The formal statement of the distributive law should read as follows:
If a, b, c and d are numbers, or algebraic expressions (same thing really) and b = c + d then ab = ac + ad
It is a by-product of the law that it tells you how to expand an expression with a bracketed factor.
In any case, what’s the big deal ?
Here is another horror which I found recently:
The distributive law of addition: a(b + c) = ab + ac (OK, it’s a definition)
The current school math explanation:
You take the a and distribute it to the b to get ab
and then you distribute the a to the c to get ac
and then you add them together to get ab + ac
I have come across this explanation in several places, and once again real damage is done to the language, and real confusion sown. “Distribute” means “to spread or share out” as in “The Arts Council distributed its funds unevenly, as usual. Opera got the lion’s share.” So it is NOT the a that is distributed. I tried to find a definition of the term in wordy form as it applies to algebra systems but failed. Heavy thinking produced the “answer”. What is being distributed is the second factor on the left.
Take 3 x 7. We know that the value of this is 21
Distribute, or spread out, the 7 as 2 + 5 . . . . . . . . the b + c
Then 3 x (2 + 5) has the value 21
But so does 3 x 2 + 3 x 5. To check, get out the blocks !
So 3 x (2 + 5) = 3 x 2 + 3 x 5 ……… The Law !
Regarding the “second” version of the distributive property, a(b – c) = ab – ac, this cannot just be stated, and you won’t find it in any abstract algebra texts. Since the students are looking at this before they have encountered the signed number system, a proof must not involve negative numbers, as a, b and c are all natural numbers. It can be done, and here it is:
set b – c equal to w (why not!)
then b = c + w
multiply both sides by a
ab = a(c + w)
expand the right hand side by the distributive law
ab = ac + aw
subtract ac from both sides
ab – ac = aw
replace w by b – c, and then
ab – ac = a(b – c)