2+2=4; Not in the Z3 algebraic group

Question

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

Answer 1

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 $3$,

Answer 2

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.

Answer 3

$\mathbb{Z}_{3}$ 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\} $.