Gerenuk said:
Actually that you didn't give a general answer, but the other way round. You gave a very specific answer which does not address the full question.
And in school no claimed they gave you proofs. Instead they gave an example how it worked. So don't use the word proof, if you're not sure what it is. And don't claim it's easily generalized to real numbers, if you don't know how.
But some people in this thread told you already. It's fine not being a mathematician, but then at least don't claim to have given an appropriate proof.
Fine, but don't claim to know how
my school taught me and the many other students who went there. You cannot generalize by saying that no school gave proofs; you don't know that, since you didn't go to my school. In fact, they didn't
just give an example, as you say. That would mean that they gave us a(b + c) and showed us how it worked with real numbers and expect us to go on using that law for the rest of our lives. (By "real numbers," I mean "actual numbers" in this case, so don't get all uptight). They didn't just give us something like 4(5+3) and show us that 4 * 8 is the same as (4*5)+(4*3). Sure, that's how it was
introduced, but later they
proved to us that it works for all integers. Later, when we got to real numbers (mathematically real), they showed us that it still worked via another proof (one that was based on the original proof, but took reals into account). So, at that point in our education we knew, absolutely, that the distributive law holds true for all integers and reals. This continued on through our education as we learned about different kinds of numbers.
Also, don't tell me
not to claim that my example is easily generalized to real numbers. It is, and I can. I just didn't do it, and I'm not going to. That wasn't the point. It can also be shown (quite easily) to work with Boolean algebra, but I didn't do that either. I also don't have to justify that multiplication works for all numbers. When introducing the distributive law, it is assumed that the person already knows that.
There are always assumptions made when a person asks a question. My assumption was that the OP wanted a very basic way to show that the distributive law works. If I gave someone a recipe for a cake and asked them to make it, I'm making several assumptions. For instance:
- they speak my language
- they understand (comprehend) what I'm asking
- they will do what I've asked
- they know how to read
- they know how to measure the ingredients
etc.
Obviously, if I'm going to get them to make that cake, they will have to know certain things first. Just as if someone is going to learn the distributive law, they should know how to add and multiply (as well as subtract and divide) first. And, they should have no reason to doubt that those functions work with all of the numbers that they are aware of, thus far (it is not important that they know that the functions work with say, irrational numbers if they have yet to learn what irrational numbers are, and currently have no need to use them). While irrational numbers exist and are very important, a child doesn't need them if he is counting apples, or for that matter slices of (fractions of) apples.
"The full question?" None of us knows what exactly what the OP was asking. Was he looking for an elementary school level answer or a post-graduate school level answer? Judging by his replies, I was right on the money as far as what he was looking for, so why don't you just leave it alone?
Not every mathematical question needs to be answered with a full, formal proof. While the OP asked for a proof, I simply gave him some information that would lead to a proof; information that would hopefully steer him in the right direction.
And, by the way, I never claimed that my method was a "proof." Mentallic asked for a proof, I gave an example of how to show it. I later referred to it as a proof, since that was what the OP was calling it. (My bad).
I would hate to have to talk to you in person. I can see it now:
Me: I'm going to the store.
You: How are you getting there?
Me: I'm driving my car.
You: Are you sure that the car is not carrying you, rather than you driving the car?
You: Can you prove that you are driving it?
You: How do you know that you're not just sitting in the car and the Earth is rotating beneath you to the point that when you get out of the car, you are at the store?
Me: I don't really care. I'm just going to the store.
Yes, you're right. I didn't prove it for all numbers. You win. I said, "uncle," now please stop "beating me up!"