Modular Addition

Let's explore the addition in modular arithmetic: i.e.,

(A + B) mod N

By changing the values of A, B, and N below, we can find various outputs. For example, if A = 7, B =4, and N =4, the output is 3. Similarly, you can calculate other values.

NOTE: We can notice here that the output ranges from 0 to N.

Next, let's observe all the outputs by adding an integer to all the values from 0 to N. Below, by changing the value of b, we can see all the outcomes in a cyclic way.

NOTE: Here, we will notice that every number added with b has a unique output. Thus, we see every arrow leaving from that number once and an arrow entering the number once. For example, when n=17, b =13, we can see an arrow goes from 1 to 14, signifying when we add 13 to 1, we get the output 14. Similarly, we see an arrow originates from 5 and enters to 1, signifying when 13 is added to 5 in (mod 17) world, we get 1 as the output.

NOTE: Let's look at the Addition table for mod n. This table will help us understand two key concepts: Closure property and additive inverse property in a group. (Note: there are two other properties, namely associative and the existence of identity element, but we are not going to discuss them.)

Closure Property: Here, when we observe each row and each column of the table, we would notice that they have values from 0 to (n-1).

Additive inverse: As we know, in the additive world, when we add two numbers resulting in 0 as the output, both the numbers are said to be additive inverse of each other.

In the table, when we consider a row and find the element 0 as an entry, the corresponding column value represents the additive inverse.

For example, when n = 10, if we consider row 3 of the table, we will find 0 in the eighth cell from the right, whose corresponding column value is 7. Thus 7 is the additive inverse of 3 in (mod 10) world.

The same goes true for a column also.

NEXT: After learning about modular addition, let's learn about modular multiplication, primarily focusing on the multiplicative inverse.

Reference: Group: Wikipedia