Math Is Fun Forum

Sven

Member

Registered: 2014-05-02

Posts: 18

Multiplication algorithm reason?

I know it seems crazy but I want to know why a( x+ y). Is ax+ay, I know it is distributive law but how do you prove this law.plz help
God bless

Bob

Administrator

Registered: 2010-06-20

Posts: 10,158

Re: Multiplication algorithm reason?

hi Sven,

In number theory, it is an axiom.  Axioms cannot be proved; they are the building blocks for the theory and are assumed to be true.  So an alternative question might be "Why have the distributive law as an axiom?"

And my answer to that would be because it seems to work well with the way numbers are added and multiplied.  Test it out with some numbers and it seems to work.  Can you try it for 'all' numbers in one go?  Well you can take an application of multiplication and test it for that:

eg.  If I want the area of a rectangle I know you multiply the width by the length.  So let's say I've got a rectangle that is x + y in length, and is 'a' wide.

Then its area is a(x + y).

But I'd like to be able to split the rectangle into two, one that is a by x, the other a by y.  Working out these areas separately gives ax and ay.  I also think that adding areas should correctly give the area of the compound shape and that area doesn't change if you working it out by splitting up the shape.  That means that
a(x + y) = ax + ay.

Every time you find other applications of multiplication where the distributive law gives sensible results, you get more evidence that the law as an axiom was a good choice.

Sometimes with axioms, results don't come out sensibly and the axioms have to be modified to fit the new evidence.

eg.  Numbers have a multiplicative inverse.  eg.  0.5 is the inverse of 2.

But problems arise if you go seeking the multiplicative inverse of zero.

So the revised axiom says that all numbers except 0 have a multiplicative inverse.

Hope that helps.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Sven

Member

Registered: 2014-05-02

Posts: 18

Re: Multiplication algorithm reason?

Thank you so much & God bless you always & forever