Is there a name for the rule a ÷ ( b × c ) = a ÷ b ÷ c?Edit, because I should have looked it up before I posted the question:Is there a name for the rule a ÷ ( b ÷ c ) = a ÷ b × c? I ran across this in Liping Ma's book, Knowing and Teaching Mathematics, and I have searched the internet for a name for this rule to no avail. It is not the distributive law, but it is rather similar. Thank you!From Ma's book, p. 59 discussing "dividing by a number is equivalent to multiplying by its reciprocal":"We can use the knowledge that students have learned to prove the rule that dividing by a fraction is equivalent to multiplying by its reciprocal. They have learned the commutative law. They have learned how to take off and add parentheses. They have also learned that a fraction is equivalent to to the result of a division, for example, 1 2 = 1 ÷ 2 . Now, using these, we can rewrite the equation this way: 1 3 4 ÷ 1 2 → 1 3 4 ÷ ( 1 ÷ 2 ) → 1 3 4 ÷ 1 × 2 → (This is the step my question is about.) 1 3 4 × 2 ÷ 1 → (and I'd like an explicit explanation of this step, too.) 1 3 4 × 2 → 1 3 4 × ( 2 ÷ 1 ) →

Question

Is there a name for the rule   a  ÷  (  b  ×  c  )  =  a  ÷  b  ÷  c?Edit, because I should have looked it up before I posted the question:Is there a name for the rule   a  ÷  (  b  ÷  c  )  =  a  ÷  b  ×  c?  I ran across this in Liping Ma&#039;s book, Knowing and Teaching Mathematics, and I have searched the internet for a name for this rule to no avail. It is not the distributive law, but it is rather similar. Thank you!From Ma&#039;s book, p. 59 discussing &quot;dividing by a number is equivalent to multiplying by its reciprocal&quot;:&quot;We can use the knowledge that students have learned to prove the rule that dividing by a fraction is equivalent to multiplying by its reciprocal. They have learned the commutative law. They have learned how to take off and add parentheses. They have also learned that a fraction is equivalent to to the result of a division, for example,       1    2    =  1  ÷  2 . Now, using these, we can rewrite the equation this way:  1      3    4    ÷      1    2    →  1      3    4    ÷  (  1  ÷  2  )  →  1      3    4    ÷  1  ×  2  → (This is the step my question is about.)  1      3    4    ×  2  ÷  1  → (and I&#039;d like an explicit explanation of this step, too.)  1      3    4    ×  2  →  1      3    4    ×  (  2  ÷  1  )  →

Haleigh Vega · Accepted Answer

In the same way as subtraction should be thought of as adding by the additive inverse, it is better to think of division as multiplication by the multiplicative inverse to avoid any potential confusion.That is to say,   a  −  b  −  c  =  a  +  (  −  b  )  +  (  −  c  ) and   a  ÷  b  ÷  c  =  a  ×      b          −      1        ×      c          −      1        =  a  ×      1    b    ×      1    c  As for why   a  ×      1    b    ×      1    c    =      a          b      ×      c      , this is an immediate consequence of how multiplication is defined for rational numbers (and fractions in general) and so likely doesn&#039;t have a name.The definition of multiplication of two fractions is       a    b    ×      c    d    :=            a      ×      c              b      ×      d      , so you have   (      a    1    ×      1    b    )  ×      1    c    =      a    b    ×      1    c    =      a          b      ×      c      You go on to say &quot;but in fraction form...&quot; implying you think something looks different about the case where the numbers are fractions instead, but I see no difference. The application of the rule is exactly the same in both scenarios.

Riley Yates · Accepted Answer

I suppose that   a  ,  b  ,  c are rational (or real) numbers. In this case your starting expression is equivalent to:            a              b        ×        c              =  a  ×            1      b        ×            1      c        =      (    a    ×                  1        b              )    ×            1      c        =                    a        ×                              1            b                              c        =                              a          b                    c      so you can see that this property does not need a special name since it is simply the application of the definition of inverse and of associativity for the product.

Is there a name for the rule a ÷ ( b × c )

Answered question

Answer & Explanation

New Questions in Pre-Algebra