In 1962, computer hardware was incredibly primitive. If a programmer wanted to draw a simple diagonal line on a digital screen, the processor had to calculate complex decimal numbers (floating-point arithmetic). This process was so mathematically exhausting that the computer would slow to an agonizing crawl just to render a single triangle. Enter Jack Bresenham, a programmer at IBM. He invented a legendary, elegant mathematical shortcut that completely revolutionized computer graphics. The Bresenham Line Algorithm allowed the computer to calculate which pixels to color in using only basic addition, subtraction, and whole numbers (integers), completely eliminating the need for processor-melting division and decimals. This technical manual deconstructs the beauty of bit-level optimization. We explore how this simple, blazing-fast algorithm became the foundational bedrock for every 2D plotting software, video game engine, and digital printer created over the next forty years. Master the logic that painted the digital world. Learn how a brilliant algebraic trick allowed the earliest, weakest computers to seamlessly render geometric reality.