Understanding the Distribution Rule in Algebra: Proof and Logic Explained"

[mtanti]

I need to know if there is any form of proof for the distribution rule in algebra. Why is it that a(b+c) = ab+ac or more simply 2(3+4)=6+8? What kind of logic makes you arrive to that rule?

 

[dmoravec]

well for basic algebra it is an axiom, which just means it is assumed, not proved. There are types of algebra (Lie algebra comes to mind) that don't have the distributive rule.

 

[mtanti]

Yes but there is a reason to believe it's true, like a/a = 1 and a-a=0 because they make sense when you try it in practice. a * 1/a means you have 'a' pieces of a whole broken into 'a' equal pieces therefore you get the whole back. But the distribution is a little odd to reason out the same way...

 

[daveb]

You can reason it out with natural numbers by imagining you have a set of m+n objects which are divided up into two groups, one group labeled m and one labeled n. Then imagine that you have L sets of these. You can rearrange the objects to have one group which contains all the objects labeled m, so you have L*m of these. Similarly, the other group contains L*n objects.

From there, it's little effort to extend the analogy to integers and then rationals. Extending it to the reals is a bit of a leap of faith I would imagine (or at least I can't think of any way to visualize it).

 

[Swapnil]

I need to know if there is any form of proof for the distribution rule in algebra. Why is it that a(b+c) = ab+ac or more simply 2(3+4)=6+8? What kind of logic makes you arrive to that rule?

I personally treat them as axioms (for now at least). A while back I was also curious as you are about this question and I found out that usually most mathematicians accept the basic laws of arithmetic (like the ditributive law) as axioms. However, you could prove the distributive law (and all the other basic laws in arithmtic) using something called Peano's axioms. Here is a little link that I found which derives all the basic laws of arithmetic using Peano's axioms. However, it does require some deep understanding of mathematics (like sets and functions) to grasp this concept (I personally am still learning ).

http://mathforum.org/library/drmath/view/51563.html

FYI: Usually, Peano's axioms are covered in a college-level introductory course to number theory.

 

[mtanti]

Axioms are really funny in mathematics considering that mathematics is used to 'prove' practically anything such as physics and computing. You are proving something using something else which is not proven. My mathematics teacher said that if I manage to disprove an axiom than the whole subject of mathematics and all other subjects using it will collapse and we will need to reprove everything from the start. Sounds exciting...

So the reasoning behind distributivity is that if you have M objects and N other objects and have L of M and L of N then you have LM+LN objects. From a little diagram...

L=2, M=2, N=3
xx xxx
xx xxx

You either have 2 columns of different width (M and N) and same height
LM+LN
Or you have 2 rows of equal width (M+N)
L(M+N)

But is this a proof? I guess I'm just paranoid... However this is inductive science at this point. You can only conclude from what you see and cannot prove it if not by trying out every possible number...

 

[Swapnil]

Axioms are really funny in mathematics considering that mathematics is used to 'prove' practically anything such as physics and computing. You are proving something using something else which is not proven. My mathematics teacher said that if I manage to disprove an axiom than the whole subject of mathematics and all other subjects using it will collapse and we will need to reprove everything from the start. Sounds exciting...

Axioms are really a necessacity in mathematics or science. I mean think about it. You can't keep asking Why? all the time. Imagine that a child asked you a question.You replied, "Well, this is because of ..." Then he asks, "Why is that?". Then you reply, "Well, because ..." He again asks "Why is that?" If he keeps this up, you will run out of explnations and at some point you would eventually have to say, "Well, that's just the way it is."

So you see, you HAVE TO HAVE axioms, they are a part of reality.

So the reasoning behind distributivity is that if you have M objects and N other objects and have L of M and L of N then you have LM+LN objects. From a little diagram...

L=2, M=2, N=3
xx xxx
xx xxx

You either have 2 columns of different width (M and N) and same height
LM+LN
Or you have 2 rows of equal width (M+N)
L(M+N)

But is this a proof? I guess I'm just paranoid... However this is inductive science at this point. You can only conclude from what you see and cannot prove it if not by trying out every possible number...

Your reasoning is quite intuitive. And you are correct in pointing out that your supposed "proof" is very limited and cannot be easily generalized.
Either way, if you ask a mathematician, he/she would surely disagree that you have a proof at your hand (geometry doesn't have such a great reputation in mathematics for some reason ).

 

[mtanti]

Oh, I didn't know geometry cannot be used. Too bad, geometry, although can be used for a limited number of cases is the only way to know the bare facts I believe... Just playing around with numbers and letters won't let you understand. At least for axioms there is no other way to 'prove' it I guess...

 

[Gelsamel Epsilon]

a(b+c) just means a times everything in the brackets, or a times b plus a times c. Now if it isn't logical that a*b is ab, and a*c is ac, that a*b + a*c = ab+ac then I don't know what is. Questioning this is like questioning the procedure of multiplication.

 

[mtanti]

a(b+c) just means a times everything in the brackets, or a times b plus a times c. Now if it isn't logical that a*b is ab, and a*c is ac, that a*b + a*c = ab+ac then I don't know what is. Questioning this is like questioning the procedure of multiplication.

You didn't understand the question... Its not about using * or not. It's about the logic used when stating that a(b+c) = ab + ac

What you said in the first part of your statement:

a(b+c) just means a times everything in the brackets, or a times b plus a times c.

Is what the whole question is!

 

[HallsofIvy]

Axioms are really funny in mathematics considering that mathematics is used to 'prove' practically anything such as physics and computing. You are proving something using something else which is not proven. My mathematics teacher said that if I manage to disprove an axiom than the whole subject of mathematics and all other subjects using it will collapse and we will need to reprove everything from the start. Sounds exciting...

So the reasoning behind distributivity is that if you have M objects and N other objects and have L of M and L of N then you have LM+LN objects. From a little diagram...

L=2, M=2, N=3
xx xxx
xx xxx

You either have 2 columns of different width (M and N) and same height
LM+LN
Or you have 2 rows of equal width (M+N)
L(M+N)

But is this a proof? I guess I'm just paranoid... However this is inductive science at this point. You can only conclude from what you see and cannot prove it if not by trying out every possible number...

No, that's not a proof, mathematically, because your original assertions:
L=2, M=2, N=3
xx xxx
xx xxx
is only an example, not a general proof and, in any case, is not how multiplication is defined.

 

[HallsofIvy]

I personally treat them as axioms (for now at least). A while back I was also curious as you are about this question and I found out that usually most mathematicians accept the basic laws of arithmetic (like the ditributive law) as axioms. However, you could prove the distributive law (and all the other basic laws in arithmtic) using something called Peano's axioms. Here is a little link that I found which derives all the basic laws of arithmetic using Peano's axioms. However, it does require some deep understanding of mathematics (like sets and functions) to grasp this concept (I personally am still learning ).

http://mathforum.org/library/drmath/view/51563.html

FYI: Usually, Peano's axioms are covered in a college-level introductory course to number theory.

That's an excellent link. I started to include a link to a paper I wrote myself on the topic but that's better.

On the question of "axioms" in general, remember that one of the strengths of mathematics is it's generality. Theorems in calculus, that were orginally developed to solve physics problems can be applied to biology, chemistry, economics, etc. In order to understand why, you need to think about another important property of axiomatic systems: "undefined terms".

Every "mathematical system" (or "axiomatic system") starts with "undefined terms", definitions, and "axioms" which state relations between the various undefined terms and definitions. Then we prove theorems from those. The "undefined terms" are "placeholders". Perhaps it is best to think of them as "templates". Just as a business man can buy generic software and fit it to his business by setting variables to specific values, so a physicist (biologist, chemist, economist, etc.) can take a mathematical system and fit it to his needs by assigning specific meanings to the "undefined terms". The only requirement be that the axioms be demonstrably (by experimentation, say) true for those meanings. Once you know that, then all theorems derived from the axioms must be true and all methods of solving problems derived from those theorems must work.
[Of course, you can't physically (biologically, chemically, economically, etc.) prove that the axioms are true. All experiments involve approximation so the best you can hope for is that the axioms are true to within the limits of approximation.]

 

[mtanti]

Then how is multiplication defined?

Is it possible to have a subject where the axioms cannot be used?

 

[Gelsamel Epsilon]

Multiplication is simply in the case of a*b that there are 'a' many groups of 'b'

 

[mtanti]

What if the 'a' many groups is a fraction? What does that mean?
(See thread 'Finding a fraction of a number' for a discussion about this)