Question 117632
The simplest explanation that I can give is this:


If you divide any number by itself, you always get 1.  {{{a/a=1}}} for all real numbers.


You can also express any negative number as the product of that number's opposite and -1.  In other words, you could write {{{-4}}} as {{{(-1)(4)}}}.


Let's say that x is some positive number {{{x>0}}} and y is some positive number {{{y>0}}}.  Then we can say that {{{-x}}} and {{{-y}}} are negative numbers.  (I hope you clearly understand why you can't just say {{{-x}}} is a negative number without qualifying {{{x}}} as positive in the first place.)


So let's divide {{{-x}}} by {{{-y}}} => {{{(-x)/(-y)}}}.  But we already said that you can also express any negative number as the product of that number's opposite and -1, so we can write: {{{((-1)x)/((-1)y)}}}.   But from the first rule we talked about above {{{(-1)/(-1)=1}}}.


Therefore {{{((-1)x)/((-1)y)=(1)(x/y)=(x/y)}}}.


Now all you have to do is prove that the quotient of a positive number divided by a positive number is positive -- or just take that one on faith.


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There is another way to do this.  Remember that division is nothing more than multiplication by the reciprocal.  A reciprocal is a number formed from an original number such that the product of the original and the reciprocal equal one.  {{{a*(1/a)=1}}}  A reciprocal is also called the multiplicative inverse.


So, if you are dividing a by b {{{a/b}}}, it is the same as multiplying a by the reciprocal of b {{{a*(1/b)}}}.  Now we can define some {{{c=1/b}}} and our division becomes a straight multiplication:  {{{a*c}}}, and our problem becomes one of proving that a negative number times a negative number yields a positive product.


Let a and b be any two real numbers.


Consider the number x defined by 
{{{x = ab + (-a)(b) + (-a)(-b)}}}


We can write 


{{{x = ab + (-a)( (b) + (-b) )}}}       (factor out -a)
{{{x  = ab + (-a)(0)}}}
{{{x = ab + 0}}}
{{{x  = ab}}}

Also, 


{{{x = ( a + (-a) )b + (-a)(-b)}}}      (factor out b)
{{{x  = 0 * b + (-a)(-b)}}}
{{{x  = 0 + (-a)(-b)}}}
{{{x  = (-a)(-b)}}}


So we have 


      {{{x = ab}}}
and
      {{{x = (-a)(-b)}}}


Hence, by the transitivity of equality, we have 
      {{{ab = (-a)(-b)}}}



Hope that helps.