 In some sense, the important difference between higher mathematics and earlier mathematics is that higher mathematics begins to ask the question why as opposed to what. So in arithmetic, we might ask the question, what is 5 plus 1? And the answer is obviously 6. When we get to higher mathematics, we begin to ask the question, why is 5 plus 1 equal to 6? Now, this may seem to be a philosophical question which is interesting to discuss intellectually but not necessarily useful. It turns out, though, in mathematics this is extremely useful. So let's consider that. Why is 5 plus 1 equal to 6? And one interpretation is that 6 is the first number after 5. Now, the reason that this is useful is that this means that as long as first thing after makes sense, we can define n plus 1. Hey, don't jump your cue, you're not on yet. So 5 plus 1 is 6 because 6 is the first number after 5. How about Monday plus 1? Well, if I think about arithmetic, Monday plus 1 doesn't make sense. But if I think about this as first thing after, the first thing after Monday, I could call that Tuesday. Or, and here's a useful cryptographic application, c plus 1. Well, that's the first thing after c. And if we regard c as a letter, the first letter after c is d. Let's focus on our weekday arithmetic to begin with. Once we've defined plus 1 as the day after, we can define larger sums. For example, Monday plus 3 is, well, that's three days after Monday. Or Thursday. Or we can define Friday plus 5, Wednesday. Or how about Thursday plus 7? And that's kind of interesting because here we took Thursday, we added 7, and we got the same thing that we started with. And so we might say that since adding 7 is like adding nothing, we say that 7 is equivalent to 0. And we can write it this way, 7 is equivalent to 0, where we don't have an equals because they are not the same thing, but we have an equivalence, so we use three lines. Now let's think about that in other context. If we allow for this sort of wrap around, we could also have, well, we have Thursday plus 7 is Thursday. If we're talking clock time, I might write something like 3 plus 12 is equal to 3. Or if we're talking about letters of the alphabet, I might write that n plus 26, well, that's n again. Or using months of the year, June plus 12 is June. And what's useful to note here is that in every case, there is a least number equivalent to 0. And this leads us to the following definition. The modulus is the least positive value equivalent to 0. Now we should write 26 is equivalent to 0 in a system where the modulus is 26. Well, that's a lot of words and mathematicians don't like writing more than they absolutely have to. So what we could write is 26 is equivalent to 0 mod 26, where because the mod 26 applies to the entire statement, we set it off with a comma. Alternatively, we may include mod 26 in parentheses because it's essentially a parenthetical statement. Well, that's what we should do. What we actually write is 26 equivalent to 0 mod 26. And especially when we write it in this form, it's very important to remember that the modulus applies to both sides of the equivalence. Again, mathematicians don't like to write down more than we have to. And once we've specified the modulus, as long as you don't change it, you can omit the specification. So, for example, if we're not going to change our modulus of 26, in later statements, we can just write the equivalence 26 equivalent to 0. Now, this equivalence is very useful. So suppose we want to find Tuesday plus 10. Well, we could count the tenth day after Tuesday, but in our weekday arithmetic, we might make the observation. Since 7 is equivalent to 0, we might observe that 10 is 7 plus 3. But since 7 is equivalent to 0, that's the same as 0 plus 3, or just 3. So the tenth day after Tuesday is equivalent to the third day after Tuesday. And so Tuesday plus 10 is equivalent to Tuesday plus 3, and that's much easier to determine. It's Friday. What if we have bigger numbers, like Tuesday plus 30? So since 7 is equivalent to 0, we might proceed as follows. 30, well, that's 23 plus 7. But since 7 is equivalent to 0, we don't need to worry about it. But wait, 23, well, that's 16 plus 7. And again, since 7 is equivalent to 0, we could ignore it. But wait, 16 is 9 plus 7. And since 7 is equivalent to 0, we could ignore it. And once again, 9 is 2 plus 7, and we could ignore the 7. And so that says that 30 is equivalent to 2, and so Tuesday plus 30 is equivalent to Tuesday plus 2, which is Thursday. And you might take a look at this and say, well, we could do that, but that's a lot of work. And so here it's helpful to remember that all progress comes from asking, can we find a better way to do this? Well, let's take a look at that next.