Modular Arithmetic

First discovered and defined by Carl Friedrich Gauss in his Disquisitiones Arithmeticae in 1801 (when Gauss was only 24 years old!), modular arithmetic is a very clever form of mathematics which works in a cyclical way, rather than linear.

What's the difference?

Linear arithmetic is the standard mathematics taught to children early on: 1 + 1 = 2, 10 + 10 = 20, 100 +100 = 200, etc. Numbers are added together and the results can be arbitrarily large or small.

Modular arithmetic, on the other hand, can probably best be described using the example of a standard analogue clock. Say, for the sake of example, that it is one o'clock. What time will it be fourteen hours later? According to linear arithmetic, in fourteen hours it will be fifteen o'clock, which, if not entirely incorrect, is at least inaccurate.

To solve this problem, one has to use modular arithmetic - in this case, it would be arithmetic with a modular (or cycle) of 12. This means that every twelve hours, the counting starts over again at one, so that in this example, as any person who knows how to tell time will understand, one plus fourteen does not equal fifteen, but rather three.

Tricks to Modular Arithmetic

While using a clock is perhaps the simplest example to understand in modular arithmetic, it is certainly not the only one. Some examples can get very complicated, and as such mathematicians, beginning with Gauss himself, have come up with many different ways of actually computing modularity.

One of the simplest ways relates directly to the standard division that is taught in elementary school, and the idea of a remainder. One might recall learning to solve such problems as 13 divided by 6, wherein the answer comes out to 2 remainder 1. This, in essence, is modular arithmetic! It is saying that there is a modular of 6 (a clock with six numbers instead of twelve), and is wondering how many times the hand has to spin around in order to count to sixteen (which turns out to be two full turns plus one extra number).

Without using this particular visualization, this idea of division can also be helpful in the reverse sense. For example, if it is one o'clock now, what time will it be in 125 hours? Easy. Simply divide 125 by the modulation: 125 divided by 12, which equals 10 remainder 5, then add the 1. It will be six o'clock!

These are only a couple of the simplest ways to understand modular arithmetic, but it provides a good start at being able to understand this form of mathematics.

Modular Applications

So, besides telling time and doing division, where is modular arithmetic used?

A great many places, actually.

In calculus, modular arithmetic used used extensively in trigonometric functions, which then carries over into the physical sciences when attempting to decipher the behavior of various waves (which are defined trigonometrically).

Also in the world of physics, modular arithmetic is also used in more complex forms in quantum electrodynamics, which uses the wave features of light and their cyclical nature to discover their amplitude under various circumstances.

In music theory modular arithmetic also has obvious applications, as the standard musical staff has only 8 notes (A through G) which are then repeated over and over. So if a certain musical composition requires a flat 13th, modular thinking is required to decipher the fact that the note in question is actually the fifth note of the second octave.

Clearly, there are a great number of ways in which modular arithmetic has been made to benefit humanity, both mathematically and practically, and while it may seem rather complex on the surface, it actually becomes a very intuitive way to look at numbers in certain instances. In fact, modular arithmetic is often used as a method of learning "tricks" for doing mathematics in their heads, as even linear arithmetic behaves in a modular sense, being that it is based on decimal units with a modulation of 10.

There is really no end to the uses of modular arithmetic, and its existence is all thanks to a brilliant young man named Carl Gauss.

A Clock Uses Modular Arithmetic

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