Question

Why we can't define "0" ?

Answer 1

Can't define 0?

Answer 2

The numeral, the number?

Answer 3

Do you mean something like: 1/0?

Answer 4

The absence of value?
Somehow, that doesn't sound right :/

Answer 5

For a second there I was 96, but now I'm back to 95... what?

Answer 6

Haha, poor wio

Answer 7

LOL If you don't tell anyone, maybe we can fix that XD

Answer 8

haha

Answer 9

It's back again, wtf? How!?

Answer 10

Are you sure it wasn't for 0 seconds, but since it's apparently undefined, it never happened?

Answer 11

Oh, wow.

Answer 12

Quick someone demedal him back down and make him beg!

Answer 13

hooray, 3 points from 99, keep it up wio.

Answer 14

...

Answer 15

LOL You guys -_-
Anyway... don't we HAVE a definition for zero

Answer 16

I saved it, 95 again.

Answer 17

There you go. You should upgrade to windows 98.

Answer 18

I have the power lol

Answer 19

But what about the question? D:

Answer 20

Zero can be defined, so this is kind of like asking, why is the sky red? It's not!

Answer 21

Zero is the additive identity.

Answer 22

Interesting, you can multiply something by 1 as many times as you like similar to how you can add 0 to anything you like a bunch of times. See:

a*1=a

ln(a*1)=ln(a)

ln(a)+ln(1)=ln(a)

Tahdah

Answer 23

Or how about: \[
0:= \lim_{n\to \infty }\frac 1n
\]

Answer 24

If u must know

Answer 25

In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as a/0 where a is the dividend (numerator). Whether this expression can be assigned a well-defined value depends upon the mathematical setting. In ordinary (real number) arithmetic, the expression has no meaning, as there is no number which, multiplied by 0, gives a (a≠0), and so division by zero is undefined. Since any number multiplied by zero is zero, the expression 0/0 has no defined value and is called an indeterminate form. Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to a /0 is contained in George Berkeley's criticism of infinitesimal calculus in The Analyst ("ghosts of departed quantities").[citation needed]
In computing, a program error may lead to an attempt to divide a number 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, or result in a special not-a-number value.

In elementary arithmetic
When division is explained at the elementary arithmetic level, it is often considered as splitting a set of objects into equal parts. As an example, consider having ten cookies, and these cookies are to be distributed equally to five people at a table. Each person would receive \textstyle\frac{10}{5} = 2 cookies. Similarly, if there are ten cookies, and only one person at the table, that person would receive \textstyle\frac{10}{1} = 10 cookies.
So for dividing by zero – what is the number of cookies that each person receives when 10 cookies are evenly distributed amongst 0 people at a table? Certain words can be pinpointed in the question to highlight the problem. The problem with this question is the "when". There is no way to evenly distribute 10 cookies to nobody. In mathematical jargon, a set of 10 items cannot be partitioned into 0 subsets. So \textstyle\frac{10}{0}, at least in elementary arithmetic, is said to be either meaningless, or undefined.
Similar problems occur if one has 0 cookies and 0 people, but this time the problem is in the phrase "the number". A partition is possible (of a set with 0 elements into 0 parts), but since the partition has 0 parts, vacuously every set in our partition has a given number of elements, be it 0, 2, 5, or 1000.
If there are, say, 5 cookies and 2 people, the problem is in "evenly distribute". In any integer partition of a 5-set into 2 parts, one of the parts of the partition will have more elements than the other. But the problem with 5 cookies and 2 people can be solved by cutting one cookie in half. The problem with 5 cookies and 0 people cannot be solved in any way that preserves the meaning of "divides".
Another way of looking at division by zero, is that division can always be checked using multiplication. Considering the 10/0 example above, setting x = 10/0, if x equals ten divided by zero, then x times zero equals ten, but there is no x that, multiplied by zero, gives ten (or any other number than zero). If instead of x=10/0 we have x=0/0, then every x satisfies the question 'what number x, multiplied by zero, gives zero?'

It is generally regarded among mathematicians that a natural way to interpret division by zero is to first define division in terms of other arithmetic operations. Under the standard rules for arithmetic on integers, rational numbers, real numbers, and complex numbers, division by zero is undefined. Division by zero must be left undefined in any mathematical system that obeys the axioms of a field. The reason is that division is defined to be the inverse operation of multiplication. This means that the value of a/b is the solution x of the equation bx = a whenever such a value exists and is unique. Otherwise the value is left undefined.
For b = 0, the equation bx = a can be rewritten as 0x = a or simply 0 = a. Thus, in this case, the equation bx = a has no solution if a is not equal to 0, and has any x as a solution if a equals 0. In either case, there is no unique value, so \textstyle\frac{a}{b} is undefined. Conversely, in a field, the expression \textstyle\frac{a}{b} is always defined if b is not equal to zero.

In calculus
Extended real line[edit]
At first glance it seems possible to define a/0 by considering the limit of a/b as b approaches 0.
For any positive a, the limit from the right is
\lim_{b \to 0^+} {a \over b} = +\infty
however, the limit from the left is
\lim_{b \to 0^-} {a \over b} = -\infty
and so the \lim_{b \to 0} {a \over b} is undefined (the limit is also undefined for negative a).
Furthermore, there is no obvious definition of 0/0 that can be derived from considering the limit of a ratio. The limit
\lim_{(a,b) \to (0,0)} {a \over b}
does not exist. Limits of the form
\lim_{x \to 0} {f(x) \over g(x)}
in which both ƒ(x) and g(x) approach 0 as x approaches 0, may equal any real or infinite value, or may not exist at all, depending on the particular functions ƒ and g (see l'Hôpital's rule for discussion and examples of limits of ratios). These and other similar facts show that the expression 0/0 cannot be well-defined as a limit.

Formal operations[edit]
A formal calculation is one carried out using rules of arithmetic, without consideration of whether the result of the calculation is well-defined. Thus, it is sometimes useful to think of a/0, where a ≠ 0, as being \infty. This infinity can be either positive, negative, or unsigned, depending on context. For example, formally:
\lim\limits_{x \to 0} {\frac{1}{x} =\frac{\lim\limits_{x \to 0} {1}}{\lim\limits_{x \to 0} {x}}} = \infty.
As with any formal calculation, invalid results may be obtained. A logically rigorous (as opposed to formal) computation would assert only that
\lim\limits_{x \to 0^+} \frac{1}{x} = +\infty\text{ and }\lim\limits_{x \to 0^-} \frac{1}{x} = -\infty.
Since the one-sided limits are different, the two-sided limit does not exist in the standard framework of the real numbers. Also, the fraction 1/0 is left undefined in the extended real line, therefore it and
\frac{\lim\limits_{x \to 0} 1 }{\lim\limits_{x \to 0} x}
are meaningless expressions.




Real projective line[edit]
The set \mathbb{R}\cup\{\infty\} is the real projective line, which is a one-point compactification of the real line. Here \infty means an unsigned infinity, an infinite quantity that is neither positive nor negative. This quantity satisfies -\infty = \infty, which is necessary in this context. In this structure, \scriptstyle a/0 = \infty can be defined for nonzero a, and \scriptstyle a/\infty = 0. It is the natural way to view the range of the tangent and cotangent functions of trigonometry: tan(x) approaches the single point at infinity as x approaches either \scriptstyle+\pi/2 or \scriptstyle-\pi/2 from either direction.
This definition leads to many interesting results. However, the resulting algebraic structure is not a field, and should not be expected to behave like one. For example, \infty + \infty is undefined in the projective line.
Riemann sphere[edit]
The set \mathbb{C}\cup\{\infty\} is the Riemann sphere, which is of major importance in complex analysis. Here too \infty is an unsigned infinity – or, as it is often called in this context, the point at infinity. This set is analogous to the real projective line, except that it is based on the field of complex numbers. In the Riemann sphere, 1/0=\infty, but 0/0 is undefined, as is 0\times\infty.
Extended non-negative real number line[edit]
The negative real numbers can be discarded, and infinity introduced, leading to the set [0, ∞], where division by zero can be naturally defined as a/0 = ∞ for positive a. While this makes division defined in more cases than usual, subtraction is instead left undefined in many cases, because there are no negative numbers.
In higher mathematics[edit]

Although division by zero cannot be sensibly defined with real numbers and integers, it is possible to consistently define it, or similar operations, in other mathematical structures.
Non-standard analysis[edit]
In the hyperreal numbers and the surreal numbers, division by zero is still impossible, but division by non-zero infinitesimals is possible.
Distribution theory[edit]
In distribution theory one can extend the function \textstyle\frac{1}{x} to a distribution on the whole space of real numbers (in effect by using Cauchy principal values). It does not, however, make sense to ask for a 'value' of this distribution at x = 0; a sophisticated answer refers to the singular support of the distribution.
Linear algebra[edit]
In matrix algebra (or linear algebra in general), one can define a pseudo-division, by setting a/b = ab+, in which b+ represents the pseudoinverse of b. It can be proven that if b−1 exists, then b+ = b−1. If b equals 0, then b+ = 0; see Generalized inverse.
Abstract algebra[edit]
Any number system that forms a commutative ring — for instance, the integers, the real numbers, and the complex numbers — can be extended to a wheel in which division by zero is always possible; however, in such a case, "division" has a slightly different meaning.
The concepts applied to standard arithmetic are similar to those in more general algebraic structures, such as rings and fields. In a field, every nonzero element is invertible under multiplication; as above, division poses problems only when attempting to divide by zero. This is likewise true in a skew field (which for this reason is called a division ring). However, in other rings, division by nonzero elements may also pose problems. For example, the ring Z/6Z of integers mod 6. The meaning of the expression \textstyle\frac{2}{2} should be the solution x of the equation 2x = 2. But in the ring Z/6Z, 2 is not invertible under multiplication. This equation has two distinct solutions, x = 1 and x = 4, so the expression \textstyle\frac{2}{2} is undefined.
In field theory, the expression \textstyle\frac{a}{b} is only shorthand for the formal expression ab−1, where b−1 is the multiplicative inverse of b. Since the field axioms only guarantee the existence of such inverses for nonzero elements, this expression has no meaning when b is zero. Modern texts include the axiom 0 ≠ 1 to avoid having to consider the trivial ring or a "field with one element", where the multiplicative identity coincides with the additive identity.
In computer arithmetic[edit]



In the SpeedCrunch calculator application, when a number is divided by zero the answer box displays “Error: Divide by zero”.


Most calculators, such as this Texas Instruments TI-86, will halt execution and display an error message when the user or a running program attempts to divide by zero.
The IEEE floating-point standard, supported by almost all modern floating-point units, specifies that every floating point arithmetic operation, including division by zero, has a well-defined result. The standard supports signed zero, as well as infinity and NaN (not a number). There are two zeroes, +0 (positive zero) and −0 (negative zero) and this removes any ambiguity when dividing. In IEEE 754 arithmetic, a ÷ +0 is positive infinity when a is positive, negative infinity when a is negative, and NaN when a = ±0. The infinity signs change when dividing by −0 instead.
The justification for this definition is to preserve the sign of the result in case of arithmetic underflow.[2] For example, in the double-precision computation 1/(x/2), where x = ±2−149, the computation x/2 underflows and produces ±0 with sign matching x, and the result will be ±∞ with sign matching x. The sign will match that of the exact result ±2150, but the magnitude of the exact result is too large to represent, so infinity is used to indicate overflow.
Integer division by zero is usually handled differently from floating point since there is no integer representation for the result. Some processors generate an exception when an attempt is made to divide an integer by zero, although others will simply continue and generate an incorrect result for the division. The result depends on how division is implemented, and can either be zero, or sometimes the largest possible integer.
Because of the improper algebraic results of assigning any value to division by zero, many computer programming languages (including those used by calculators) explicitly forbid the execution of the operation and may prematurely halt a program that attempts it, sometimes reporting a "Divide by zero" error. In these cases, if some special behavior is desired for division by zero, the condition must be explicitly tested (for example, using an if statement). Some programs (especially those that use fixed-point arithmetic where no dedicated floating-point hardware is available) will use behavior similar to the IEEE standard, using large positive and negative numbers to approximate infinities. In some programming languages, an attempt to divide by zero results in undefined behavior. The graphical programming language Scratch 2 used in many schools returns Infinity or -Infinity depending on the sign of the dividend.
In two's complement arithmetic, attempts to divide the smallest signed integer by -1 are attended by similar problems, and are handled with the same range of solutions, from explicit error conditions to undefined behavior.
Most calculators will either return an error or state that 1/0 is undefined, however some TI and HP graphing calculators will evaluate (1/0)2 to ∞.
More advanced computer algebra systems will return an infinity as a result for division by zero; for instance, Microsoft Math and Mathematica will show a ComplexInfinity result.
Historical accidents
On September 21, 1997, a division by zero error on board the USS Yorktown (CG-48) Remote Data Base Manager brought down all the machines on the network, causing the ship's propulsion system to fail.

Answer 26

@***[ISURU]***

Answer 27

Man I wish I could blithely copy and paste as well as you rather than only hyperlinks.