first, this is going to get into mathematics that i’m not going to know or understand. so really, this is a doofus going into it. establishing this, let’s dive.
5 - 0 = 5
if you have five apples and take away none - you will still have five apples.
5 x 0 = 0
if you have five apples but multiply it by nothing - you would have gained nothing.
so here is my question, why is it that for subtraction we are left with the amount we started with as the answer but for multiplication we rely on the fact that we gained nothing?
what i’d like is this: to not have the mathematics here be seen like a theory of convenience (as in, because x 0 = 0 would work better in mathematical equations). maybe the fault here is using the apples as an example? are they irrelevant?
In the first example, you have a set of five from which you subtract nothing leaving you with a set of five.
In the second example, you have five sets of nothing, leaving you with nothing.
5 X 3 is the same as 3 X 5. Either 5 sets of 3 or 3 sets of 5.
A x B = B X A
5 X 0 = 0 X 5.
It’s not just five multiplied zero times. It’s also zero multiplied five times. Which is zero, and makes more intuitive sense when spoken. Five times nothing is just a lot more nothing.
Here’s one way to look at it. Take one you know, like 5 x 2. You can read that as “What is the answer when you add five twice?” Likewise, 5 x 20 is “What is the answer when you add five twenty times?” So, 5 x 0 is “What is the answer when you don’t add any fives at all?”.
It’s because zero is the identity element for addition (among the reals), but not for multiplication (that is one).
In terms of apples you could rephrase it as how many apples in total you’d have if you have zero groups (I use the word non-technically) of five apples. The answer is obviously zero apples.
The formal definitions of addition and multiplication rely on the notion of the successor of x, denoted as x’. What exactly that means is irrelevant, but we define 1 = 0’, 2 = 0’’, and so on and so forth.
Addition is defined with x + 0 = x and x + y’ = (x + y)’. Multiplication is defined with x * 0 = 0 and x * y’ = x + (x * y).
From this, it’s easy to see that x + 1 = x’.
vasyachkin, are you talking about a boolean algebra, or Z[sub]2[/sub]2? Be careful: there is no - operator in a boolean algebra.
Another thing to consider is that 5 x 1 = 5. If 5 x 0 = 5 was also true, then you could cancel out the fives in both equations and prove that 1 = 0, an obvious fallacy.
I think you’re starting to see why we math nerds like math. Pretty cool, isn’t it?