So, why is the distributive property always true? If we check any example such as 2(3+4) with a calculator, we will always find that the result is what the property says it should be. We can either add the pieces inside the parentheses, and then multiply, or multiply each number inside the parentheses by the number outside, and then add, and we'd get the same result.
Here's a simple proof for the case when the multiplier is a natural number.
As you know from arithmetics, multiplication by a natural number simply means repeated addition. If you multiply some number by a natural number, you simply add it up as many times as is the natural number. With this knowledge at hand, we can easily prove the distributive property.
Let's say that you multiply a natural number N by a sum of a and b. Note that a and b do not have to be natural numbers.
Rewrite the product as a repeated sum
N(a+b) = (a+b) + (a+b) + (a+b) .... + (a+b). The addition repeats N times.
Move all a's to front and all b's to the back
N(a+b) = a+a+a+a...+a + b+b+b+b+...+b, where both a's and b's are added N times each.
Rewrite repeated sums as products again
Since a+a+a+a+...+a N times is simply N*a (as we know from arithmetics), and b added N times is N*b, we now have
N(a+b) = N*a + N*b.
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