What is Division by zero: Definition and 27 Discussions

In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as


{\displaystyle {\dfrac {a}{0}}}
where a is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number which, when multiplied by 0, gives a (assuming



{\displaystyle a\neq 0}
), and so division by zero is undefined. Since any number multiplied by zero is zero, the expression


{\displaystyle {\dfrac {0}{0}}}
is also undefined; when it is the form of a limit, it is an indeterminate form. Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to


{\displaystyle {\dfrac {a}{0}}}
is contained in Anglo-Irish philosopher George Berkeley's criticism of infinitesimal calculus in 1734 in The Analyst ("ghosts of departed quantities").There are mathematical structures in which


{\displaystyle {\dfrac {a}{0}}}
is defined for some a such as in the Riemann sphere (a model of the extended complex plane) and the projectively extended real line; however, such structures do not satisfy every ordinary rule of arithmetic (the field axioms).
In computing, a program error may result from an attempt to divide by zero. Depending on the programming environment and the type of number (e.g. floating point, integer) being divided by zero, it may generate positive or negative infinity by the IEEE 754 floating point standard, generate an exception, generate an error message, cause the program to terminate, result in a special not-a-number value, or a crash.

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  1. fresh_42

    Insights Why Division by Zero is a Bad Idea

    Continue reading...
  2. A

    B Confusion about division by zero in sets

    So the confusion here is that division by zero is often said to be undefined. So whereas, the point (0,0) certainly appears in the set of values where x=y, does the point (0,0) appear in the set of values where 1=y/x. Why or why not? In other words are the set of points where x=y the same as...
  3. AtoMick-u235

    Why does a division by zero produce an error or undefined message?

    A division by zero is a mathematical oddity, , if a computer or calculator etc performs a 1\0 they would crash, the OS prevents the execution, and displays ERROR or UNDEFINED, , , some versions of windows calculator display positive or negative infinity, , try it Any ideas on this ?
  4. D

    B Why 1 / ∞ = 0 but ∞ * 0 is not equal to 1?

    As we know those relations are true: if a/b = c, then a = b*c and b = a/c Therefore if 1/ ∞ = 0, ∞ * 0 should be equal to 1 and 1/0 = ∞
  5. N

    I Is a Vertical Line the Correct Depiction for this Equation?

    Recently, after work beers with a colleague went down a bit of rabbit hole as he attempted to "red pill" me on a couple of topics. One of those happen to relate to the below self-published paper, which was written by someone he went to college with...
  6. D

    B Division by Zero: Is it Defined?

    Hi. x2 = x3 / x but x2 is defined for all x and equals zero at x=0 but what happens for x3 / x at x=o ? Is it defined at x=o ? Does it equal zero ? If not what is causing this anomaly ? Thanks
  7. Mr Indeterminate

    I Where has this proof gone wrong? ∞= 1/0

    Now I expect that most of you on this forum would be familiar with the equality between point nine reoccurring and one: 0.999...=1 If your not familiar please review https://en.wikipedia.org/wiki/0.999... Now this equality can be used to imply something else, which is rather heterodox...
  8. M

    MHB Why is dividing by zero impossible in math?

    We know division by zero is not possible but what is the math reason why it is impossible to divide by zero?
  9. I

    I Can Division by Zero Be Solved in Mathematics?

    First of all, I didn't know where to put this in general math or differential equations. Let's start with the basic, x/0 = α. Where α is every number and decimal number from -∞ to +∞, by rearranging, we get x = 0α, x= 0, therefore only 0/0 = α. Now, we can integrate this with graphs. Take the...
  10. Why dividing by zero is undefined | Functions and their graphs | Algebra II | Khan Academy

    Why dividing by zero is undefined | Functions and their graphs | Algebra II | Khan Academy

    Watch the next lesson: https://www.khanacademy.org/math/algebra2/functions_and_graphs/undefined_indeterminate/v/why-zero-divided-by-zero-is-undefined-indeter...
  11. Why zero divided by zero is undefined/indeterminate | Algebra II | Khan Academy - YouTube

    Why zero divided by zero is undefined/indeterminate | Algebra II | Khan Academy - YouTube

    Multiple arguments for what we could get when we divide zero by zero. We will later see that this can be considered indeterminate Watch the next lesson: http...
  12. U

    Fourier's method, division by zero

    Homework Statement Solve the BVP for a vibrating string with Separation of Variables/Fourier's method. \frac{\partial ^2}{\partial ^2 t} u(x,t) = c^2 \frac{\partial ^2}{\partial ^2 x} u(x,t) The string is of length L with each end fixed, ie u(0,t) = u(L,t) = 0 The Attempt at a Solution I...
  13. S

    Divide by Zero Error: Solving Pendulum Radius

    I have data from a pendulum and I am using it to work out the radius of the pendulum. I have acceleration in the x and y directions and so thought this would be easy enough. Simply I determine the (velocity in the x direction)^2/acceleration in the y direction. However when I use python to give...
  14. Z

    Fortran [FORTRAN] Division by zero allowed?

    Hi, I have an issue where I am squaring a very small complex number 1e-20 (FALFA2) which seems to make the value zero (both the real and imag part) and then dividing by it, the problem line is (all the numbers are complex): GALFA = ( FBETA * FAA / FALFA2 ) - ( FAB / FALFA )...
  15. P

    Simple Proof That Division By Zero Is Impossible

    x / y = z, so z * y = x 1 / 0 = x, so x * 0 = 1 But 0 does not equal 1, so x / 0 is unsolvable. Oh, and I'm new to forums so if this shouldn't be here you can delete it.
  16. S

    Division By Zero: Why Do People Call it Infinity?

    In most cases division be zero ends up with not defined , but why do people sometimes call it infinity ?
  17. Beer w/Straw

    Why is division by zero undefined?

    This question seems to befall everyone at one point or another. So much so I begin to get deliberately silly when it is asked http://www.wolframalpha.com/input/?i=Abs%5B1%2F0%5D Anyway, I'm wondering if there is a sticky present on these forums that addresses it specifically. Something...
  18. DocZaius

    Unphysical Division By Zero for Zero Field Location

    Unphysical Division By Zero for Trivial Case of Zero Field Location Homework Statement Q1 and Q2 are two positive charges a distance s apart. Find the distance x from Q1 where the field is zero. Homework Equations E=\frac{kq}{r^2} The Attempt at a Solution Let's put Q1 at the...
  19. A

    Measurable functions and division by zero

    Consider the function g(t) = f(t)/t on [0,1], where f is measurable on [0,1]. Does it follow that g is measurable on [0,1]? I know there's a problem -- namely, division by zero -- only on a set of measure zero -- namely, \{0\} -- and that g agrees with the measurable function g_0 = g|_{(0,1]}...
  20. B

    Can Division by Zero Be Defined in a More General Number Group?

    I assume everyone who saw this thread first rolled their eyes. Sorry for that. My question is an innocent one, as I am completely uneducated in number theory. My understanding is that numbers are grouped in a sort of Russian doll fashion, with each successive group encompassing all previous...
  21. V

    Converting from Cartesian to Cylindrical coords - but division by zero

    Converting from Cartesian to Cylindrical coords - but division by zero! Homework Statement Let's say I want to convert the point P(0, -4, 3) to cylindrical. To convert from Cartesian to Cylindrical coordinates, one must use the formulas listed below. Homework Equations...
  22. A

    Division by Zero: What Does it Mean?

    Good morning all I want to know why 0 divide anything = zero, and why division by zero is undefined? regards Blog Insight: Division by zero
  23. J

    Imaginary Number to Indicate Division by Zero?

    Forgive me if I'm being ignorant, but this recently occurred to me. We all know division by zero is undefinfed, but \sqrt {-1} used to be undefined too, until i was created. Has anyone ever proposed an imaginary number to indicate the result after division by zero?
  24. E

    Divide By Zero: Is It Ever Logical Not To?

    Can somebody please tell me any case where it is logical to NOT divide by zero? I know division by zero is illogical itself, but usually when you divide by zero the result should be zero anyway. :rolleyes:
  25. Shahil

    Division by Zero - Can it be Done?

    Division by ZERO! Hey guys! Was just pondering... ALL my mathematical subject lecturers go on and on and on about trying to avoid division by zero. Fine - after all, it makes logical nonsense to divide by zero but my pondering thoughts led me somewhere... Is it possible to do some...
  26. Shahil

    Heaviside Method Division By Zero

    Help! :confused: Here I am busy doing some Laplace Transforms for my Maths 2 paper on Monday when suddenly to my surprise, an apparent "mistake" appears! GASP! :eek: It's got to do with the Heaviside "cover up" method. To work out a problem, you need to multiply the opposing side by...
  27. abstract

    Starting from 2 to remove division by zero from the equation.

    The number 2 What if the number 2 is the origin of whole numbers? (Instead of 0) The criteria for a whole number would contain that a whole number would primarily need to be wholly divisble by 2. does 2/2 = 2 as there are 2 combinations of the halves of 2 I think this can be...