Why does multiplying by 0 equal 0?

HallsofIvy
Homework Helper
But that is a statement about apples, not mathematics. it shows that that particular interpretation of "0 times 100 apples" is "0 apples". It does NOT show "0 * 0= 0".

Hope it's not bad form to post to a thread that is over a year old. I also am interested in a proof of why multiplying a number by 0 gives 0. Both statdad and poutsos.A gave nice proofs in this thread that I understand clearly. However I am using the book "Introduction To Analysis" by Maxwell Rosenlicht and he uses a slightly different proof (in chap 2, 'The Real Number System'). I can't follow his final reasoning and hope someone here can help to explain it. I will outline his proof here: (in the following, the astirix (*) is the multiplication symbol)

Proof of a*0 = 0

Start by assuming 5 field axioms of real numbers as follows:
1. commutativity
2. associativity
3. distributivity
4. existence of neutral elements (definition of 1 and 0)
5. existence of additive and multiplicative inverses (definition of -a and 1/a)

He then proves several theorems, one of which I will state here because he uses it in the proof of a*0 = 0.

Theorem F3. For any a,b belonging to R the equation x+a = b has one and only one solution, namely x = b+(-a). He uses the uniqueness of this solution in his proof of a*0 = 0.

Now here is his proof of a*0 = 0 :

a*0 + a*0 = a * (0+0) = a*0 = a*0 + 0, so that a*0 and 0 are both solutions of the equation x + a*0 = a*0, hence equal, by F3.

That's it. I can follow all of his equalities but he loses me when he states that a*0 and 0 are both solutions of the equation. If they are solutions I can understand that they would be equal by F3.

Can anyone fill in the blanks between his equalities and the statement that a*0 and 0 are both solutions of the equation x + a*0 = a*0 ?

Sincere Thanks,
Bill

Landau
a*0 + a*0 = a * (0+0) = a*0 = a*0 + 0, so that a*0 and 0 are both solutions of the equation x + a*0 = a*0, hence equal, by F3.

(...) he loses me when he states that a*0 and 0 are both solutions of the equation. If they are solutions I can understand that they would be equal by F3.
The author has shown that a*0 + a*0 = a * (0+0) = a*0 = a*0 + 0. In particular:
(1) a*0 + a*0 = a*0
(2) a*0 + 0 = a*0

Consider the equation a*0 + x = a*0. By (1) we see a*0 is a solution. By (2) we see 0 is a solution.

Beautiful...

Thank you very much, Landau.

Bill

It helps to remind yourself that zero is an additive identity, while one is a multiplicative identity.

Let's avoid the whole "but what is nothing" issue, call the additive a, and the multiplicative m.

n+a=n
n*m=n

Now, n*a can't equal n, it can't equal m either, or there is no use assigning identities in the first place.

If adding a to n, a+n, doesn't change n, then multiplying a by n, a*n, won't change a.

If multiplying n by m, n*m, doesn't change n, and adding m to n, m+n, can't produce the same result as a+n, then n+m=r, which we can call the successor to n.

m*m=m
a*m=a

m+a=m
a+a=a

When you see a zero, think "doesn't change when adding", and one is "doesn't change when multiplying".

Assigning a value is less important than the axioms you use in most cases.

I have only one thing to say about this discussion: I opened it up, read the problem, thought of my answer, then looked at the first post and realized I just thought of the same response I posted two years ago. Obviously, little progress has been made on the math front ;)

adding, you add 100 to zero. that's 100 * 1. one instance of 100. zero instances of 100 is none.

One possible conceptual way to think about it is to say, multiplying 2 * 100 is like having 2 groups of 100 cookies on the table. Multiplying 1 * 100 is like having 1 group of 100 cookies on the table. Multiplying 0 * 100 is like having no groups of 100 cookies on the table, hence no cookies at all.

However, I agree that this is may not be very intuitive. When dealing with 0, it is sometimes more difficult to match mathematical situations to real life situations. It is probably better to understand 0 * any number = 0 just as a consequence of several properties of numbers that we take for granted.

0 * 100 is 0 groups of 100, or 100 groups of 0

if you take 0 to mean nothing, then you end up with 0

otherwise, if you don't assign a meaning to 0, then you end up with:

0 * 100 = 0 groups of 100 = 0 groups of 100 = ... ad infinitum

so the only sensible thing to do is assign it with a meaning (nothing)

also, this helps explain why you can't divide by 0

this is really a physical "proof" of the idea though

EDIT: I think the French don't have this problem …
they distinguish between (pardon my French! ) …
"multiplier par rien" and "ne multiplier par rien"
[/QUOTE]

very true :)

oops i failed with the quote thing :(
but you get the idea!
:)

Last edited:
This is far too natural to be confusing. Remember that multiplication is repeated addition. If you add 100, ZERO times, you have ZERO. If you want 100 as result, then you must add 100 ONE time.
Hi all.
Just saw this post and wanted to follow up with a question that will address this point: the issue is in the order, if you think about it. What if one sees the issue as not adding 100 zeros, but as adding 100 zero times (read this carefully, as it confuses people even if I speak it in person). Basically the statement that if you have 100 already and you don't add anything to it (as you add 100 zero times, thus add nothing or anything really). So the order of numbers comes to play, in a way, and so when lets say 8*5 is the same as 5*8, when it come to zero it's not interchangeable (or at least one can argue that perception may be argued). It is partially a language issue, I def agree with that. But it also seems to be the issue of view point and perception of the order in the world.
I’m not a mathematician by trade at all but whole zero concept always got to me. I should say that I was always somewhat bright when it came to math and I liked to think deeply about it when I was in high school. Comes from having a grandfather physicist I guess and ironically this issue was actually seeded in me by the same grandfather (though I think he only wanted to mass with me at the time and make me think). In reality I understand that the concept works and all, but it still bugs me that in practical applications there is certain way of perceiving multiplication by zero that does create some confusion.
In any event, if someone looking for some helpful links, I found these very straight forward:
http://mathforum.org/library/drmath/view/61435.html
http://www.math.utah.edu/~pa/math/0by0.html