2

I was reading the article/wiki here When I came across this quote

ObviousFact?: examples:

2+2=4 for most people

Those with higher mathematical knowledge may disagree - not in the Z3 algebraic group. No, 2+2 is still 4 in Z3, it just also happens that 4=1. But this is really insignificant, since 4 is usually defined to be 2+2 or 3+1.

Could someone give me a rough idea for a layman what this person meant when they said that 2+2 != 4 in the Z3 algebraic group?

I'd like to understand the reference some so I can use it one day

AnotherUser
  • 349
  • 2
    Google "modular arithmetic" – DonAntonio May 30 '14 at 19:03
  • 3
    That doesn't make any sense. It's self-contradictory. One moment it's telling you that $2+2$ isn't $4$ in $Z_3$, the next it concedes "well, $2+2$ is still $4$, but it's is also $1$" in $Z_3$ (which is correct). Also, while groups are part of modern aka abstract algebra, the term algebraic group is much more sophisticated - all we want to do here is call it a group. – anon May 30 '14 at 19:03
  • 2
    @seaturtles Self-contradictory? Maybe you shouldn't overstate your case. It's just "popular math", written in a way to attract attention. You should read this cum grano salis, I think. – user144248 May 30 '14 at 20:54
  • 1
    In reference to 2+2=4 it says "not in the Z3 algebraic group" and also says "2+2 is still 4 in Z3." Regardless of writing style, that's a contradiction. – anon May 30 '14 at 21:09
  • 2
    @seaturtles Yes, if read literally, then that's a contradiction. If interpreted literally, many everyday propositions do not satisfy mathematical rigor. (Do you happen to know the famous work Logic and conversation?) I think what these guys wanted to say is, "$2+2\neq 4$ - but no, hold, $2+2=4$ but also $4=1$". In any case, to say that this is a "self-contradictory text" is a bit far-fetched. – user144248 May 30 '14 at 21:39
  • @user144248 I really don't understand you one bit. – anon May 30 '14 at 22:08
  • @seaturtles But it's really not so difficult ... What they're saying is not a theorem. They're formulating their idea in a colloquial, non-mathematical way. They wanted to say (and should have said) "$2+2\neq 4$ - but no, $2+2=4$, but also $4=1$" (I'm repeating myself ...). As I said, there's a very famous work of philosopher HP Grice on how logic in conversation differs from logic in mathematics. If you don't understand it, I'm fine with this. – user144248 May 30 '14 at 22:59
  • Anyway, there's an additional level of abstractness that you appear to have missed here. They do not even claim that $2+2\neq 4$ in the first place. They merely claim that those with "higher mathematical knowledge may [falsely] disagree - not in the Z3 algebraic group." Note how they do not write that those with higher mathematical knowledge do disagree (but merely (false) may). – user144248 May 30 '14 at 23:14

3 Answers3

5

Think of $Z_3$ as a clock with only $3$ hours, (i.e. $0,1,2$). So if you are at $2$ o'clock and you go $2$ hours forward, you will be back at $1$ o'clock. This is the best way to think of it in my opinion.

Rocket Man
  • 2,433
  • 2
    That is $\mathbb{Z}_4$. $\mathbb{Z}_3$ is ${0, 1, 2}$. – Caleb Stanford May 30 '14 at 19:03
  • oopsy, I'll edit it. I said $Z_3$ and then provided an example in $Z_4$. Such is life. – Rocket Man May 30 '14 at 19:04
  • So this Z group. This just means the only numbers available to use within that mathematical framework are all integers below n? – AnotherUser May 30 '14 at 19:07
  • Yes, non-negative integers, including $0$, since any negative integer can be congruent to some element in $\mathbb Z_n = {0, 1, \cdots, n-1}$, modulo $n$. – amWhy May 30 '14 at 19:09
  • So when I see "algebraic group" it just means a mathematical framework using a restricted set of integers? – AnotherUser May 30 '14 at 19:12
  • 2
    @AnotherUser You are allowed to use integers above $n$, as long as you understand them to be alternative names for smaller integers. For example, in $Z_3$ you can use 4 as much as you like, as long as you understand that it is another name for 1. It is a remarkable fact that you can do this and never get mixed up. For example, what is $4×4$? It is 16, but this must be the same as $1×1=1$, so 16 must be another name for 1. But 16 is another name for 1 in this system, because if you start at 0 and go 16 hours forward on the clock, you end at 1. – MJD May 30 '14 at 19:17
  • @AnotherUser It means that the set is closed under the given property (i.e. $+$), the set contains an additive identity (i.e. $0$), that every element has an additive inverse (this means that you can always find an element in your set such that if you add it to your given element, you get back the additive identity), and that the set is associative (i.e. $(a+b)+c=a+(b+c)$). – Rocket Man May 30 '14 at 19:18
  • @AnotherUser “Group” means that it is a system with something like addition and subtraction, but not necessarily anything like multiplication or division. Multiplication and division can be added to $Z_3$ to yield a more complicated structure called a field; it is a rather interesting fact that $Z_2, Z_3,$ and $Z_5$ can all be extended in this way, but $Z_4$ cannot. – MJD May 30 '14 at 19:27
2

Addition in $\mathbb Z_n$ is modulo $n$. In your case, $n = 3$. So $2 + 2 = 4 = 1$ modulo $3$. That means that $1$ is the remainder of $4$ when divided by $3$,

amWhy
  • 209,954
  • 1
    The elements of $\mathbb Z_n$ are by convention the smallest non-negative integers that exhaust all possible remainders when each integer in $\mathbb Z$ is divided by $n$. $$\mathbb Z_n = {0, 1, 2, \cdots n-1}.$$ Every integer is equivalent to one of these elements, modulo $n$. – amWhy May 30 '14 at 19:30
  • Note that even in this case you are taking representatives of cosets and $4$ is in the same coset as $1$ – Mark Bennet May 30 '14 at 19:48
2

$\mathbb{Z}_{3}$ has exactly $3$ elements.

You can denote them e.g. as $\overline{0},\overline{1},\overline{2}$ where $\bar{i}$ stands for $i+3\mathbb{Z}=\left\{ i+3n\mid n\in\mathbb{Z}\right\} $.

In this context $\overline{4}=\overline{1}$ and a nice way to describe the addition on $\mathbb{Z}_{3}$ is simply $\overline{i}+\overline{j}=\overline{i+j}$.

Then $\overline{2}+\overline{2}=\overline{4}=\overline{1}$ and for convenience the bars are quite often left out: $2+2=4=1$.

The symbols $1$ and $4$ should be interpreted here as labels that cover exactly the same mathematical object: set $\left\{ 1+3n\mid n\in\mathbb{Z}\right\} =\left\{ 4+3n\mid n\in\mathbb{Z}\right\} $.

drhab
  • 151,093