).

To generate these keys, RSA uses prime numbers in the following way:

Step 1 - Choose two large prime numbers, say p and q.

Step 2 - Multiply them together to get n = p × q. The number n is used as part of both the public and private keys.

Step 3 - Calculate Euler's Totient Function: This function, denoted ϕ(n) represents the number of integers less than n that are relatively prime to n. It’s computed as:

ϕ(n) = (p−1)(q−1)

*ϕ is phi, pronounced "fye" or "fee"

*Two numbers are said to be relatively prime (or coprime) if their highest/greatest common divisor/factor (GCD or GCF or HCF) is 1. In other words, they share no common factors other than 1. For example, 8 and 15 are relatively prime because their GCD is 1. The factors of 8 are 1, 2, 4, 8, and the factors of 15 are 1, 3, 5, 15. They only have 1 as a common factor.

Step 4 - Choose an encryption exponent e such that it is coprime to ϕ (n) (this means e and ϕ(n) share no common factors other than 1).

Step 5 - Compute the decryption exponent d, which is the modular inverse of e modulo ϕ(n). This means:

e × d ≡ 1 (modϕ(n))

*This step is critical because the value of d will be used to decrypt the message.

This is a glimpse of how prime numbers ensure security. The security of RSA is based on the fact that factoring a large number n (which is the product of two primes) into its prime factors p and q is computationally difficult.

While it’s easy to multiply two large primes together, it’s extremely hard to reverse the process (factor n into p and q) when n is large. This is called the factorization problem, and it is the foundation of RSA's security.

Mathematics as Beauty: A huge part of number theory and pure mathematics is that it’s simply beautiful. The patterns, the symmetry, the way that simple questions - like “How do prime numbers work?” or “What is a repunit?” - can lead to centuries of thought and discovery. Math, in this sense, becomes a form of art, where the true joy lies in the exploration. For some learners, the thrill isn't just in the answers, but in discovering how deep the rabbit hole can go - uncovering unexpected connections, patterns, and mysteries along the way.

4. The Principle of Discovery Through Knowledge and Application

I think through multiple occurrences and epiphanies, I may have coined a new principle: The Principle of Discovery Through Knowledge and Application.

At its core, the principle is this: The more a concept or method is known and the more frequently it is applied, the more likely it is to reveal itself through additional insights, connections, and broader applications. In other words, repeated engagement with a concept leads to deeper understanding and the discovery of its relevance in diverse contexts. In simplest form, “When you use something more, you find it more often.”

Imagine you begin banking with "Golden Ratio" Bank. At first, you only notice the branch near your home or work or school where you opened the account. Then, you discover its ATM in your nearest mall. Slowly, as you continue using their services – paying bills, withdrawing cash, and using online banking – you start to notice more "Golden Ratio" Bank branches popping up around the city, more ATMs in previously unknown and unexpected places, and even partner businesses that offer discounts if you're a "Golden Ratio" Bank customer. The more you engage with Golden Ratio Bank, the more you realize how expansive their network is.

The idea that "when you use something more, you find it more often" can be understood in mathematics through the lens of familiarity, practice, and application. Essentially, when we repeatedly engage with a concept, its relevance and interconnectedness reveal themselves. This principle - "the more you use something, the more you see it" - is particularly evident in mathematics:

Repetition Builds Fluency: Practicing quadratic equations, for instance, not only sharpens problem-solving skills but also helps recognize their application in physics, economics, and other fields.

Interconnected Concepts: Mastering algebra paves the way for understanding calculus, trigonometry, and beyond. Each concept builds upon the last, creating a web of understanding.

Recurring Tools: Techniques like factorization appear across disciplines, from polynomials to cryptography.

Problem-Solving Growth: Mathematics trains the mind to dissect complex problems systematically, fostering logical thinking applicable across domains.

Generalization: Recognizing patterns across diverse problems leads to broader applications, whether calculating areas or optimizing engineering designs.

Proofs and Logical Structures: Engaging with mathematical proofs enhances logical reasoning, applicable even in fields like computer science.

5. When They Ask, “When Will I Ever Use This?”

This is a tough one, but a good response could be:

“You might not use this exact concept directly in your life or job yet, but what you’re learning right now is how to think logically, how to approach problems systematically, and how to work through abstract ideas. These skills will help you no matter what you do. Whether you’re a scientist, a musician, a writer, or even an entrepreneur, being comfortable with abstract thinking is an incredibly valuable skill. And sometimes, when you know the fundamentals of something, you find that it shows up in the most unexpected places.

For example, you might be a chef and need to adjust a recipe because you're cooking for a different number of people. The original recipe calls for 3 cups of flour for 4 people, but you’re cooking for 6. And, you use ratios. Or, you might be playing a board game with your friends, and you’re deciding whether to risk landing on a dangerous space or play it safe. You know that a 6-sided die has a one-sixth chance of landing on any given number. Without thinking about it in formal terms, you’re already using probability. You’re considering your odds of rolling a 6 (or any other number) and making a decision based on those odds."

Another great way to answer is: “You might not use this directly, but understanding how to find patterns and how to solve problems step-by-step will help you out in situations you can’t even imagine yet. You might end up working in professions where this kind of thinking is exactly what you need.”

6. The Real Joy

When you shift from just "applying" math to understanding and appreciating its pure form, you get to experience the joy of discovery. You might not be applying Pythagoras' theorem to measure the distance from your ladder to the wall, but when you’re trying to prove the Goldbach Conjecture (which says that every even integer greater than 2 is the sum of two primes), you’re diving into a world of fascinating, open-ended questions that have engaged mathematicians for centuries. Just as playing Sudoku or doing the crossword enables thinking and processing.

Mathematics isn’t just about finding answers – it’s about seeking out truths, unraveling mysteries, and experiencing the thrill of problem-solving.

So, yes, not all math has an apparent and visible "real-world" application, but that’s where the beauty lies. Through pure math like number theory, learners are developing a mindset that will serve them in ways they might not even realize until later in life. And while they might not use prime numbers directly, the act of engaging with math, seeing the world through the lens of patterns and logical structures, will open up opportunities in ways that are far more profound and lasting than just solving an applied problem.

By teaching the joy and beauty of pure mathematics, we help learners see math not as a series of disconnected facts, but as a tool for understanding the world - and beyond.

By the way, have you figured out the secret code?

References

RSA Algorithm Rivest, R. L., Shamir, A., & Adleman, L. (1978). A Method for Obtaining Digital Signatures and Public-Key Cryptosystems. Communications of the ACM, 21(2), 120-126. [https://doi.org/10.1145/359340.359342]

Cryptography and Prime Numbers Katz, J., & Lindell, Y. (2020). Introduction to Modern Cryptography (3rd ed.). CRC Press.

Mathematics in Cryptography Aumasson, J. P. (2017). The Security of RSA and Public-Key Cryptosystems. Springer.

Prime Numbers in Mathematics Hardy, G. H., & Wright, E. M. (2008). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press.

Beauty of Pure Mathematics Stewart, I. (2015). Mathematics: A Very Short Introduction (2nd ed.). Oxford University Press.

Number Theory in Cryptography Boneh, D., & Shoup, V. (2021). A Graduate Course in Applied Cryptography. Available online: https://crypto.stanford.edu/~dabo/cryptobook/

Math as a Way of Thinking Polya, G. (1957). How to Solve It: A New Aspect of Mathematical Method. Princeton University Press.

The Educational Value of Mathematics Stewart, I. (2006). The Joy of x: A Guided Tour of Math, from One to Infinity.

Applications of Geometry and Algebra Devlin, K. (2017). The Language of Mathematics: Making the Invisible Visible. W. H. Freeman and Company.

Math and Questioning Dewey, J. (1938). Experience and Education. Macmillan.

The Role of Mathematics in Cognitive Development Piaget, J. (1972). The Principles of Genetic Epistemology. Routledge & Kegan Paul.  

About the Author

Jaspreet has always been a deep thinker, a prolific writer, and a poet at heart. From an early age, she was drawn to the world of words, often expressing her thoughts and emotions through the written word. What started as personal journal entries, a safe space for her to process life’s complexities, has now evolved into published journals and thoughtful works that resonate with readers, audiences, and subscribers. Over the years, her writing has grown from intimate reflections into a powerful tool for sharing insights, sparking conversations, and inspiring others. The journey from those private musings to more formal publications is a testament to her growth as a person and her dedication to using words as a means of connection and expression.