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Hello everyone, and welcome 
to Carmen's Core Concepts.

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My name is Carmen Bruni, 

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and this is Week 7 Part 2.

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So I felt the need to break 
off this last section because

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it is fairly abstract that's for 1.

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For 2, the other video's getting

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quite long, so I thought I would cut it 
off there and start a new video here.

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And thirdly, there's a lot of parts that I'm 
going to talk about in this video that are…

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what do I want to say… sort 
of above and beyond what a

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Math 135 student should know 
at the end of the course,

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but at the same time, I feel like 
these concepts are very important

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and that there's no reason 
not to include them

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in Math 135.

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So that being said,

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those factors combined gave 
us a Week 7 Part 2 video

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that is the following.

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So wall of text, but

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what am I going to 
say here? A ring is

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a very common structure. You 
actually know what a ring is

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without really knowing 
what a ring is.

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The idea behind this slide and
this definition is to give you...

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a way to group 

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sets that you know 
together basically.

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So let's describe
what I mean by this.

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So definition: “a commutative ring is a 
set R along with two closed operations,

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addition and multiplication…”

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by closed operations we mean
if I add 2 elements of the set R,

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I stay inside R, and if I multiply 2 
elements of the set R, I stay inside R,

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“…and these two
operations satisfy for all

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for all a, b, c inside 
R that they are...

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that the operation is associative…”
so both operations are associative.

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So a plus b plus c is 
equal to a plus b plus c,

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and same with multiplication. So here
I've denoted - I’m not going to always use

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the dot for multiplication. I'm 
just going to use juxtaposition

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for multiplication so if there are 
two things right beside each other

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that implies that they're 
multiplied together.

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So a times b then times c is 
the same as a times b times c.

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Commutative,

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a plus b is the same as b plus a,

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and a times b is the same as b 
[times] a, in a commutative ring.

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So multiplication doesn't always 
have to be commutative in a ring,

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but in a commutative ring it is.

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We have identities; so there 
are distinct elements 0 and 1

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such that a plus 0 is 
a, and a times 1 is a.

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I've written “distinct” in 
brackets. Some books

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don't require this, some books do.

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Take your pick. It 
doesn't matter that much.

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Additive Inverses, there exists 
an element minus a such that

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a plus negative a is 0.

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So every element inside your set

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has an additive inverse.

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And the distributive property, a times b 
plus c is equal to a times b plus a times c.

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As I said at the beginning, well

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a commutative ring is something 
that you already kind of know.

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You already know examples 
of commutative rings:

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the integers, the rational 
numbers the real numbers,

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all of these are
commutative rings.

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The natural numbers don't 
form a commutative ring.

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They are associative, 
they are commutative,

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they don't really 
have an identity, right?

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The additive identity isn't inside N. 
Remember our natural numbers start at 1.

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And additive inverses are missing. This is the 
key one that's missing from the natural numbers.

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There's no - if I have 1, I don't have
negative 1 inside the natural numbers.

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But I do inside the integers, and that's
what forms a commutative ring.

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Okay, so here's the definition. Again 
these are things that you know, right?

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So I'm just kind of grouping 
together a set of objects that you

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already know have
these properties

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and I'm giving them some sort 
of classification name, okay?

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And I haven't defined a 
field yet. What's a field?

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Maybe you can guess, 
but maybe you can’t.

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So here we have additive inverses, 
a field adds multiplicative inverses.

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So if, in addition, every non-zero 
element has a multiplicative inverse,

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that is an element a inverse,

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so we denote it by a to 
the power of negative 1,

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and this element satisfies 
a times a inverse equals 1,

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we say that R is a field.

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Okay? So

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every field is a ring,

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every field is a 
commutative ring,

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but not every ring is a field.

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So for example, Q and R,

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these are fields and 
these are commutative rings,

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but the integers, for 
example, is not a field

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because elements like 2, 3, 4

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their inverses, or their 
multiplicative inverses,

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1 over 2, 1 over 3, 1 over 4,
those aren't inside the integers.

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So that's why again so maybe this 
is giving you a little bit of context

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as to why we have all these 
symbols and definitions and

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how this all came about.

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That's sort of what I'm trying 
to give you with this slide.

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It's just a way to describe

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sets that have special properties, and 
we'll see in the next couple of slides,

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we're going to define a
new commutative ring

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and that's going to be
the integers modulo m,

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and it's kind of what you think based 
on what we've done this week

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but we'll talk about that 
in a minute, okay?

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So here's the definition of a
commutative ring and fields.

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We’ll also see fields later on. 
We're going to talk about the

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set called the complex numbers, and we'll 
see that the complex numbers form a field.

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But again, I'll save that for later.

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Congruence classes, so now let's bring 
this back down to where we were, okay?

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So rings, fields, we have these things 
in our back pocket when we need them.

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What is a congruence class?

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A congruence, or equivalence 
class modulo m of an integer a

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is the set of integers 
defined as follows: so

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we use square brackets 
and put a in the middle,

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and we define this to be… 
so this “colon equals” means

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“defined as” or 'defined to be' 
something along those lines.

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[square bracket] a is defined to be 
the set of integers x such that

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x is congruent to a mod m.

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So for example, if m 
were 7 and my a were 1,

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the set square bracket 1
would consist of the elements

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1, 8, 15, dot dot dot,

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and on the other side it would also consist 
of negative 6, negative 13, negative 20,

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and all those as well. Okay?

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So there's an example of the
congruence class of 1 mod 7.

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Further, we define this Z m notation.

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So the integers modulo m, 
sometimes it's denoted by

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a Z with a slash m Z.

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In this course, we're going to 
use this first description, this Z

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with the little subscript m.

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Later on in your mathematical careers, 
you're more likely to see this notation,

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for reasons which I
won't get into now,

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but it's a good notation 
so you should know it.

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And this is defined to be what? 
Well this is defined to be

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the set consisting of the sets

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of 0, 1, all the way 
up to m minus 1,

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and again these sets 
are taken modulo m.

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You could argue that maybe we should 
decorate these square brackets with an m.

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We're not going to do that, 
just to keep it simple.

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Notice that you don't need to 
always use the symbols from

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0, 1, all the way up to m minus 1. Remember 
that the equivalence class consisting of

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0 is the same as the equivalence 
class consisting of m.

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So if you wanted, you could write the set 
as 1, 2, 3, m minus 1, all the way up to m,

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instead of - ignoring 0.

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You can also use different elements as
well. These elements are just the most

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convenient ones to 
describe Z mod m. Okay?

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So that's our congruence classes.

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How do we put this all together? So 
I had this previous slide defining

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rings and fields and I have this 
slide defining congruence classes.

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Well the merger
between the two is

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given by giving the set Z m

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a ring structure, okay,

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and the ring structure is going 
to be what you think it is,

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but I'm still going to
define it formally.

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So the ring Z m. So we're 
going to turn this set

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into a ring by defining 
addition and subtraction

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and multiplication 
as follows, okay?

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So subtraction, again, you can think 
of subtraction as the additive inverse.

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Now how do we do this? 
We take a plus or minus b,

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and if I'm adding these -
so remember, this is the set… 

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this is the congruence - this a 
is the congruence class of a,

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and this b is the congruence class 
of b. These two things are sets.

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So what we're trying to do is we're 
trying to define an operation on sets.

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And on these two sets, how 
do we define the operation?

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Well we take a plus b

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to be the congruence 
class of a plus b, okay?

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That doesn't look like we're doing very much, 
right? It looks like we're just pushing symbols,

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but there's some going on here, right? 
We're adding two sets together.

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How do we claim that we
add the two sets together?

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Take the arguments and add them.

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And in the same way, we're going 
to multiply two sets together,

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and how are we going to multiply
these two sets together?

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Well take the two 
arguments, a and b,

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multiply them together, and 
look at that congruence class.

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Something I'll get you to check is that 
this makes 0 the additive identity

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and 1 the multiplicative identity.

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And again, notice
that a plus b means 

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add and then reduce modulo m. That's 
basically what's happening here.

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You're adding first, and 
then reducing modulo m.

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That's how the addition is 
defined on this set Z m.

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And that's going to turn this into a ring. 
This is not something I'm going to

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prove, but maybe this is
something that you should…

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well this is definitely 
something that you do.

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Test, go back to the definition of 
a ring and try it out with this set,

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and make sure you understand 
why this operation makes it a ring.

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The members are sometimes
called representative members,

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so the members 0, 1, all the way up to m minus 1 
are sometimes called representative members.

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So that's like I said, right, 
we can change the 0 to m

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and it doesn't really change 
the definition of the set, 

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it's still the same thing.

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Something else to
note, when m is a prime,

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let’s call it p, the ring 
Z p is also a field

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as non-zero elements have 
multiplicative inverses,

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they're invertible.

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We'll see this later.

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We will see this later.
I'll leave it at this for now,

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but this is why I want to introduce 
the concept of a ring and a field

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right now because
it fits in quite nicely.