How to make sense of multiplication in the case of negative times positive?












4














Multiplication, most fundamentally, means that when there are two or more equal numbers to be added together, the expression of their sum can be abridged:



$2+2+2+2+2+2$ can be abridged as $6times 2$ (which essentially means the repeated addition of $2$ for $6$ times)



$(-8)+(-8)+(-8)+(-8)+(-8)$ can be abridged as $5times(-8)$ (which essentially means the repeated addition of $-8$ for $5$ times)



Conversely one can conclude from $4times 2$ the repeated addition of $2$ for $4$ times $(2+2+2+2)$ and from $2times 4$ the repeated addition of $4$ for $2$ times $(2+2)$ and one can further discover the commutative property for the multiplication.



Till this things make sense but how to make sense of $(-3)times 4$ (repeated addition of $4$ for $-3$ times!) and also how to establish the commutative property for the same case?






arithmetic







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  • What kind of answers are you looking for?
    – Somos
    Dec 22 at 22:04






  • 1




    I don't know if this is really the sort of thing you want, but you might find some of the ideas in the answers here helpful.
    – Stahl
    Dec 22 at 22:07






  • 3




    Multiplication isn't repeated addition. If you are having trouble making sense of multiplication by negative numbers, the right course of action is almost certainly to think in terms of a better model of multiplication (personally, I tend to think of it as a scaling, where the sign indicates an orientation).
    – Xander Henderson
    2 days ago






  • 1




    Possible duplicate of Why is negative times negative = positive?
    – Xander Henderson
    2 days ago










  • Hmm, I could have sworn we had a question for the more specific case of "negative times positive", but the search function is uncooperative.
    – Henning Makholm
    2 days ago
















4














Multiplication, most fundamentally, means that when there are two or more equal numbers to be added together, the expression of their sum can be abridged:



$2+2+2+2+2+2$ can be abridged as $6times 2$ (which essentially means the repeated addition of $2$ for $6$ times)



$(-8)+(-8)+(-8)+(-8)+(-8)$ can be abridged as $5times(-8)$ (which essentially means the repeated addition of $-8$ for $5$ times)



Conversely one can conclude from $4times 2$ the repeated addition of $2$ for $4$ times $(2+2+2+2)$ and from $2times 4$ the repeated addition of $4$ for $2$ times $(2+2)$ and one can further discover the commutative property for the multiplication.



Till this things make sense but how to make sense of $(-3)times 4$ (repeated addition of $4$ for $-3$ times!) and also how to establish the commutative property for the same case?






arithmetic







share|cite|improve this question
























  • What kind of answers are you looking for?
    – Somos
    Dec 22 at 22:04






  • 1




    I don't know if this is really the sort of thing you want, but you might find some of the ideas in the answers here helpful.
    – Stahl
    Dec 22 at 22:07






  • 3




    Multiplication isn't repeated addition. If you are having trouble making sense of multiplication by negative numbers, the right course of action is almost certainly to think in terms of a better model of multiplication (personally, I tend to think of it as a scaling, where the sign indicates an orientation).
    – Xander Henderson
    2 days ago






  • 1




    Possible duplicate of Why is negative times negative = positive?
    – Xander Henderson
    2 days ago










  • Hmm, I could have sworn we had a question for the more specific case of "negative times positive", but the search function is uncooperative.
    – Henning Makholm
    2 days ago














4












4








4







Multiplication, most fundamentally, means that when there are two or more equal numbers to be added together, the expression of their sum can be abridged:



$2+2+2+2+2+2$ can be abridged as $6times 2$ (which essentially means the repeated addition of $2$ for $6$ times)



$(-8)+(-8)+(-8)+(-8)+(-8)$ can be abridged as $5times(-8)$ (which essentially means the repeated addition of $-8$ for $5$ times)



Conversely one can conclude from $4times 2$ the repeated addition of $2$ for $4$ times $(2+2+2+2)$ and from $2times 4$ the repeated addition of $4$ for $2$ times $(2+2)$ and one can further discover the commutative property for the multiplication.



Till this things make sense but how to make sense of $(-3)times 4$ (repeated addition of $4$ for $-3$ times!) and also how to establish the commutative property for the same case?






arithmetic







share|cite|improve this question















Multiplication, most fundamentally, means that when there are two or more equal numbers to be added together, the expression of their sum can be abridged:



$2+2+2+2+2+2$ can be abridged as $6times 2$ (which essentially means the repeated addition of $2$ for $6$ times)



$(-8)+(-8)+(-8)+(-8)+(-8)$ can be abridged as $5times(-8)$ (which essentially means the repeated addition of $-8$ for $5$ times)



Conversely one can conclude from $4times 2$ the repeated addition of $2$ for $4$ times $(2+2+2+2)$ and from $2times 4$ the repeated addition of $4$ for $2$ times $(2+2)$ and one can further discover the commutative property for the multiplication.



Till this things make sense but how to make sense of $(-3)times 4$ (repeated addition of $4$ for $-3$ times!) and also how to establish the commutative property for the same case?






arithmetic




arithmetic






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 days ago









bof

50k457119




50k457119










asked Dec 22 at 21:35









user596245

373




373












  • What kind of answers are you looking for?
    – Somos
    Dec 22 at 22:04






  • 1




    I don't know if this is really the sort of thing you want, but you might find some of the ideas in the answers here helpful.
    – Stahl
    Dec 22 at 22:07






  • 3




    Multiplication isn't repeated addition. If you are having trouble making sense of multiplication by negative numbers, the right course of action is almost certainly to think in terms of a better model of multiplication (personally, I tend to think of it as a scaling, where the sign indicates an orientation).
    – Xander Henderson
    2 days ago






  • 1




    Possible duplicate of Why is negative times negative = positive?
    – Xander Henderson
    2 days ago










  • Hmm, I could have sworn we had a question for the more specific case of "negative times positive", but the search function is uncooperative.
    – Henning Makholm
    2 days ago


















  • What kind of answers are you looking for?
    – Somos
    Dec 22 at 22:04






  • 1




    I don't know if this is really the sort of thing you want, but you might find some of the ideas in the answers here helpful.
    – Stahl
    Dec 22 at 22:07






  • 3




    Multiplication isn't repeated addition. If you are having trouble making sense of multiplication by negative numbers, the right course of action is almost certainly to think in terms of a better model of multiplication (personally, I tend to think of it as a scaling, where the sign indicates an orientation).
    – Xander Henderson
    2 days ago






  • 1




    Possible duplicate of Why is negative times negative = positive?
    – Xander Henderson
    2 days ago










  • Hmm, I could have sworn we had a question for the more specific case of "negative times positive", but the search function is uncooperative.
    – Henning Makholm
    2 days ago
















What kind of answers are you looking for?
– Somos
Dec 22 at 22:04




What kind of answers are you looking for?
– Somos
Dec 22 at 22:04




1




1




I don't know if this is really the sort of thing you want, but you might find some of the ideas in the answers here helpful.
– Stahl
Dec 22 at 22:07




I don't know if this is really the sort of thing you want, but you might find some of the ideas in the answers here helpful.
– Stahl
Dec 22 at 22:07




3




3




Multiplication isn't repeated addition. If you are having trouble making sense of multiplication by negative numbers, the right course of action is almost certainly to think in terms of a better model of multiplication (personally, I tend to think of it as a scaling, where the sign indicates an orientation).
– Xander Henderson
2 days ago




Multiplication isn't repeated addition. If you are having trouble making sense of multiplication by negative numbers, the right course of action is almost certainly to think in terms of a better model of multiplication (personally, I tend to think of it as a scaling, where the sign indicates an orientation).
– Xander Henderson
2 days ago




1




1




Possible duplicate of Why is negative times negative = positive?
– Xander Henderson
2 days ago




Possible duplicate of Why is negative times negative = positive?
– Xander Henderson
2 days ago












Hmm, I could have sworn we had a question for the more specific case of "negative times positive", but the search function is uncooperative.
– Henning Makholm
2 days ago




Hmm, I could have sworn we had a question for the more specific case of "negative times positive", but the search function is uncooperative.
– Henning Makholm
2 days ago










2 Answers
2






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oldest

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2














Repeated addition of $4$ for $-3$ times means repeated subtraction of $4$ thrice.






share|cite|improve this answer





























    1














    As noted elsewhere, you could consider the last example as a case of multiple subtraction.



    Another approach is a graphical one. Imagine that multplying by a positive number stretches the number line. Multiplying by a negative number rotates the number line 180 degrees in addition to the stretch.



    So $4 x (-3)$ would see you start on $-3$ and stretch the number line by a factor if 4 so you would end up at $-12$. $(-3)×4$ would have you start at $4$, undergo the stretch, then rotate the number line to end up at $-12$.



    This geometric interpretation may seem forced, and unnecessarily complicated, but you'll be truly thankful for it when you see how immensely it simplifies working with Complex numbers. Then, you will be rotating the number line through any amount of arc, and plotting numbers on the plane, and not only the number line.






    share|cite|improve this answer





















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      2 Answers
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      2 Answers
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      Repeated addition of $4$ for $-3$ times means repeated subtraction of $4$ thrice.






      share|cite|improve this answer


























        2














        Repeated addition of $4$ for $-3$ times means repeated subtraction of $4$ thrice.






        share|cite|improve this answer
























          2












          2








          2






          Repeated addition of $4$ for $-3$ times means repeated subtraction of $4$ thrice.






          share|cite|improve this answer












          Repeated addition of $4$ for $-3$ times means repeated subtraction of $4$ thrice.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 2 days ago









          Martund

          1,349212




          1,349212























              1














              As noted elsewhere, you could consider the last example as a case of multiple subtraction.



              Another approach is a graphical one. Imagine that multplying by a positive number stretches the number line. Multiplying by a negative number rotates the number line 180 degrees in addition to the stretch.



              So $4 x (-3)$ would see you start on $-3$ and stretch the number line by a factor if 4 so you would end up at $-12$. $(-3)×4$ would have you start at $4$, undergo the stretch, then rotate the number line to end up at $-12$.



              This geometric interpretation may seem forced, and unnecessarily complicated, but you'll be truly thankful for it when you see how immensely it simplifies working with Complex numbers. Then, you will be rotating the number line through any amount of arc, and plotting numbers on the plane, and not only the number line.






              share|cite|improve this answer


























                1














                As noted elsewhere, you could consider the last example as a case of multiple subtraction.



                Another approach is a graphical one. Imagine that multplying by a positive number stretches the number line. Multiplying by a negative number rotates the number line 180 degrees in addition to the stretch.



                So $4 x (-3)$ would see you start on $-3$ and stretch the number line by a factor if 4 so you would end up at $-12$. $(-3)×4$ would have you start at $4$, undergo the stretch, then rotate the number line to end up at $-12$.



                This geometric interpretation may seem forced, and unnecessarily complicated, but you'll be truly thankful for it when you see how immensely it simplifies working with Complex numbers. Then, you will be rotating the number line through any amount of arc, and plotting numbers on the plane, and not only the number line.






                share|cite|improve this answer
























                  1












                  1








                  1






                  As noted elsewhere, you could consider the last example as a case of multiple subtraction.



                  Another approach is a graphical one. Imagine that multplying by a positive number stretches the number line. Multiplying by a negative number rotates the number line 180 degrees in addition to the stretch.



                  So $4 x (-3)$ would see you start on $-3$ and stretch the number line by a factor if 4 so you would end up at $-12$. $(-3)×4$ would have you start at $4$, undergo the stretch, then rotate the number line to end up at $-12$.



                  This geometric interpretation may seem forced, and unnecessarily complicated, but you'll be truly thankful for it when you see how immensely it simplifies working with Complex numbers. Then, you will be rotating the number line through any amount of arc, and plotting numbers on the plane, and not only the number line.






                  share|cite|improve this answer












                  As noted elsewhere, you could consider the last example as a case of multiple subtraction.



                  Another approach is a graphical one. Imagine that multplying by a positive number stretches the number line. Multiplying by a negative number rotates the number line 180 degrees in addition to the stretch.



                  So $4 x (-3)$ would see you start on $-3$ and stretch the number line by a factor if 4 so you would end up at $-12$. $(-3)×4$ would have you start at $4$, undergo the stretch, then rotate the number line to end up at $-12$.



                  This geometric interpretation may seem forced, and unnecessarily complicated, but you'll be truly thankful for it when you see how immensely it simplifies working with Complex numbers. Then, you will be rotating the number line through any amount of arc, and plotting numbers on the plane, and not only the number line.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 2 days ago









                  Adam Hrankowski

                  2,050828




                  2,050828






























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