A lot of words in mathematics are just regular English words. You know, I'll take
pizza and divide it into thirds. I'll eat the remainder. I'll take the
difference of two things. I'll, I'll associate with my friends. I'll commute
to work. You know, my identity. Mathematics mostly takes these words, and then imbues
them with some with some richer meaning. Contrast that situation with the word
"modulo." Modulo is a word that English didn't have, but needed! Mathematics
provided it to us. But what does the word "modulo" actually mean? Well, if you've done
some computer programming, you may be seen the word the word modulo in the
context of the remainder operation. So here's what I mean: by thinking of it
as a remainder operation, so, you know, you might do a calculation like 17 mod 7, and
here, you know, I'm thinking of mod as if it were an operation on numbers, the same
way that I might multiply or divide two numbers. Now, if I took 17, and divided it
by 7, I get 2 with a remainder of 3, and that's what this mod operation then is
asking me to think about. It's asking me to think about the remainder. When I
take the first number and divide it by the second, so 17 mod 7, you might say, is
equal to 3 if you're thinking of modulo as being this this kind of remainder
operation. Maybe to make it a little bit more exciting, you know, you can do the
same kind of calculation just with bigger numbers. Right, so if I took number
like a thousand, and I divide it by 17, 17 goes into a thousand 58 times with a
remainder of 14, so consequently, you know, you might, you might write down if you're
doing some programming, you might say 1000 mod 17 is equal to 14, just to
record the fact that when I took 1000 and I divided it by 17, I got a remainder
of 14, right, 17 goes into a 1000
58 times with with this remainder. Now this is often what people mean, or
maybe the experience that people have had, with "mod," but this is emphatically
not what I mean here! Right, I don't want you to think of modulo as just being an
operation between two, between two numbers. So what do I mean? As the
procedure "module five" really does just mean to divide by five and then find the
remainder, but I don't want you to think about it as a procedure. I want you think
about it differently. Think of working modulo five as working in some kind of
context where the number five is really regarded as being zero. Okay, so let's
work modulo five. That means we're gonna regard five as if it's equal to
zero, and if we're doing some calculations modulo five, then I could
get away with writing down say 7 = 2 which maybe seems like
nonsense, but we're working modulo five. What does that mean? Well seven always
means five plus two, that's what seven means, but if 5 = 0 then
5 + 2 the same thing as 0 + 2 which is just 2,
so 7 is the same thing as 2 modulo 5, once I regard 5 as being the same thing
as zero. Or here's another example. All right, let's think about 13. 13, well,
that's 2 times 5 plus 3, but now if five is equal to zero, because we're working
modulo 5, then 2 times 5 is the same thing as 2 times zero, plus 3 but 2 times
0, that's 0, plus 3, that's 3, so 13 is the same thing as 3 modulo 5. And, you know,
certainly there's some connection between this and thinking about
remainders from before, because if I took 7 and I divide it by 5, and I ask what's
the remainder, it's 2, and if I took 13 and I divide it by 5, and I ask what's
the remainder, the remainder is 3. Maybe this seems like a distinction without a
difference, but there is a difference. Well the difference is, is that if I'm
working modulo 5, I not only can say things like 7 is equivalent to 2, or 13
is equivalent to 3 modulo 5, I can also say things like 13 is equivalent to 28
modulo 5. I mean, after all, 13 plus 5 plus 5 plus 5 is equal to 28, but that's
thing modulo five is thirteen plus zero plus zero plus zero, so 13 is the same, in
some sense, as 28 modulo five. If I'm regarding five as if it were zero, now
what does this mean about the meaning of "modulo." Well, you know, when we're thinking
about about modulo, your past experience about modulo might have led you to think
that it's it's really a remainder operation, right, but that's that's not
what I mean. Right, I really want you to think of modulo as
being this new kind of equivalence. Now in the case of when we're working modulo
five, right, and we're thinking about this, it is the case that when I take seven
and I divide it by five, and I ask what's the remainder, it's two, and when I take
13, and I divide it by five and I ask what's the remainder, it's, it is three,
but with an example like 13 being equivalent to 28 modulo five, I mean, it's
definitely not the case that when I take 13 and I divide it by five the remainder
is not 28, and yet I want to regard 13 and 28 as if they were the same once I
throw away the fives. This idea that that two things can be the same after
ignoring some piece of them, I mean, that comes up all the time in daily life. I
often want to be able to compare two objects up to modulo some other component.
You know, you might say something like, "these two restaurants are equally as
good modulo their prices" or "a grape is a raisin modulo water" or "a latte is an
espresso modulo milk." So to be more careful about this, I want some notation
so we can talk about this this precisely. So we're gonna write this; we're gonna
write that "a is congruent to b modulo m" to mean that a is the same thing as b
when we regard m as if it was zero, right, this is when we're working modulo m. I
want to keep track of what it is that we're working modulo when we write down
these equivalences. Now contrast this with just regular old equality. Right, if
I say that a is equal to b, I mean that's really the same thing as saying that the
difference of a and b vanishes. Now I can try to think about this over here as
well, I mean, if I want to think about a as being equivalent to b, when I'm
reguarding m as being zero, well that should be the same thing as saying that
the difference of a and b is zero modulo m. But what does it mean to be zero modulo m? I
mean to be equal to zero modulo m as as good as being a multiple of m because m
is being regarded as if it's zero. Any multiple of m is also zero, so to ask
whether or not a - b is congruent to zero modulo m is the same thing as asking
whether or not a - b is a multiple of m. If you want a little bit more
notation, instead of writing down this whole statement "is a multiple of," I could
instead write that m divides a - b. So that's what I mean by "modulo" in
this context of congruence. Right, we're defining precisely "a is congruent to b"
modulo m" and often we will use that equivalence relation. We'll explicitly
write that down for some integers a, b and m, but at other times we'll do our
work or some some computation in the context of a ring like Z mod 5. Now in
this case, we'd say that we're working in Z mod 5. Now this is some older notation.
I think if you were looking at newer sources you'd see Z / 5Z to mean Z mod 5,
but we'll use this this older notation. What I really mean by that is that Z mod
5 is, you know, it's like the integers but I'm identifying integers that are
congruent modulo 5, so I'm actually regarding those integers as being the
same in some sense in Z mod 5, so I mean, I might think of the number 4 as
being a name for an element of Z mod 5. I might think of the number 2 as being the
name of an element of Z mod 5, and then I have a computation like 4 times 2 is 3
if I'm doing that computation inside Z mod 5 where I'm regarding 4 times 2,
I mean secretly I know that's really 8, but 8 is the same thing as 3 if I'm
identifying integers that are congruent modulo 5.
What's delicate here is that the elements of Z mod five are not
themselves integers, the elements of Z mod 5 are named by integers, so Z
mod 5 consists of 5 elements, and those elements have the names 0, 1, 2, 3, and 4.
Now Z mod 5 is supposed to be just like the
integers, but I'm identifying integers that are congruent modulo 5,
so integers that might have been different as integers are now the same
in Z mod 5. And we saw that here, right, I mean 4 times 2 is really 8, but I'm
regarding 8 as being the same thing as 3, or maybe another name for 3, in Z mod 5,
but once you write down something like this, you're potentially tempted to think
of Z mod 5 as being a subset of the integers. After all the symbols 0, 1, 2, 3,
and 4 are also symbols that we would use for
the integers. So does this mean that Z mod 5 is a subset of the integers? Well, I
want to emphatically say, "No!" I don't want you to regard Z mod 5 as being a subset
of the integers, I mean, after all I could have regarded Z mod 5 not as being
labeled by 0, 1, 2, 3, 4 but instead as being labeled by 5, 16, -3, 28, and
44. I mean, it is also the case that every integer is congruent to one of these
five integers modulo 5. So, okay, I mean if you tell me that Z mod 5 is a subset of
the integers, I'm gonna push back on you and say, well, what five integers are you
thinking of as being Z mod 5 inside Z? I mean, why is this collection any less
good than choosing this, admittedly, more natural looking looking collection?
Indeed rather than thinking of Z mod 5 as being a subset of the integers, I'd
prefer that you think about a function from the integers to Z mod 5. I'd prefer
that you think of Z mod 5 as, as being exactly like the integers but where
integers that were different are now the same.
It's a gluing together as opposed to some sort of a subset. There's some real
tension here, I mean, sometimes when we say modulo we
might mean an operation between integers. We might mean just find the remainder. At
other times, we'll use modulo in the context of an equivalence relation, where
we're giving a new kind of equality, say between two integers. We'll say that a is
congruent to b modulo m. And in other times we're using modulo in the, in the
context of an algebraic object like, like a ring. And the purpose of this video, and
the purpose really of all these videos, is to heighten and showcase some of
those tensions, that the same word, the same idea is appearing in multiple
different places, and I want you to reflect on that think about what that
means. You know, how far can you push this idea
of modulo. I mean, if you can work modulo five, can you work modulo the square root
of two? Can you work modulo pi? Can you work modulo one? What would that even
mean? You know, can you come up with even, even some crazier ideas?