Can anyone help me? I am confused as to whether 0 is a multiple & indeed a common multiple of every integer. In one of my text books it clearly shows 0 as a common multiple, but in others it is not mentioned. If not, why not? Several people I've asked have said 0 is not a number. In which case what is the number between 1 and -1? I need some simple, child friendly explanations.

There was a popular-science "craze" a little while back where people would run up to mathematicians in the street and go "Ha! Zero is not a number! I read it in a magazine!". Hmmmm. Suffice that 0 IS a number just as good as 1 or -1.

As to the multiples, it's really a question of definition. For example, 2x3=6, hence 6 is a multiple of 2. 2x10 = 20 so 20 is a multiple of 2. So 2x0 = 0 and 0 is a multiple.

However, some people don't let you multiply by zero in their definition of multiples, so you really have to go with which ever definition your book is using at the time.

I wouldn't worry too much about it (unless your entire homework sheet is based on it...), it's really not that essential!

I'm sorry if this is a little vague- perhaps some number-theorist here can give you a clearer answer!

Anton.

The simple answer to this is that you would be suprised how many things in maths do not have a fixed definition, and many of these definitons are involved with 0s and 1s. Is 0 a multiple? Is 1 a prime? Should a semi-positive definitive matrix have at least one zero? (sorry, my work at the moment..)

In these case it really is up to you to define it how you want. The only important thing is to state which definition you are using when you quote your result..

So either of

• 0 is a multiple, so it is very common

• 0 is not a multiple, so it isn't common at all

I would suggest that 0 is certainly a multiple.

This is obvious since
0 x n = 0
for all numbers, n.

You could certainly define 0 not to be a multiple, but I don't really see the point. (I wouldn't do it, but other people might.) Probably the reason that some textbooks don't mention it is that it is
"trivially" a multiple of everything so it's not really very interesting compared with e.g. the fact that 12 is a multiple of 1, 2, 3, 4, 6 and 12, or that 13 is a multiple of 1 and 13 only.

Jonathan

This reminded me of my earth-shattering discovery of the possible values of 0/0:

2 x 0 = 3 x 0

implies

0/0 = 3/2...and infinite other possibilities

An even worse one is evaluating i as zero:

2 x 0 = 3 x 0 does not usually imply 0 / 0 = 3 / 2 since division by zero is conventionally undefined...

Also: a divides b if & only if there exists c such that

a x c = b

If a = 0 then a x c = 0 for all values of c, so 0 does not divide anything
except itself.

Jonathan

I still think that 0/0 has an infinite number of values. The definition of a / b is:

a / b = c

is equivalent to

a = b x c.

Let a = b = 0 and you have:

0 = 0 x c

which any value of c can satisfy. But c = 0/0. Therefore 0/0 is multivalued and returns the set of complex numbers.

Michael

Just a few points to add...

saying: ''a divides b if & only if there exists c such that a x c = b'' is simply one definition of a divides b. There is no 'fixed' one, although I will admit that is the most used one.

Also, a / b is defined to be a x c
where c is the number such that b x c = 1
(c is called the 'inverse' of b)
When b=0, there is no c such that bc=1, so you cannot divide by zero!

After what I said before, this is one point where the definiton is absolute!!

I will quote you one of my favourite alternative maths.

---------------------
Alternative Maths

Axiom 1: Same as normal maths
Axiom 2: You can divide by 0

Theorem 1: all numbers a and b are such that a=b
Therefore all numbers are solutions to all equations.
END
----------------
Would make maths so much simpler don't you think?