Zero Divided by Zero Conceptualized
0 / 0 == infinity But my question is I have 0 chocolates and
I want to distribute for 0 children and hence 0 children got 0 chocolate
each in hand.... In this case 0 / 0 == 0 ? How can explain my question
leads to infinity?
0/0 is not infinity, and division by zero does not lend itself to very
satisfying conceptualizations. Chocolate does not help.
But think about this:
x/0 = -infinity for all x less than zero, +infinity for all x greater
Also, 0/x = 0 for all x except x exactly zero, and
x/x = 1 for all x.
0/0 does not equal infinity. It is undefined. There is not enough
information to determine the ration. The value depends on what happens
"near" zero. You must have a numerator and denominator that in some way
depend on each other. From a mathematical point of view, this can be
two functions of the same variable, such as f(x)=2x and g(x)=3x. From a
real-life point of view, this can be two quantities that depend on one
another, such as the distance traveled by two racecars in the same race.
As a simple example, consider the mathematical example. At x=0, both
f(x) and g(x) equal zero. The ratio of the two, f(x)/g(x), equals
(2x)/(3x)=2/3 as x approaches zero.
For another example, let h(x)=x^2 (i.e. x-squared). At x=0, both f(x)
and h(x) equal zero. The ratio of the two, f(x)/g(x), equals
(2x)/(x^2)=2/x equals infinity as x approaches zero.
0/0 all by itself does not give enough information to define the ratio.
Dr. Ken Mellendorf
Illinois Central College
Actually, zero divided by zero is not necessarily infinity. ANY number
qualifies as zero divided by zero. It is when you get to dividing
NON-zero numbers by zero that you confront infinities.
Think of this in terms of the definition of division. A divided by B
means: How many times must you subtract B from A to reach zero?
For A divided by zero, where A is any number except zero, the number is
not even infinity, because infinity itself is not big enough. No matter
how many times you subtract zero from, say, five, you will never, ever
reach zero. So even infinity is not big enough to be 5/0.
What does this tell us about zero divided by zero? Well, how many times
must you subtract zero from zero in order to reach zero?
Zero times? Sure. That works.
One time? That works too.
Two times? Yes. If you subtract zero from zero twice, the result is
Pi times? Again, if you subtract pi zeroes from zero, the result is
We can do this with ANY NUMBER THERE IS, even zero. So, zero divided by
zero is truly a special way to define a number. The answer can be
infinity, or it can be zero, or absolutely anything else. All numbers
satisfy the operation.
Department of Physics and Astronomy
University of Wyoming
You have started with an inaccurate assumption, that is: 0/0 = infinity.
The ratio 0/0 is called "indeterminate" because it is defined in terms of
the limit (as x ---> 0) of the numerator N(x) divided by the limit
(as x --->0) of the denominator D(x). If N(x) approaches zero "faster" than
D(x) the ratio is zero. If D(x) approaches zero "faster" than N(x) the ratio
approaches infinity. They may approach zero at different, but finite, rates.
If they approach zero "at exactly the same rate" you have to apply the
The rule(s) for determining the limit of a function of the form: N(x) / D(x)
is called L'Hopital's rule, also spelled
L'Hospital's rule -- I think the reason for the difference is that the "s" is
silent in French, but my French is limited.
You will find the mechanics of the application in most introductory calculus
texts. It involves knowing how to determine derivatives of functions, so it
is not treated at levels lower than introductory calculus.
Actually any division by zero, to a mathematician, is simply undefined.
Your example makes sense, but runs smack into the unyielding definitions
of mathematics. Mathematicians it seems are not very flexible on this
The definition of division states: a/b = c if and only if c x b = a. In
other words, if you cannot reverse a division by multiplication it does
not fit the definition. It is a problem.
Division by zero fails the definition because, if b =0, then any c will
do since b x c = 0 and you can't get back to the original, a.
4/0 = anything. Anything x 0 = 0 and we can never recover the 4, even
if the answer were infinity, so division by zero is outside the
definition or is undefined.
As for 0/0, you can use any number for the answer, c, and it will
satisfy the definition. You may say infinity, and I will say 11 43/52.
Can we both be right? (infinity x 0 = 0 and 11 43/52 x 0 = 0) And
anyway, isn't anything divided by itself supposed to equal 1?? Oh oh.
Since the conflicts cannot be solved, division by zero is ignored as
being "undefined". And in many cases it does violate the definition of
division as we see above.
Having said that, 1/x approaches infinity as x decreases to near zero,
but if x ever exactly gets to equal zero, the answer becomes undefined.
This may seem like nit-picking but, from my experience, the ideas of
division by zero and infinity are stumbling points to some Calculus I
Oklahoma State University - Okmulgee, OK
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Update: June 2012