What is zero divided by zero?

Answer 1

In ordinary mathematics, you may not divide by zero. It is considered undefined.

Consider the two situations:

For the inverse of multiplication 0/0 = a, a could be any number to satisfy a x 0 = 0. At the same time, division of any nonzero number, a/0 = b, there is no number a such that b x 0 = a.

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In nearly every known algebraic structure, 0/0 is an undefinable term. This means that, based on the rules that govern most of the mathematical systems we use, there isn't just one, single, definable value for the term 0/0, and believe it or not, the reason for this isn't because we're dividing by zero, it's because the division relation is defined by another relation, multiplication. You see, when we talk about "divide," what we really mean is "multiply by the inverse." For example, x/y actually means, x*y-1 where y-1 is the inverse of y. The inverse of a number is defined to be the number which, when multiplied by the original number, equals one; e.g. x*x-1 = 1. Now, in the algebraic structures we're all familiar with, any number multiplied by zero is defined to be equal to zero; e.g. 0*x = 0. So, using these definitions, what does dividing by zero, which actually means multiplying by the inverse of zero, equal? In other words, x*0-1 = ? Well, to isolate x, you would need to cancel out 0-1, but how? As anyone who's taken any sort of algebra knows, the method of isolation in these cases would be to multiply 0-1 by 0 because, as stated above, x*x-1 = 1, therefore 0*0-1 = 1. But wait, didn't I also just say that 0*x = 0? That would mean that 0*0-1 = 0, which would mean that 0 = 1. That, my friends, is called a contradiction. Zero does not equal one; therefore the term 0-1 can't be defined.

This answer may seem unsatisfactory to some people. There's got to be a way to work around this pesky contradiction, right? Actually, there is! In the branch of mathematics called abstract algebra, there exists an algebraic structure called a wheelwhich is required to have division defined everywhere within it. Therefore, in this particular algebraic structure, 0/0 must exist or else the structure isn't a wheel.

But wait, 0/0 is undefined, right? How could you ever satisfy this requirement for a wheel then?

That's easy; all you have to do is define it! Specifically, you give this quantity, 0/0, some specific algebraic properties, and then, if it ever comes up in an equation, you manipulate it within the equation using the properties you've given it. Isn't that convenient?!

"Preposterous!" you may say. "You can't simply make something up which has no tangible or rational analogue, that's cheating!" Well my dear skeptic, may I direct your attention to the following little marvel, √(-1), otherwise known as "the imaginary number," or i. That's right, I said imaginary, as in, "doesn't exist." You see, nothing multiplied by itself in our nice little world of mathematical rationality can possibly be a negative number. Unless, of course, you define something to be as such. Then...Presto! The absurd is now reality!

Let's talk about imaginary numbers for a moment. Our newly defined yet still rather imaginary friend, i, was apparently not content on simply having a nice, comfy little existence within the realm of obscure mathematics, oh no no no. It decided to defy logic and become a fairly common number; popping up all over the place, even in (you're going to love this) actual, real-life applications. For example, anyone who's ever done some form of electromagnetic wave analysis, through the fields of engineering, physics, etc., LOVES i and will gladly bow down and kiss its feet upon command (God bless ei(ωt-kr)). Why? Because of the very thoughtful relation that it's given to us known as "Euler's formula:" eiθ = cos(θ) + isin(θ). Step back a minute and look at that. The irrational, real number, e (2.71828...) exponentiated to the product of a real number, θ, and the imaginary number, i, is equal to a simple trigonometric expression involving two basic functions. In fact (you may want to sit down for this), if the value for θ happens to be π (3.14159...), another irrational, real number mind you, the trigonometric expression on the right hand side of Euler's formula reduces to exactly -1. Let's write that out: eiπ = -1. We call that "Euler's identity," although it should really be called, "THE MOST INCREDIBLE MATHEMATICAL EXPRESSION, EVER!"

But enough about i, let's get back to our newest friend, 0/0. As stated earlier, the problem with 0/0 isn't the fact that we're dividing by zero, it's the fact that the division relation is defined by multiplication. Well, how do we fix that? Simple! Change the definition of divide! Instead of x/y = x*y-1, it's now going to equal x*/y, where "/" is defined as a unary operation analogous to the reciprocal operation.

OK, another quick aside. A unary operator is an operator that only needs one input to work. For example, you only need one number to perform the operation of negation. For instance, negating the number 1 is simply -1. This is opposed to a binaryoperator. Binary operators include many of the guys we're all familiar with; like addition, multiplication, subtraction, etc. To make this clearer, consider the addition operation. It would make no sense to write 1 +. You need another number after the "+" to satisfy the operation; 1 + 2, for example, hence the term binary.

So, with our trusty new unary operator "/" in hand, we're going to look at the number 0/0 again. 0/0 is no longer defined as 0*0-1 like it was before. Now, it's defined as 0*/0, and in our world, not only does /0 ≠ 0-1, but 0*x doesn't have to equal 0 either. Isn't abstraction fun?! Ok, so 0/0 is officially defined, now let's give it some properties!

How about, x + 0/0 = 0/0 and x*0/0 = 0/0. Awesome! Why not go ahead and make a more general rule as well: (x + 0y)z = xz + 0y. OK! Well, we're certainly off to a good start, I'd say. I'll leave the complete derivation of the algebraic structure known as the wheel to the experts, please see the corresponding link below.

I'll end this answer with a final note for those who think that this entire concept of "defining the undefined" is ridiculous. Consider the following sets of numbers: The prime numbers, P; the set of all real numbers with exactly two natural number factors.

The natural numbers, N; the set of all integers greater than or equal to 0.

The integers, Z; the set of all real numbers without remainders or decimals.

The rational numbers, Q; the set of all real numbers that can be expressed as an integer divided by a non-zero integer.

The Irrational Numbers, I; the set of all real numbers that aren't rational.

Now consider this:

The imaginary number, i, is undefined in I.

The ratio pi, or π (3.14159...), is undefined in Q.

The common fraction 1/2 is undefined in Z.

All of the negative numbers, including -1, are undefined in N.

The number 4 is undefined in P.

Yet, these "undefined" numbers are hardly mysterious to us. We just broadened our definition of definable to include the "undefined" ones, and life became good again. 0/0 is not quite, but nearly, the same idea.

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I once asked one of my professor lecturers at University this and his answer was any value you want (or need).

0/0 is used as a limit in Calculus.

Consider any curve y = f(x)

Take a point (x, f(x)) on that curve.

The slope of that point is the slope of the tangent at that point.

The slope of the tangent is close to the slope of a small chord between the point (x, y) = (x, f(x)) and a point a small distance h away (x+h, f(x+h)), which can be found by: m = (f(x+h) - f(x))/((x+h) - x) = (f(x+h) - f(x))/h

The smaller the value of h, the closer the chord is to the tangent and the closer the slope of the chord is to the slope of the tangent and thus the slope of the curve at that point.

As h tends towards 0, f(x+h) tends towards f(x) and the expression m = (f(x+h) - f(x))/h tends towards 0/0.

In other words, 0/0 is the limit of (f(x+h) - f(x))/h as h tends towards 0.

But as this chord tends towards the tangent at the point (x, f(x)) on the curve y = f(x), 0/0 must be the slope of the tangent.

Clearly not every point of a non-linear curve has the same slope, thus 0/0 is any value you want (or need).

As the chord tends towards having zero length (when h = 0), (f(x+h) - f(x))/h will tend towards a constant value, a limit, which is the slope of the tangent.

The "trick" that calculus uses is that as h never reaches 0 but tends towards 0 it is possible to divide by h, and then see what happens when h becomes 0, ie when the original expression became 0/0, since (f(x+h) - f(x))/h = (f(x+0) - f(x))/0 = (f(x) - f(x))/0 = 0/0 when h = 0.

For example, take the curve y = x³ - 2x² + 5x + 3; what is the value of the slope of that line?

slope = lim{h→0} (f(x+h) = f(x))/h

= lim{h→0} ((x+h)³ - 2(x+h)² + 5(x+h) + 3 - (x³ - 2x² + 5x + 3))/h

Expanding the brackets:

= lim{h→0} (x³ + 3x²h +3xh² + h³ - 2x² - 4xh - 2h² + 5x + 5h + 3 - x³ + 2x² - 5x - 3)/h

Simplifying:

= lim{h→0} (3x²h +3xh² + h³ - 4xh - 2h² + 5)/h

Since h ≠ 0, it is possible to divide by h:

= lim{h→0} 3x² +3xh + h² - 4x - 2h + 5

Now the limit can be found by letting h = 0:

= 3x² - 4x + 5

Thus the slope of y = x³ - 2x² + 5x + 3 is given by m = 3x² - 4x + 5 at any value for x.

This value m, which is normally written as f'(x) is the first derivative of f(x), also written as dy/dx.

The slope of any line y = f(x) is given by y = f'(x).

Answer 2

You can't any number (0 or otherwise) by zero. It just doesn't make sense: Simply by assuming that you can get a sensible answer (i.e, simply by allowing the operation), you will get weird results, for example when solving equations.The Wikipedia article "Division by zero" can provide you with a more detailed explanation.

Answer 3

Either 1 or 0 because any number divided by its self equals 1, but this might be 0 because it is 0 divided by 0 and 0 times 1 and 1 times 0 is 0.

Answer 4

When something vanishes off of the face of the universe. How often does nothing go into...

Something?

Answer 5

I used to think that 0/0 would be undefined but then I remembered the rule that any number divided by itself will always equal 1 so I'm not sure what the answer to this is

Answer 6

its 0 i actually solved it