I have seen some heavy handed ways of explaining the identity

a – (b + c) = a – b – c

Let us use algebra. Give the left hand side a name, say d .Then

a – (b + c) = d

This is an equation, so add (b+c) to each side and get

a = d + (b + c), then a = d + b + c as the parentheses are now superfluous.

Now subtract b from each side

a – b = d + c

Now subtract c from each side

a – b – c = d

so a – (b + c) = a – b – c

or is this too simple ? Look, no messing with p – q = p + -(q) stuff,

and no appeal to the famous distributive law.

You can do this, and other stuff, with numbers as well.

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Love it! Thanks for thinking out side the distributive box!

When I was doing maths we never heard about the three laws until starting with groups, rings and fields, I guess because they were considered as “what you can obviously do with numbers” and “algebra is all about unspecified numbers so it has the same rules”. This attempt to be “mathematical” is clearly a result of the input from mathematicians, or students of math who wanted to put their oar in. I looked up “non-associative algebras” in Google once, and the example was so far-fetched it was funny.

I really want to know what is the matter with :

“It doesn’t matter what order you write the sum of 2, 3 and 4, and it doesn’t matter where you start the calculation, because when the three oiles of things are together in a box and stirred up the total number of things does not change”.

Oh, and I love your comments.

for “oiles” read “piles”.