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How to logically understand the multiplication of two integers ?

Eg:

3 x 4 = 12 (is understandable)

-3 x4 = -12 (is also somewhat understandable)

But , 3 x -4 = -12 (is NOT understandable)

-3 x -4 = 12 (is also NOT understandable)

Now , at this we must assume that commutative property may not be true .

What is the logical explanation for something being multiplied negative times ?

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  • $\begingroup$ If you want proof, you need to start with definitions. What is your definition of multiplication of two integers? (There are many to choose from, and the exact form of the proof will depend heavily on exactly which one you use.) What does $(-3)\times 4$ and $3\times(-4)$ mean to you? Once you can answer that, applying that definition to $(-3)\times (-4)$ shouldn't be that hard. $\endgroup$ – Arthur Oct 29 '18 at 16:22
  • $\begingroup$ As @Arthur noted above, you need to start with the definitions. Often in elementary school multiplication is introduced via multiple additions. Using such definition, it's hard to explain what multiplying by a negative integer or by a fraction means. $\endgroup$ – Vasya Oct 29 '18 at 16:28
  • $\begingroup$ Well, then as I see it, your question is really "What is multiplication?", not "Why is $(-3)\times (-4) = 12$?" And as I said, there are many different ways to define multiplication. Most people only have a vague sense of what multiplication is, and to them, proving something like $(-3)\times(-4) = 12$ is really tough. $\endgroup$ – Arthur Oct 29 '18 at 16:28
  • $\begingroup$ A negative times a negative must be a positive in order for the basic rules of arithmetic that we know and love to remain true. It's not hard to give a proof from the axioms for the integers. Maybe you could also think of it intuitively like this. If I have -3 of something that means I owe someone 3 of those things. If I have -3 of -4 it means I owe someone 3 debts of 4 dollars. Once I give those debts to him he will have 3 debts of 4 dollars, so he will owe me 12 dollars. Once he pays me I will have 12 dollars. $\endgroup$ – littleO Oct 29 '18 at 16:30
  • $\begingroup$ @Arthur , When we talk about addition "a + b" we are sure that both a and b are same objects ( eg: you can add 4apples and 3 apples , but not 4apples and 3 oranges). But we talk about multiplication "a x b" , " a " can be any object , but " b " has to be number of times we are adding " a ". So , shouldn't we not use communicative property for multiplication ? $\endgroup$ – NothingIsIrrational Oct 29 '18 at 16:41
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Long comment

$(3 \times 4)=12$ --- agreed.

$3 \times (4-4) = 3 \times 0 = 0$ --- agreed ?

$3 \times (4-4) = (3 \times 4) + (3 \times (-4)) = 12 + ? =0$ --- compute.

And finally : $(-3) \times (4-4) = ((-3) \times 4) + ((-3) \times (-4)) = ? + ?? =0$ --- having computed $?$ above, we are able to compute also $??$


Rules are part of a system of rules : if we start accepting some of them, we are forced (by "logic") to follow other rules implied by the firs ones in order to ensure the consistency (i.e. proper working) of the system.


And see the post : Historical roots of the justification for the rule for multiplication of negative numbers.

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$$(-4)\cdot(-3)=(-1)(4)\cdot(-1)(3)=(-1)^2(3\cdot4)=12$$ Because $(-1)^2=1$

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  • $\begingroup$ But why is -1 X -1 = 1 ? $\endgroup$ – NothingIsIrrational Oct 29 '18 at 16:24
  • $\begingroup$ @NothingIsIrrational If you answer the question I have asked you in a comment to your question post, we can probably sort it out. Until then, there's basically nothing we can do that will really help you. $\endgroup$ – Arthur Oct 29 '18 at 16:25
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You can use the distributive law as follows:

$0=(3-3)\times 4=3\times 4+(-3)\times 4=12+(-3)\times 4$ whence $(-3)\times 4=-12$

Then also $0=(-3)\times (4-4)=(-3)\times 4+(-3)\times (-4)=-12+(-3)\times (-4)$ whence $(-3)\times (-4)=12$

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