How to interpret multiplication of a number with zero

Question

I am sorry if this question looks stupid. When we multiply $2\times2$ it means we are adding $2$ two times, i.e. $(2+2)$. Or $3\times2$ means $(3+3)$ or $(2+2+2)$. Then if we multiply $3\times0$ it means we are $3$ zero times. Does this mean that we are not adding $3$ at all?

Answer 1

Yes, your intuition is correct. For a "real-world" interpretation:

If you have two $3$-dollar bills, you have $6$ dollars. If you have one $3$-dollar bill, you have $3$ dollars. How many dollars do you have if you have zero $3$-dollar bills?

(Equivalently, how much money do you have if you have three $0$-dollar bills?)

Answer 2

When we multiply 2*2 it means we are adding 2 two times.(2+2)

Along the same line $\,2 \cdot \color{red}{1}\,$ means we are adding $\,2\color{red}{\text{ one}}$ time. But adding it to what? The implicit assumption here is that all these additions start at $\,0\,$ since that's the neutral element for addition i.e. $\,n + 0 = n\,$ for all $\,\forall n\,$, so $\,2 \cdot \color{red}{1} = 0 \,\color{red}{\underbrace{+2}_{\text{1 time}}}\,$, and $\,2 \cdot \color{red}{2} = 0 \,\color{red}{\underbrace{+2+2}_{\text{2 times}}}\,$ etc. It then makes sense to state that $\,2 \cdot \color{red}{0}\,$ means adding $\,2 \color{red}{\text{ zero}}\,$ times, and therefore $\,2 \cdot \color{red}{0} = 0 \color{red}{ \underbrace{\;}_{+\;\text{nothing}}} = 0\,$.