ATRACTOR

(Modular Arithmetic)

Modular Arithmetic is also known as Circular Aritmética. Why?

Consider the number \(0\). Adding up \(1\) gives \(1\). Adding up \(1\) once again gives \(2\) and adding again \(1\) gives \(3\). What will happen if we repeat this indefinitely? In the case of natural numbers the sum will increase forever. What about Modular Arithmetic? The answer is different. Take the example of arithmetic modulus \ (12 \). At step eleven the sum will be \(11\). What happens next? When we add \(1\) to \(11\) module \(12\), the result is zero and thus we return to the beginning of the process. When we get back to \(11\) at step \(23\), in the next step we come back to zero again, and so on. Of course, this is not an exclusive property of modulus \(12\).

How can we use this property to compute the modular sum of two numbers \(a\) and \(b\)? Consider, for example, the case of arithmetic modulus \(12\):

• Draw a circular arc corresponding to \(\frac{a}{12}\) of the circle.
• Draw another circular arc corresponding to \(\frac{b}{12}\) of the circle.
• Juxtapose the two arcs in order to form a single arc (possibly this new arc may give more than a turn in the circle).

Then, the modular sum of numbers \(a\) and \(b\) corresponds to the latter arc (after deleting the turn in the circle, in case it applies).

To understand it better, check this app.