Your question can be understood as a basic question or as a complex question. I will begin by understanding it as a basic question.Originally Posted by Waveform
For example you have 2X3=6 but why that means adding 2 by itself 3 times OR adding 3 by itself 2 times,
Note that is makes no difference how you mutiply integers thus
That is how we can on a basic level visualize mutiplication.
Since I do not know if you are asking a deep question, I will answer it in another way, by beginning to ask a question. What is an integer? This question is exceptionally difficult. Because you can provide a basic non-mathematical meaning to it but how do you on a mathematical level define such a concept? The answer are sets. Through this basic concept the foundations or mathematics lay. The person who succeded defining natural numbers was Guisseppe Peano.
Now assuming you taken some set theory, here is where it gets fun.
One way to construct integers is like this,
Then define the integers as,
You can define addition/multiplication on these as,
You can define is there exists
which is injective.