0 divided by 0?

Question

At work we got into a discussion about 0 divided by 0.

The classic arguments;
0/a = 0 because a * 0 = 0 (associative)

a/0 = infinity.
Why?
a / 1 = a * 1
a / .1 = a * 10
a / .01 = a * 100
a / .001 = a * 1000
As you approach dividing by 0, the number becomes exponentially larger.
Until you reach 0 where a/0=infinity.

Unfortunately, the associative property does not seem to jive with the infinity solution.

if a / 0 = b then a = b * 0

But that's only because we are making the assumption that b * 0 = 0.

I contend that multiplying by zero is not that easy, and perhaps we need to treat 0 as an imaginary number instead of a real number.

So I guess my question is, "Is the simplest form of 0 * a actually equal to 0?" or is it equal to a0, similar to how a * i (square of negative 1) = ai?

Should we treat 0 as an imaginary number when multiplying and dividing?

Thanks

Answer 1

0 is the additive identity, which means that:



a + 0 = a (which implies that a - a = 0)



Multiplying both sides by b gives:



ab + 0b = ab



Subtracting ab from both sides gives.



0b = ab - ab



But because a - a = 0, we also know that



0b = ab - ab = 0



So,



0b = 0



Multiplying by 0 is straightforward.



However,



If 0/0 = a



Then 0 = 0a.



This is true for all a, so 0/0 is not well-defined.



In your example, suppose you had:



1 / 1 = 1

.1 / .1 = 1

.01 / .01 = 1

.001 / .001 = 1



As you approach dividing 0/0 this way, the number always stays the same. There are really many ways to approach dividing 0/0.

Answer 2

0/0=0

Answer 3

Anything divided by 0 is actually undefined, not infinity. When you think about how that works, it makes sense:



Dividing a numerator by smaller numbers makes your number larger. Divide 4 by 2, you get 2. Divide 4 by 1, you get 4. Divide 4 by 0.5, you get 8. And so on until you are dividing by a number so small it's pretty close to 0. The overall number gets larger and goes off into infinity.



But, if you divide a negative number in the same manner, your number gets smaller. -4 by 2 is -2, -4 by 1 is -4, -4 by 0.5 is -8, etc. As you approach 0, your numbers go off into negative infinity.



You'll have the same weird effect if you divide by negative numbers - a positive divided by an increasingly smaller negative will yield negative infinity, and a negative divided by an increasingly smaller negative will yield positive infinity.



Your argument works fine when you consider dividing by a positive, yet progressively smaller number. But considering that you can divide by a negative number just the same, you can also end up with a negative infinity for your same problem. Because of this problem with having a dual value (you can't be both positive and negative infinity), dividing by 0 creates an undefined number. So a/0 = undefined, and there's not much you can do with that equation.



BTW, I always thought that the associative property had to do with grouping numbers - i.e. a(bc) = (ab)c and (a + b) + c = a + (b + c).

Answer 4

This is a Frequently Asked Question; the following is my previous answer:



This question may be asked so frequently (along with the infamous 0.999...) because no concrete axioms or definitions are ever given for the real number system until one studies higher algebra or real analysis; it is assumed that everyone knows what a real number is until then. The proliferation of these questions show that this assumption is no good.

For the sake of conciseness, I will not define what a real number is here, as only the algebraic structure of the real numbers is necessary to answer this question. The algebraic structure of the real numbers is what we call a field. A field is an algebraic structure that follows a finite set of axioms. These axioms define what statements are true and false in this system based on the laws of logic. You can find the field axioms here: http://mathworld.wolfram.com/Field.html .

Each line is preceded by "For all real numbers a, b and c, the following is true...". Note that the proper form of the additive identity and multiplicative identity is that it prescribes the existence of some number 1 and some number 0 such that the equations hold. These same numbers are then used to prescribe the existence of additive and multiplicative inverses.

Division a/b is just shorthand for a*(b^(-1)). So in your question, you are looking for 0*0^(-1). It is easy to show that there is no such number as 0^(-1) in any field. Suppose 0^(-1) exists.

Then 0*0^(-1) = 1. But multiplying both sides by 0^(-1) gives 0 = 0^(-1), which means 0*0 = 0 = 1. This is obviously false.

Suppose 0 = 1. 0*a = a for all numbers a. But 0*a = (1 + (-1))*a = a + (-a) = 0. Thus 0 is not equivalent to 1 and our sole assumption that 0^(-1) exists must be false. Tus the expression 0*0^(-1) is undefined in a field.

Proof by contradiction is very powerful for many types of statements.

As to the limit argument, note that you are evaluating a limit over a plane of the function f(x,y) = x/y. As we approach (0,0) along the x-axis, this limit is 0, but as we approach (0,0) along the y-axis, the limit diverges. Thus, the limit of f(x,y) at (0,0) does not exist, so f(0,0) remains undefined with no continuous way to complete it. The limit is not indeterminate for this function; it does not exist.

The expression 0/0 is said to be indeterminate as a limiting form because one can have many functions that approach the form 0/0 that have convergent limits of different values. For example, the limit as x approaches 0 of [sin(x)]/x yields the indeterminate form 0/0, but L'Hopital's rule reveals the limit to be 1. The expressions listed on this page: http://mathworld.wolfram.com/Indeterminate.html are called indeterminate because they do not have a fixed behavior when studied as limits of expressions.

Answer 5

Mathematically 0/0 is said to be indeterminate

its neither 1(a/a=1) nor infinity(a/0=infinity) nor 0(0/a=0)

it can be spoken in terms of reaching zero but not actually zero.that is a/a of the form such that a tends to zero and has a very small positive or negative value in that case it can be said to be 1.sin X=0 when X=0 so (Sin X)/X is indeterminate but as X-->0 (sin X)/X is 1

Answer 6

Normally any number divided by 0 equals infinity. But if you have nothing it is still 0 when divided by 0!

Answer 7

0/a = 0 is true



a/0 is undefined. the limit as x -> 0 of a/x is unbounded. (that means it approaches 'infinity')



0/0 is also undefined. the limit as x -> 0 of x/x is 1.



"if a / 0 = b then a = b * 0



But that's only because we are making the assumption that b * 0 = 0."



No, that's because you're ignoring the fact that to reach the second equation, you're multiplying 'a/0' and 'b' by zero.



"treat 0 as an imaginary number instead of a real number."



so ... the number of elephants in my sock drawer is imaginary? heh



"Should we treat 0 as an imaginary number when multiplying and dividing?"



No, but you should treat 'i' as an imaginary number.

Answer 8

Confusing actual numbers with function limits is a common mistake.



TRUE: when x tends to zero, lim(1/x) = infinity.

FALSE: 1/0 = infinity.



Both in the field of real numbers (R) and complex numbers (C), The division by zero is an *undefined* operation, whatever the dividend is. Zero itself (as in 0/0) is no exception.

Look, this is not something you may subvert: it's an axiom of the algebraic structure that we call "field". Both R and C are fields, therefore there's no escape.

Answer 9

Mathematically dividing by 0 is undefined !!!! ( not undetermined but undefined !!!!)

In mathematics the definition of division by number "a" is multiplying by the inverse number of "a".

And the definition of the inverse number is :

if x*y = 1, then we call y as the inverse of x.



Well 0 has no inverse number ,for there is no number x such that 0*x=1.

Hence we cannot define the division by 0.

Answer 10

Zero divided by zero does not exist. By the way, infinity does not exist either, so if you want to define it as that, go ahead. It won't hurt anything.



I like the idea of special treatment for multiplication by zero. I haven't pondered it but there could be some merit there.

Answer 11

Bottom line? You're thinking too much. Take yourself back to junior high math...



Zero times anything is zero. That only stands to logical reason. If you've got two rows of five chairs, you have ten chairs, but if you have zero rows, you haven't got any chairs. But you cannot divide anything by zero. Something divided by zero is not infinity. It doesn't exist. This is one of the cardinal rules of mathematics: THOU SHALT NOT DIVIDE BY ZERO!

Answer 12

manojlds is correct. 0/0 is one of the 7 indeterminates in mathematics. (Infinity divided by infinity is another.) Although indeterminate and undefined basically mean the same thing, the correct terminology is indeterminate.

Answer 13

Errrr... I'd rather think in practical terms:

You have no cake and you want to divide it by nobody. How many no slices does nobody get?

err... I mean, you can't divide nothing by nobody. Or maybe you can, but nobody gets nothing.



ERRRRRRRRRRRRR



I better shut up

Answer 14

why to complicate things? 0 / 0 = 0, period.

Answer 15

its not infinity or 0 or anything..its called undefined or indeterminate as someone else said..and thats the truth.

Answer 16

uhh i dint get anything you just said



but hmm YEH!.



*nods head with a big smile*



:)

Answer 17

Alrighty then... i think you should get out more

Answer 18

haha..i dunno,personally not good in maths...(."<)