0/0? 0:0? What will I get if I divide ZERO by ZERO? 0:0? 0/0? What's the mystery behind it?

Question

What's the answer actually? Zero? Infinity? Undefined?
How do you prove your answer?

Answer 1

For any nonzero number divided by zero, the answer is undefined.
Division is the inverse operation to multiplication.
We know that 6 ÷ 3 = 2, because 2 × 3 = 6.

Okay, fill in the blank:
2 ÷ 0 = __. This means the same as __ × 0 = 2.

But this causes a big problem! Anything times zero is zero (not 2), and there is no number that you can use to fill in the blank. That's why 2 ÷ 0 is undefined.

You can do this with any nonzero number. 6 ÷ 0 is undefined... -37 ÷ 0 is undefined... any nonzero divided by zero is undefined.

The case of 0 ÷ 0 is quite different, though. Try the "fill in the blank" technique:
0 ÷ 0 = __. This means the same as __ × 0 = 0.

In this case, any real number makes the multiplication correct, and that causes a different problem. When a contradiction in math causes multiple possibilities for answers, this is called an indeterminate form. Indeterminates come up from time to time in the calculus when finding limits and derivatives. You don't have to worry about that 'til you get there, though. For the time-being, this should suffice:

n ÷ 0 is undefined for any n ≠ 0.
0 ÷ 0 is an indeterminate form.
In either case, dividing by zero is a definite math no-no!

Answer 2

As you've probably noticed, there's no way to divide zero by zero. When dividing 12/3, for instance, we're asking how many 3's added together make 12. The answer is: four 3's added together make 12, and therefore we say that 12/3 = 4.

But the problem when using zeroes is that it's just as accurate to say that *five* 0's added together make 0 as it is to say that *nineteen* 0's added together make 0. Or one hundred 0's. Or none at all.

Since any number, therefore, *could* be the result of 0/0, we say that 0/0 is "indeterminate." More generally, we teach young math students that dividing by zero in any circumstance is a no-no, without going into the details (most young math students are stressed enough ;-) ).

Hope that helps!

Answer 3

1) Math definition of division says: a/b , with b different from zero, is a real number.
2) So, we cannot divide a number by zero. 5/0 or 0/0 are not numbers
3) When we are studying limits of functions we say that the limit is a number when the function value is getting near and near that value. So we say that the limit of x^2 - 1 is zero when x tends to 1, because the function value is getting near and near zero... that do not require the value will be exactly zero.
4) When we have a fractional function like (x^2-)/(x-1) and we do x tend to 1, then both numerator and denominator tend to zero... it will no be zero... it will be near and near zero. In Math this situation is represented by the symbol 0/0. It is no a number, it is just an information. The limit of a function in such situation may exist or not... so Math understand 0/0 as an information that you do not know yet the result, or undetermined formula.

Answer 4

If you think about it, to say that a/b=c is the same thing as saying that c*b=a. Using that relation, we'll set 0/0=x, then relate it to 0x=0. The question to answer is what number could you substitute in for x, so that the equation holds? What number could you multiply by 0 to get 0? Of course the answer is that ANY number would fit the equation as x. Technically, 0/0 could equal anything. By convention, we say that 0/0 is "indeterminate".

Likewise, using something like 1/0=x, that relates to 0x=1, and here no number could be multiplied by 0 to get 1, so we say that 1/0 is "undefined" because nothing works.

Answer 5

0/0 or 0:0 is not a valid operation. Contrary to the belief that you can perform any arithmetic operation with any two numbers, there are some operations which can be performed between some, but not all, the numbers.

To be even more precise, not even 2/0 (or anything/0) is a valid operation. Many people think it is infinity (and symbolically, it is), but infinity is not a real number (rather, it is an extended real number).

Answer 6

X,Y,Z Real numbers,
Dividing X by Y is defined as multiplying X with the inverse of Y.
the Inverse of Y is defined as an element Z such that Y*Z = 1
suppose you allow 0 to have an inverse X
thus 0 * X = 1, since 0 times any number = 0 there is no such X.

What you can do is define a completely new Number that is not a Real number, denote it with INF

You extend the set of real numbers with INF
with rules INF + INF = INF
n/0 = INF
n/INF = 0
INF - INF undefined....


good luck

Answer 7

Logically, if you divide zero into zero, your answer is an infinite set of numbers. Anything you multiply by zero gives you zero, so anything can be the answer to 0/0.

{all real numbers}

Answer 8

Well 0/0 is generally undefined, in that it can have many values, some of which may not exist(like infinity)
To give it meaning you need to assign functions to the numerator and denominator.
say f(x)= x
and g(x)= x
then f(x)/g(x) as x->0 is a 0/0
but there's a useful result in analysis that for differentiable functions the ratio of functions equal the respective ratio of derivatives. This is easily proven and is known as L'Hopital's rule.
by this f(x)/g(x) = f'(x)/g'(x) = 1/1=1(as x goes to zero)
but say f(x) = sin x - cos x
and g(x)= x - pi/4
when x->pi/4
f(x)/g(x)= f'(x)/g'(x)= (cos x + sin x)/ 1
as x-> pi/4
the 0/0 is cos pi/4 + sin pi/4 = 2/ V2 = V2

Answer 9

Infinity/ Undefined.

Answer 10

undefined, you cannot divide anything by zero....

ex.. you can divide a number like 0/1 , but not 1/0.....