Zero Divided by Zero Conceptualized ```Name: Srinivasarao Status: student Age: N/A Location: N/A Country: N/A Date: N/A ``` 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? Replies: 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 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 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 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 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 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 Click here to return to the Mathematics Archives

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