A precise definition
Arithmetic and Number Theory study the integers — the whole numbers — and the deep structural properties they possess. Which numbers are prime? How are primes distributed among the integers? What integer solutions do polynomial equations have? What is the additive structure of the natural numbers? These questions are easy to state and extraordinarily hard to answer. The Riemann Hypothesis, the Goldbach conjecture, and the Twin Prime conjecture have each resisted solution for over 150 years.
The problem it was invented to solve
Euclid proved the infinitude of primes around 300 BC. Gauss, the 'Prince of Mathematicians', called number theory the queen of mathematics — the branch whose beauty lay in its own intrinsic depth, not in its applications. The irony is that number theory, developed for centuries with no application in mind, turned out in the 20th century to be the foundation of all modern cryptography.
Where you find it in the world — including South Africa
These are not contrived textbook examples. Each application below is currently in use, driven by real institutions, and producing real outcomes.
RSA cryptography: securing every internet transaction
Every HTTPS connection — your bank login, your WhatsApp messages, your email — is protected by RSA encryption, which rests entirely on the difficulty of factoring large numbers (a number theory problem). The product of two 300-digit primes can be computed in milliseconds; factoring it back takes longer than the age of the universe.
SA banking: SARS and NPC transaction security
Every EFT, every credit card transaction, every SARS eFiling submission uses cryptographic protocols built on number theory. The National Payment System (NPC) processes over 200 million transactions per day, each secured by prime numbers.
Error-correcting codes: Vodacom, MTN, and SABC broadcasts
When your phone receives a 5G signal or you watch satellite TV, the digital data has been encoded using error-correcting codes (Reed-Solomon, BCH codes) that are built on finite field arithmetic — a branch of number theory.
You've already encountered this
Goldbach's conjecture (1742) states that every even number greater than 2 is the sum of two primes. It has been verified for all even numbers up to 4 × 10¹⁸. It remains unproved. This is the character of number theory: elementary to state, centuries deep to resolve.
Where it connects in the map of mathematics
Related topics and institutions
Number theory begins with the integers you've counted since childhood.
The Continuum builds number sense and algebraic fluency that makes the deeper structure of integers accessible — connecting your Grade 10 algebra to the mathematics that protects every financial transaction in South Africa.
No card required. South African curriculum. Grade 8–12.