Lord Boxtop Posted:

0 / 0 = infinity

I win.

Name: Srinivasarao

Status: other

Grade: other

Location: IN

Question: 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?

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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

than zero.

Also, 0/x = 0 for all x except x exactly zero, and

x/x = 1 for all x.

—

Tim Mooney

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Srinivasarao,

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

Physics Instructor

Illinois Central College

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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

zero.

Pi times? Again, if you subtract pi zeroes from zero, the result is

zero.

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.

Richard Barrans

Department of Physics and Astronomy

University of Wyoming

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You have started with an inaccurate bumumption, 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

test again.

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.

Vince Calder

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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

point.

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

students.

Bob Avakian

Oklahoma State University – Okmulgee, OK

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