Why do we have $ab=ba$? Why is it that
$$ab=ba$$
for positive integers $a,b$? Is there an intuitive explanation or we just need to accept it as a given fact?
 A: The number $ab$ represents the number of unit squares in an $a\times b$ rectangle.  The number $ba$ represents the number of unidt squares in a $b\times a$ rectangle.  Can you see that they have the same number of $1\times 1$ squares in them?
A: Imagine if you have a rectangle with $a$ rows and $b$ columns. If you want to calculate the area of the rectangle, then you can calculate it as rows x columns, which gives $ab$. You can also calculate it as columns x rows, which gives $ba$. Since the area of the rectangle didn't change at all here, you have $ab=ba$.
A: It comes from the associativity & commutativity of addition, and the Peano definition/construction of the positive integers.
You can prove that $ab=ba$ for all positive integers $a,b$ by induction on $a$.
$P(a=1)$ is just the definition of $b$, and the inductive step is easy.
A: suppose you have a grid of points with $a$ rows and $b$ columns. The total number of points will be $ab$. 
Now if you count it row-wise: each row has $b$ points and there are $a$ such rows. Thus the number of points is $ba$.
Now if you count it column-wise: each column has $a$ points and there are $b$ such colums. Thus the number of points is $ab$. 
But both should be equal because you are counting the same set of points.
