13.4. Addition Modulo k

Now let us see some other interesting examples of groups. So let k be a positive integer and let ℤk be the set of integers 0 to k−1. Basically, it is the set of all possible remainders which you can obtain by dividing any integer, it could either be positive or negative, by k.

This chunk introduces the concept of addition modulo k. In this context, k is defined as a positive integer, and the set ℤk consists of integers ranging from 0 to k−1. This range represents all possible remainders you get when any integer is divided by k. For example, if k = 5, then ℤ5 would be {0, 1, 2, 3, 4}. Understanding this helps us visualize how numbers wrap around when we exceed the value of k.

Think about hours on a clock. If it's 10 o'clock and you want to know what time it will be in 5 hours, you wrap around after reaching 12: you will arrive at 3 o'clock. This is similar to how addition modulo works, where once we reach the maximum (e.g., k = 12), we start counting from 0 again.

Now I define a new form of addition over this set called addition modulo k, which is denoted by +k. So, addition modulo k of a and b is defined as follows: I add a and b and then take modulo k, the result will be called as the result of addition of a and b modulo k i.e., a +k b = [a + b] mod k.

In this chunk, we learn how addition modulo k is performed. To add two numbers a and b using addition modulo k, you first add them together as you normally would. Then, you take the result and find the remainder when divided by k using the modulo operation. This means that if the total exceeds k, it wraps back around. For instance, in ℤ5, if you add 3 + 4, the sum is 7, and the modulo operation tells you to take 7 mod 5, which gives a remainder of 2.

Imagine you're at a party with 5 chairs and 7 guests. If each time a guest occupies a chair, another has to get up if no more chairs are available. After 5 guests sit down, the next one (6th) will have to go back to the 1st chair. This wrapping around illustrates addition modulo in a relatable way.

So, now my claim is that this set ℤk, which is a finite set because k is a positive integer, constitutes a group with respect to this operation of addition modulo k. So, let us see whether the 4 properties are satisfied or not.

This chunk states that ℤk forms a group under the addition modulo k operation. To confirm this, we need to verify four key properties: closure, associativity, identity, and inverses. The closure property ensures that if we take any two elements from ℤk and add them using modulo k, the result is still within ℤk. Associativity means that the grouping of numbers does not affect the outcome. The identity element is 0, since adding 0 to any element leaves it unchanged. Finally, the inverse of any element a in ℤk is found by taking k - a, so that when added results in 0.

Consider a group of friends at a concert standing next to each other in a circle (like the numbers 0 to k-1). If you hold hands and skip over one friend (addition), you still end up within the circle, corresponding to the closure property. Moreover, no matter how you group your friends when skipping, you still form the same pattern (associativity). The friend at position 0 represents the one who always lets you stay in the same place (identity), while for every other position, there is a friend you can skip back to 0 (inverse).

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