I don't understand how distributive property works in case like this:
$(x_1 \cdot y_1) + (x_2 \cdot y_2) + (x_3 \cdot y_3) + \dots + (x_n \cdot y_n)$
So, how do I calculate this differently? I want to calculate $x$'s and $y$'s separately.
I know this:
$ 2 \cdot ( 5 + 3 ) = ( 2 \cdot 5 ) + ( 2 \cdot 3 ) $
but why it doesn't work here:
$ 2 + ( 5 \cdot 3 ) \ne ( 2 \cdot 5 ) + ( 2 \cdot 3 ) $
I hope I'm explaining this right. I'm $11$. Can you explain how this works in simple words? thank you.
Well, think of $m\ast n$ as a shorthand for $$ \underbrace{n+n+\cdots +n}_{m\ \mathrm{times}}. $$ Then, from your examples, $$ 2\ast(5+3)=2\ast8=8+8=16 $$ and $$ (2\ast 5)+(2\ast 3)=(5+5)+(3+3)=10+6=16 $$ so the rule works out fine, while $$ 2+(5\ast3)=2+(3+3+3+3+3)=2+15=17 $$ and $$ (2\ast5)+(2\ast3)=(5+5)+(3+3)=16 $$ so the rule doesn't work.
Another way to look at multiplication is to take $m\ast n$ as the number of squares in a chessboard with $m$ rows and $n$ columns.
This explains right away why $m\ast n=n\ast m$ and the property $$ a\ast(b+c)=a\ast b+a\ast c $$ can be understood as joining together two chessboards with the same number of rows, while the other way around has no neat geometric interpretation: when you do $$ a+(b\ast c) $$ you are adding $a$ extra squares to a chessboard with $b$ rows and $c$ columns and there's no way to rearrange them in a neatly way that works every time.
