Carl Friedrich Gauss

Modular Arithmetic: Driven by Inherent Beauty and Human Curiosity

By Richard Taylor 

In modular arithmetic, one thinks of the whole numbers arranged around a circle, like the hours on a clock, instead of along an infinite straight line. Here we have seven “hours” on our clock—arithmetic modulo 7. To add 3 and 5 modulo 7, you start at 0, count 3 clockwise, and then a further 5 clockwise, this time ending on 1. To multiply 3 by 5 modulo 7, you start at 0 and count 3 clockwise 5 times, again ending up at 1.
Modular arithmetic has been a major concern of mathematicians for at least 250 years, and is still a very active topic of current research. In this article, I will explain what modular arithmetic is, illustrate why it is of importance for mathematicians, and discuss some recent breakthroughs.
For almost all its history, the study of modular arithmetic has been driven purely by its inherent beauty and by human curiosity. But in one of those strange pieces of serendipity which often characterize the advance of human knowledge, in the last half century modular arithmetic has found important applications in the “real world.” Today, the theory of modular arithmetic (e.g., Reed-Solomon error correcting codes) is the basis for the way DVDs store or satellites transmit large amounts of data without corrupting it. Moreover, the cryptographic codes which keep, for example, our banking transactions secure are also closely connected with the theory of modular arithmetic. You can visualize the usual arithmetic as operating on points strung out along the “number line.”
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