What Does It Mean to Distribute

Date: 08/25/2007 at 22:59:01
From: Andrea
Subject: distributive property

I understand the basics of the distributive property but I don't
understand all the different parts of it, like there's the 
distributive property of multiplication over addition and over 
subtraction and the distributive property of division over addition.  
And something with square roots distributing over multiplication and 
over addition.  I have a worksheet and I have no idea. 

I just don't understand all the different parts of the property. 
What does something distributing over something else mean? (I.e. 
division distributing over addition.)

Date: 08/25/2007 at 23:50:22
From: Doctor Peterson
Subject: Re: distributive property

Hi, Andrea.

I wouldn't call these different "parts" of one property; they are
equivalent properties of different operations.

In general, we say that one operation, "@", distributes over another,
"#", if

  a @ (b # c) = (a @ b) # (a @ c)  for all a, b, c

and, on the other side,

  (a # b) @ c = (a @ c) # (b @ c)

That is, doing @ on the result of # gives the same result as doing @
first to EACH element and then combining them with #.  The idea of
distribution is just like "distributing" money to a group of people:
we give the same amount to EACH of them.

Now, we have to consider each pair of operations separately; some
operations distribute over others, and some don't.  To test it, write
the statements above using the given operations, and decide whether
each is always true.  Among the pairs that do have this property, as
you mentioned, are

  multiplication over addition:       a*(b + c) = a*b + a*c
                                      (a + b)*c = a*c + b*c

  multiplication over subtraction:    a*(b - c) = a*b - a*c
                                      (a - b)*c = a*c - b*c

Among those that DON'T distribute (on the left) are

  division over addition:             a/(b + c) = a/b + a/c -- no
  exponentiation over multiplication: a^(b * c) = a^b * a^c -- no
  exponentiation over division:       a^(b / c) = a^b / a^c -- no

But these do distribute on the right:

  division over addition:             (a + b)/c = a/c + b/c
  exponentiation over multiplication: (a * b)^c = a^c * b^c
  exponentiation over division:       (a / b)^c = a^c / b^c

We don't usually bother talking about this property for operations
that are not commutative, because this sort of detail is confusing.

Since a square root is really the 1/2 power, you can talk about it
distributing:

  root over multiplication:         (a * b)^(1/2) = a^(1/2) * b^(1/2)
                                    sqrt(a * b) = sqrt(a) * sqrt(b)

If your assignment is to test different combinations of operations and
see if they distribute, I've probably answered a lot of them; the
non-commutative operations are messy, and I'm not sure how you'd be
expected to handle them.  It might help if you show me the actual
wording of the assignment.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/26/2007 at 00:58:54
From: Andrea
Subject: Thank you (distributive property)

Thank you so much - you've made my day!  And I totally get it now! 
This makes so much more sense!  Thank you so so so...much!