Math Study Tips: Introduction to Number Theory

Number theory is foundational to a strong understanding of advanced mathematics, but that's not to say it's easy to learn.

Many students trip up on proofing and on expressing the correct answer succinctly. It's also often difficult to wrap your head around the purely abstract nature of many of the lemmas and corollary's present in your average number theory textbook, which can make scoring well in a number theory class highly difficult.

But like most difficult things, it's not impossible.

Doing well in number theory mostly depends on your ability to understand a few basic concepts, and then extrapolating from there.

In this guide, we give you a brief, effective set of math study tips to introduce you to number theory and help you maximize your ability to solve problems.

 

1. Foundational Number Theory Principles

Number theory has long been called the Queen of Mathematics, for the simple reason that ‘every other mathematical subject must first listen to her'.

The basic, underlying concepts behind number theory are integral to understanding more advanced subjects like algebra, calculus, encryption, computer science, and more.

At its core, number theory deals with integers (1, 2, 3) and their negatives (-1, -2, -3), and how they relate to prime numbers, numbers that are only divisible by themselves and the number one.

The reason prime numbers are so important is because they are the core features that mathematicians use to work with novel mathematical constructs; prime numbers share characteristics across their set, and these characteristics help inform us about the nature of new numbers.

Common questions that number theory deals with are things like ‘is this a prime number?' or ‘what's the best way to generate massive prime numbers?'.

One of the foundational theories in number theory is the Foundational Theorem of Arithmetic, which shows that every positive integer greater than one can be written as the product of primes.

Many algorithms have been created to verify prime-ness with high computational efficiency, and proponents of number theory use these algorithms to help create novel encryption schemes, search for patterns in prime numbers, and use integers to study other types of numbers (rationals, reals, etc).

At its core, though, number theory is foundationally abstract and is really the study of numbers for the sake of studying numbers. If you like numbers, you're definitely in the right class.

 

2. Practice Makes Better

The best way to improve your understanding of number theory (in terms of math study tips) is through consistently answering practice problems.

Active problem solving engages areas of the mind that are not engaged while passively reading, and allows you to ‘work out' the conceptual frameworks that underlie your understanding.

We've compiled a few short practice problems below. This list is in no way exhaustive, and we highly recommend consulting additional sources such as a maths teacher or lecturer while preparing for a test or exam.

Q1: Is 16 a prime number? Why or why not?

A1: No - 16 can be divided by 4, making it not a prime (primes are only divisible by themselves or 1).

Q2: Is 17 a prime number? Why or why not?

A2: Yes - 17 is only divisible by itself or 1, making it a prime.

Q3: Find the greatest common divisor of 159 and 45.

A3: GCD(159, 45) = 3. This can be deduced using the Euclidean Algorithm.

159 = 3 x 45 + 24

45 = 1 x 24 + 21

24 = 1 x 21 + 3 ← this is the last remainder above the 0, and is our GCD.

21 = 7 x 3 + 0

Q4: Is every integer a rational number?

A4: Yes - every integer is a rational number because integers can be written as rationals, i.e an integer n can be written as n/m, where m = 1.

The above questions will help get you started on your path to number theory mastery - search others out and improve your understanding bit-by-bit until you're confident you can tackle anything.

 

3. Take Consistent Breaks

Because number theory is such a logic-heavy subject, it engages the problem-solving aspect of your brain quite substantially (more so than most other subjects).

It may seem counterintuitive at first, but taking regular breaks from learning will help keep your brain sharp and prevent burnout when in the midst of a longer studying session.

These breaks can be as short as five minutes, and as long as several hours - don't be afraid to stop studying and go outside for a walk if you're feeling stifled.

When you get back, you'll likely find that the next time you need to reason through a problem in number theory, it will come easier for some reason. Your brain needs rest just like the other parts of your body - don't neglect it.

 

4. Spend Time With Friends

Number theory isn't something that everyone enjoys doing, so make sure you spend time with the people who enjoy it and are excited by it!

You might even want to join a club or two - there's plenty of information available online about clubs and organizations around various schools and the world.

In a way, number theory attracts a kind of 'cult following', due to its esoteric nature and abstraction-fueled reasoning.

Mathematics doesn't occur in a vacuum. Oftentimes, the best way to solve a problem is by reasoning through it out loud with a keen listener.

The more like-minded people you meet and friends you get, the better equipped you'll be to tackle various higher level number theory questions and improve your understanding. Plus, it's fun!

 

5. Arrive Early on Test Day

One of the most important things you can do to prepare for your test is arrive on test day early, with everything you'll need. This means you should have all your books and study materials ready for you in the morning (or the night before even).

Why is this useful?

The morning or afternoon of a test is usually stressful and involves numerous time-crunches. And the more anxious your brain is, the worse you typically perform on highly abstract, elaborate subjects like number theory.

Instead, you want to be in a state of slightly heightened arousal, but not too much - and the best way to get yourself into this state is by arriving early on test day and giving yourself significant time to relax.

 

6. Visualize Mathematical Concepts

A key difference between people that typically excel in abstract mathematics and people that suffer is their ability to visualize mathematical functions and ‘see’ the numbers in their head. Humans aren’t de-facto abstract reasoners, after all.

We evolved in a practical, concrete environment, and most of our brain’s architecture was created to solve real-world problems.

This is certainly easier said than done - numbers don’t really owe themselves to inherent visualization, after all - but if you can learn how to do this, you’ll allow the abstract to become the concrete, and this will put math on easy mode for the rest of your life.

A good exercise to get started is to pick a simple function, say f(x) = x^2, and attempt to visualize the graph for that function.

Then, disassociate from the graph and attempt to more fundamentally visualize exponential growth. Do this with f(x) = sqrt(x), f(x) = log(x), and other common functions as well.

You can also attempt to visualize the distribution of primes using models like the Ulam spiral or other frameworks, and this will go a long way towards allowing you to conceptualize relationships that you might not have been aware of previously.

 

7. Understand the Bigger Picture

At the end of the day, achieving 100% in number theory isn't the make-it-or-break-it point in your life.

Understand that there are a lot of very, very bright and successful people out there who simply do not have a strong background in abstract mathematical reasoning or number theory, and that’s okay (think: theoretical mathematicians, famous industry professionals, heads of big corporations, etc).

Instead of working yourself up over not being able to get every question right, zoom out and ask yourself what your end goal is; then ask yourself how number theory fits in to that goal.

Do you want to design complex machine learning systems?

Become a theoretical mathematician?

Or how about simply pass math class?

Whatever your goal is, knowing that goal is the first step in efficiently moving towards it, and it also helps take a load of stress off your (probably already over-stressed) mind.

 

Perhaps the most important thing you can do to achieve your best is understand the bigger picture.

All that time you're spending studying now is time you're not spending doing other stuff - like hanging out with friends and loved ones, having fun, working on other school assignments, or just relaxing.

Like everything else, you need to strike a balance between work and enjoying the rest of your life, and this goes a long way towards improving your enjoyment of math more generally.

Still having a headache with Maths? Check out other help topics:
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