Math & Number Theory: Overview
Math & Number Theory: The DNA of Logic
The Eternal Truth
Software changes every year. Frameworks die every decade. But the properties of numbers havenāt changed in 2,500 years. If Euclid found a way to find the āGreatest Common Divisorā in 300 BC, that algorithm is still the fastest way to do it today.
In engineering, we donāt study math to solve āmath problems.ā We study math to solve Trust and Efficiency problems. Without Number Theory, there is no RSA encryption (Trust). Without Modular Arithmetic, there are no Hash Tables (Efficiency).
The Strategy of the Atom
In this section, we look at the building blocks of the digital universe:
| Concept | The Soul / Metaphor | Representative | Best Forā¦ |
|---|---|---|---|
| Primality | The Atoms Primes are the unique building blocks of all numbers. | Sieve of Eratosthenes | Cryptography RSA, DH, and secure keys. |
| Harmony | The Great Harmonizer Finding the common rhythm between two different cycles. | GCD / Euclidean | Simplification Aspect ratios, fractions, and timing. |
| Looping | The Digital Clock A world that wraps around, keeping values within bounds. | Modular Arithmetic | Hashing & Crypto Distributing data and hiding secrets. |
| Switching | The Switchboard Speaking the native language of the machine: 0 and 1. | Bit Manipulation | Performance Flags, compression, and ultra-fast math. |
The Three Laws of Digital Math
- Precision is King: In math, 0.999ā¦ is 1. In computers, floating-point errors (0.1 + 0.2 != 0.3) can crash a rocket or bankrupt a bank. Number theory deals with Integers, where truth is absolute.
- The Large Number Wall: Modern security relies on the fact that multiplying two 1024-bit primes is easy, but factoring the result is practically impossible.
- Cyclic Logic: Most computer systems donāt go to infinity. They go in circles (Modulo). Understanding the circle is the key to understanding the machine.
Summary
In this section, we will strip away the UI, the frameworks, and the APIs to see the raw, pulsing logic of numbers. We will learn that the oldest algorithms in history are often the most relevant ones in our pocket today.
Letās start with the atoms of the universe: Primes.