Would any of you be willing to try and explain Abstract Algebra or Group Theory or Ring Theory to me? My mathematical background is moderately limited - I took 2 years of Algebra (8th and 9th grade), 1 year of Geometry (10th), and a year of watered-down trigonometry (11th).
Specifically, my question is how can 2+2 not equal 4 in the “Z3 or Z4” realm of numbers? And do these number systems serve a practical purpose?
Group theory is just a structure on a set with an opperation.
The opperation must be a binary opperation. When any two elements of a set are combined, there must be a result, and the result must be in the original set.
The structure of a group is as follows,
You can think of it as the bare essentials of addition and subtraction. But lots of groups have no relation to the numbers you are used to other than their similar group stucture. The group of integers modulo an integer are also a group. Some are even fields. Z4 is the group of integers modulo 4. It is a finite group with 4 elements, 0,1,2,3. Ring theory adds a “multiplication” with the following properties,
There are lots of intermediate structures along the way, but finally there is what is called a Field, which is a ring basically where we can do all the math we are used to, like divide(field is a commutative division ring, the opperations commute, and every element has a multiplicative inverse, a@1/a=1, where 1 is the multiplicative identity).
So z3 is a group, with elements 0,1,2. You can think of each of the elements of the group as being the remainder when divided by 3, you have 3 choices. For example 4 is congruent to 1 modulo 3 (I’m careful to say “congruent” and Modulo but often times people just write equal or = if it is clear you’re talking about z3, it’s not the “normal” =, we’re in a different group)because 1 and 4 have a remainder of 1 when devided by 3. 1=3x0+1, 4=3x1+1. In z3, 2+2=1 because if you like, any two numbers that leave a remainder of 2 when divided by three, their sum will have a remainder of 1 when divided by 3. It’s easy to see using normal addition and multiplication, take a=2 mod 3, and b=2 mod 3 (now a and b are any numbers that leave 2 as a remainder of 2 when divided by 3)then a=3xc+2, and b=3xd+2 (where x is normal multiplication) so a+b=3xc+2+3xb+2=3xc+3xd+4=3x(c+d)+4=3x(c+d)+3(1)+1=3x(c+d+1)+1, so a+b leaves a remainder of 1 when divided by 3.
For example look at 11, and 14, 11=3x3+2, 14=3x4+2, and 11+14=25=3x(8)+1,
so 11=14=2 mod 3 and 11+14 leaves a remainder of 1 when divided by 3 so 11+14=25=1 mod 3 and also 2+2=1 mod 3, (4 leaves a remainder of 1 when divided by 3).
So 2+2=4 when you are working with the group of integers and standard addition, 2+2=1 when you are in z3. You could also just say if you like that any two numbers that leave a remainder of 2 when divided by 3, their sum will have a remainder of 1.
Nitpick: Every element except the additive identity has a multiplicative inverse.
Whoops.
You don’t really need much in the way of abstract algebra to understand this.
Z3 is the set {0, 1, 2} with addition and multiplication defined so that 3 = 0. So 2 +2 = 4, but that 3 + 1, so it’s 1. 1 + 2 and 2 + 1 are both 3, so they equal 0. You can construct the addition and multiplication tables that way.
Z4 is basically the same, except that it contains 3, and 4 is 0.
What we’re looking at is the remainders upon division by 3 (or 4). If you take two numbers of the form 3k + 2 and add them together, you’ll get a number of the form 3n + 1. Z3 is just a convenient way of representing this type of operation.
Z4 isn’t very interesting because not all of its non-zero elements have a multiplicative inverse. But in Z3, they all do, and in fact, addition and multiplication work pretty much like you’d expect them to in the real numbers.
In general, Zp[sup]n[/sup] has that property, where p is any prime, and n is a positive integer.
These do have applications, but not in anything you’d probably deal with. Z2 was very important in designing the CD audio format.
Is this clear?
But not where p = n = 2. 
You know, that had never even occurred to me. Z8 is also not a field.
That raises an interesting question: When is Zn a field?
For every prime p and integer n there is some field of characteristic p^n, but only for primes (n=1) those are isomorphic to Z mod pZ, as far as I remember.
(and I hope my guessed translations of the German terms are not too far off)
OK, so it turns out that it’s easy to show that Zn is not a field when n is not prime. Just wish I’d realized that before.
Almost. There is a field with p^n elements, the characteristic is prime for a finite field.
Yeah, Zn is a field if and only if n is prime.
All finite fields have order p[sup]n[/sup] for some prime p and some positive integer n, but the fields Zp where p is prime are obviously only some of these (for each p[sup]n[/sup], somewhere out there is a field of that order).
To construct a field of order p[sup]n[/sup], p prime and n > 1, start with Zp, which is a field. Take the polnomial ring Zp, and mod out the ideal generated by an irreducible polynomial of degree n, and you’ve got your field.
For example, to construct a field with 4 elements, start with the polynomial ring Z2. x[sup]2[/sup] + x + 1 is irreducible in Z2. Then Z2 / (x[sup]2[/sup]+x+1) is a field of order 4.
Ahh, thank you all for your replies, I’m not going to lie and say I understood all of that…but I got enough of it to satisfy my curiosity.
Incidentally, this all came about when I made a statement to a friend of mine and compared it to the, I thought, immutable fact that 2+2=4. Unfortunately for me, she happens to be the sister of a fellow who majored in advanced mathematics and he had related to her once that there were occasions in which 2+2 does not equal 4…
Anyway, thanks again!
I would disagree with that. First of all, it’s really just a matter of definition. I think we all know (intuitively, at least) what the individual symbols “2”, “+”,… and so on mean. Certainly we can define those symbols in an alternate manner so that 2 + 2 is no longer 4, but I don’t think that’s a valid objection to your original claim. When we say, “2+2=4 is an immutable fact”, we’re not speaking of any intrinsic meaning contained in the symbols themselves, we’re speaking directly of the concept related by the sentence “2+2=4”. That concept is an immutable fact, independent of the symbols used to express it; if we then change the meaning of the symbols, we haven’t altered the truth value of the concept originally expressed by them.
That’s partly what’s going on here, if you’re speaking of the integers mod n. To elaborate on some of what’s been said before, the integers mod 3 (for example) are constructed by grouping the integers into three classes:
{…,-6,-3,0,3,6,…}
{…,-5,-2,1,4,7,…}
{…,-4,-1,2,5,8,…}
These sets I just listed are the elements of Z3. However, a shorthand is used. In Z3, when we write 2 + 2 = 1, what we really mean is:
{…,-4,-1,2,5,8,…} + {…,-4,-1,2,5,8,…} = {…,-5,-2,1,4,7,…}
In other words, “2+2=1” means: If you take the class containing 2 and add it to the class containing 2, the result is the class containing 1. “2+2=1” is just a shorthand way of writing this, by picking an element from each class and using it as a symbol to represent that entire class. To illustrate a bit further:
-4 + 8 = 7
8 + 2 = -5
2 + (-4) = 1
8 + 8 = 7
Each of these expresses the same equation represented by the original 2+2=1; I’ve only picked a different representative from each class, but you should see that each equation really expresses:
{…,-4,-1,2,5,8,…} + {…,-4,-1,2,5,8,…} = {…,-5,-2,1,4,7,…}
To get to the point, in the integers mod 3 (or mod any n), we’ve changed the meaning of “2”. Previously, “2” referred to the integer two; in Z3, “2” refers to the class {…,-4,-1,2,5,8,…}. As I argued originally, this doesn’t alter the fact that the integer two plus the integer two equals the integer four; we’ve only changed the meanings of the symbols “2”, “4” and “1”.
Furthermore, you may notice, in fact, that even in the integers mod 3, we still have 2+2=4 (i.e., {…,-4,-1,2,5,8,…} + {…,-4,-1,2,5,8,…} = {…,-5,-2,1,4,7,…}, and I picked 2 from each of the first two classes, and 4 from the last). In the integers mod n (any n), we still have 2+2=4, even though, as I mentioned before, we changed the meanings of those symbols. In the integers mod 3, it is true to say any of the following:
2+2=1
2+2=7
2+2=10
2+2=13
2+2=-2
2+2=-5
2+2=-8
and, of course,
2+2=4.
That’s essentially the stance that I took. I wasn’t going to mention that because I didn’t want to bore anyone with the details of one of my petty squabbles, but armed with this new understanding (tenuous, though it may be) my statement stands. I said that her argument for 2+2 not always equalling 4 was grasping at mathematical, semantic straws. I mean, you don’t even need to go into obscure numerical domains to come up with instances in which 2+2 does not equal four. For instance, you could use different numbering systems where the arabic numerals we use are simply ascribed different values…in the way that 10 in binary does not equal 10 in decimal.
Anyway, this all reassuring and my faith in the sanctity of mathematics is maintained. All throughout my elementary years I said that I preferred math over anything else because there were no exceptions. 2+2 always equals 4, but i does not always follow e.