MODULUS

What time is it? What is the date today? The day is 24 hours, why did you say 2:25 pm or why did you say the month and day but probably not 14:25 or 734142^nd day after Christ?

Does time exist in the natural world or is it an invention of the human beings? For example, is the motion of the world around the sun determine time? Does the motion of the moon around the earth, or the world around itself? Actually, our calendar is organized upon the motion of the world around the sun and we name it as year, the world itself as day. And similarly there is a lunar calendar which was being used by Chinese and Arabians.

What are the devices used to measure the time? Sundial, sand glass, Quartz minerals, Cesium atom, else? All of the devices used are based on a motion in one way or another.

Hence, in a circular statement we can say that “we use time to measure motion and we use motion to measure time.”

But which motion are we interested in? Can we consider the motion of a comet for measuring time? What is common for the motions mentioned above? If I say these motions are ‘periodic’ or simply ‘circular*’, does it make sense? And in all of the above cases, we always calculate the measurement of the smallest period. (Say ‘year’ for one turn of the world around the sun)

That is where the modulus comes from: it means ‘small measure’ in Latin. Gauss used this term in 1801 to introduce modular arithmetic; mod meaning ‘according to modulus’ and arithmetic ‘the subbranch of mathematics, dealing with operations on numerals’.

That is why we say that today is January 18^th, 2008 in place of 734142^nd day. We use small measures like day, month, year, etc.

Number Circle instead of Number Line

So, we have a motion which is circular and we are interested in the smallest measure. For example, let’s consider the usual clock based on 12 hours. The modulus (small measure) is 12.

To show 13 in normal arithmetic we use number line:

To show 13 in modular arithmetic we use number circle such as:

That is why modular arithmetic is also referred as clock arithmetic.
Modular/clock arithmetic is used in many fields. Besides calendars and clocks it is also the arithmetic of the computers, it is used in music, cryptology, banks and in lots of other places for security reasons. And it will be kept using as long as the humankind loves both simplifying and complicating the things.

*The term ‘circular’ is used to indicate that the motion’s starting point and ending point is the same. According to this definition, the motion of an oscillating arc is considered also circular.