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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?

John Doe
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aneesh cool
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2 Answers2

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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 $\,2 \cdot \color{red}{0} = 0 \color{red}{ \underbrace{\;}_{+\;\text{nothing}}} = 0\,$.

dxiv
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  • how can we visualise a fractional number multiplying with another fractional number??. – aneesh cool Apr 27 '18 at 05:06
  • @aneeshcool That's an entirely different and unrelated question. You could ask it separately, but you'd first need to define visualize better. There is the $,(1/2)\cdot(3/5)=(1 \cdot 3)/(2 \cdot 5)=3/10,$ argument, and then there is the three 5ths of half-a-pizza story. – dxiv Apr 27 '18 at 05:15
  • it was easy to say 22 is adding 2 twice.what does it look like when we do 0.50.5.?? – aneesh cool Apr 27 '18 at 05:19
  • @aneeshcool what does it look like when we do 0.5*0.5 It looks like halving twice. – dxiv Apr 27 '18 at 05:21
  • iam sorry.But the question is under same topic.Anyways thanks for your explanation. – aneesh cool Apr 27 '18 at 05:24
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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?)

Carl Schildkraut
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  • “Equivalently, how much money do you have if you have three 0-dollar bills?” — Since there are no official 0-dollar bills, all 0-dollar bills must be counterfeit, and thus you might get sued for using counterfeit money. So their net value is negative. :-) – celtschk Apr 27 '18 at 04:44
  • @celtschk For a while, old Zimbabwean currency was essentially 0-dollar bills... but this is not meant to be taken literally. We can replace 0-dollar bills with pieces of broken pencil graphite or anything with just about no value. Anyway, you get the idea. – Carl Schildkraut Apr 27 '18 at 04:56
  • Thank you all for the explanation. – aneesh cool Apr 27 '18 at 05:04
  • Also how can we visualise a. fractional number multiplying with another fractional number.In probability y do we multiply for AND and additional for OR. – aneesh cool Apr 27 '18 at 05:07
  • @CarlSchildkraut I would have said that the Zimbabwean dollar was a model for the mathematics of infinitesimals, rather that that of zero. – Angina Seng Apr 27 '18 at 05:35