Why can't numbers be divided by zero?

You can think of division as: how many pieces this big does it take to make that?
For example, 8 divided by 2 means how many pieces "two big" does it take to make 8. It takes four pieces "two big" to make eight: 8/2 = 4

Dividing by zero means: how many pieces "zero big" do you need to make some other number? It can not be done. You can add as many pieces "zero big" as you want and you will never get anywhere. Division by zero is not defined for any number - even zero can not be divided into zero pieces. Mathematicians have never defined what this means.

Looking at 0/0 another way, every number works because zero times anything makes zero (e.g. 34.6 pieces "zero big" makes zero). Anyway, they decided not to define it because no one answer makes sense.

The concept of division can be illustrated by this simple example. The following metaphor breaks down in complex situations like imaginary numbers. Partially full bags and partial marbles can be a little challenging, but the conceptual definition of division is accurate.

There are two ways of conceptualizing division. Using a marbles and bags metaphor for set theory:
I have 8 marbles that I must store in bags. 8 divided by 2 means:
Construct 1: 2 equal parts - I have 2 bags. How many marbles will be in each bag if each bag must have the same number of marbles?
Construct 2: Groups of 2 - How many bags do I need to store my marbles if only 2 marbles fit in each bag?

Of course the answer is 4 in each conceptual construct. Now let us try to divide 8 by zero.

Construct 1: 0 equal parts - you have no bags. How many marbles do you put in each of zero bags to store all your marbles in bags? The answer is not a number because it can not be done. You can not put your marbles in a bag if you have no bag. Even the English sentence is nonsense - you can't separate something into zero equal parts.
Construct 2: Groups of 0 - how many bags do I need to store my marbles if I am not permitted to put any marbles, not even minute fractions of marbles, into bags? Once again, even the statement of the problem is illogical: you can't store your marbles in bags if you can't put any marbles in bags.

Both constructs are valid conceptual representations of division. In both cases division by zero is nonsense and illogical. This is the mathematical concept of division by zero.

The repetitive subtraction method of finding the answer to a division problem only works if you complete the method. For example, 25 divided by 2 is not 0 with a remainder of 25. It is not 11 with a remainder of 3. While these answers are mathematically consistent, it is not what we mean when we say "divided by." You must continue the process till the remainder is *less than* the number you are dividing by: e.g. 24 divided by 2 is not 11 with a remainder of 2. While it is true that you can subtract 0 from 24 infinity times, subtracting zeros can never give a remainder less than zero so the method is invalid - it yields no meaningful result.

As we know, division is seeing how many times a number goes into another.

For example:
12/3=4
Twelve divided by three equals four.
Three goes into twelve four times.
It then makes sense that 3*4=12.

Assume 8/0=0
If that was true, then 0*0=8.
When you multiply a the divisor (the quantity by which another quantity is being divided) and the quotient (the result of the division), you should get the divident (the number being divided in the first place)

No matter what, with zero, you'll always get zero, not the original number being divided.


IT'S EASIER TO SEE VISUALLY!


If 20/4=5 than 5*4=20
If 20/0=0 than 0*0=20

you should change the wording. Instead of saying "2 divided by 0" say "2 divided 0 times". It is the same math problem but you can see the error. If you had 10 and divided it 5 times, each division would be 2. If you had 10 and divided it 0 times...you aren't dividing it.

You can divide a number by zero - it's just that the answer is not a number. A positive number divided by zero equals positive (+) infinity and and a negative number produces negative (-) infinity. Dividing by zero does not produce a numerical result, since infinity is a limitless concept, but it does produce a mathematical answer. Anyone who has studied calculus will have encountered the concept of values approaching infinity.

Since an infinite number cannot be expressed in computer-land, this type of operation is usually classified as a numerical overflow or divide by zero error. Dividing zero by zero will also generate an overflow.

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Re: "Division by zero is not defined"
Among others, the IEEE defines infinity for use in numerical analysis. An example of this can be demonstrated using MATLAB, a de facto standard for scientific and engineering analysis:

"MATLAB represents infinity by the special value 'inf'. Infinity results from operations like division by zero and overflow, which lead to results too large to represent as conventional floating-point values. MATLAB also provides a function called 'inf' that returns the IEEE arithmetic representation for positive infinity as a double scalar value."

>> 1/0
Warning: Divide by zero.
ans = Inf

>> -1/0
Warning: Divide by zero.
ans = -Inf

Calculating the tangent of an angle may also produce a divide by zero situation, since the tangent is defined as rise/run or sine(x)/cosine(x). At angles of 90 (pi/2) and 270 (3*pi/2)degrees, where the run equals zero (0), the tangent is equal to +infinity and -infinity, respectively.

Infinity has some interesting properties, such as:
1+infinity = infinity
Going further into infinity is meaningless.

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And to quote Wikipedia:
"The IEEE floating-point standard, supported by almost all modern processors, specifies that every floating point arithmetic operation, including division by zero, has a well-defined result. In IEEE 754 arithmetic, a/0 is positive infinity when a is positive, negative infinity when a is negative, and NaN (not a number) when a = 0."

Think of dividing as splitting things into parts. You can split 4 into 2 parts, each of which are 2, but you cant split 4 into 0 parts.

division is the reverse process to multiplication.

A divided by B is defined as a solution to the equation for X:

B multipled by X = A

So in the expression: 2/0, X would be some number which would fit in place of X in the equation:
0 multiplied by X = 2

But there is no such number, so there is no solution to the division.

If it had been 0/0 you would get
0 multiplied by X = 0
now there are an infinite number of solutions, which is equally useless.

So, to be consistant with multiplication, division by zero cannot be allowed. If you make up some answer, it will lead to nonsensical statements.

The most famous of these nonsensical statements is usually a more complicated version of this:

Let x = 1
Then (x-1) = 2(x-1)
divide by (x-1)
1 = 2

a=b/c
Multiply both sides by b
a*c=b

Nice and simple, right?
Lets put 0 in for c, and 1 in for b.
a=1/0
a*0=1

Whoops! NOTHING multiplied by 0 will give 1. Therefore, anything divided by 0 is undefined, as it cannot be done.

GarikZ Dec, 17 2008 at 08:40 AM
The y such that x=y*0 most certainly exists when x=0. This is correct though that in this case y can be any number, which means infinitely many potential results of division - not a practically useful answer...


You are most certainly wrong.

We can prove that: the y such that x=y*0 cannot exist!

There cannot be a unique answer when there is more than one answer.

In a fraction, the smaller the denominator (bottom number) gets, the larger the overall number becomes:

10/10 = 1
10/5 = 2
10/2 = 5
10/.05 = 200
10/.0005 = 20000

As you approach zero, the overall number approaches infinity which is not able to be expressed in numbers.

Because there is no unique number as a result.

"In mathematics, a division is called a division by zero if the divisor is zero. Such a division can be formally expressed as a/0 where a is the dividend. 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.
In computer programming, integer division by zero may cause a program to terminate or, as in the case of floating point numbers, may result in a special not-a-number value "

"When division is explained at the elementary arithmetic level, it is often considered as a description of dividing a set of objects into equal parts. As an example, consider having 10 apples, and these apples are to be distributed equally to five people at a table. Each person would receive 10/5 = 2 apples. Similarly, if there are 10 apples, and only one person at the table, each person would receive 10/1 = 10 apples.
So for dividing by zero — what if there are 10 apples to be distributed, but no one comes to the table? How many apples does each "person" at the table receive? The question itself is meaningless — each "person" can't receive zero, or 10, or an infinite number of apples for that matter, because there are simply no people to receive anything in the first place. So 10/0, at least in elementary arithmetic, is said to be meaningless, or undefined.
Another way to understand the nature of division by zero is by considering division as a repeated subtraction. For example, to divide 13 by 5, 5 can be subtracted twice, which leaves a remainder of 3 — the divisor is subtracted until the remainder is less than the divisor. The result is often reported as = 2 remainder 3. But, in the case of zero, repeated subtraction of zero will never yield a remainder less than zero. Dividing by zero by repeated subtraction results in a series of subtractions that never ends. This connection of division by zero to infinity takes us beyond elementary arithmetic"

"In higher mathematics:
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."
Source and further information:
http://en.wikipedia.org/wiki/Division_by_zero

Who actually gives a damn? Seriously... Sort it out..

Numbers cant be divided by zero because zero is an logical implication to an impossible phenomenon XD.
Zero was originally implied to represent that there is nothing or symbolize that you have none of the certain object like ten or hundreds place.

105 or 1096 and the like

However, if you get philosophically into it you get into a time paradox which you get killed by your inability to grasp an impossibly easy yet complex matter which makes you build a time machine to kill yourself, just kill yourself or make you depressed for the rest of your life, like the meaning of life does =(

So the point is dividing by nothing puts stuff into a non-existant(How do you spell that?) plane which is unable to be explained by math or philosophers and isn't put into a symbol for daily use making it impossible to divide by zero.

Besides, why would you want to divide by zero? Theres no point!

Because you are actually dividing them by nothing (0) therefore you are not actually dividing at all!
Zero is a very complex phenomenum and many ancient cultures didn't even use it!

Division is an operation inverse to multiplication. This means that if the result of multiplying A by B is C, i.e., A*B=C, by means of division we can retrieve the value of B if the values of A and C are known: B=C/A.

Please note that according to the formula above division allows to retrieve the second factor, which is B. However, since multiplication of numbers is known to follow the commutative law, the first factor A can be retrieved equally easily: A*B=B*A, and now the number A becomes the second factor. (This is a reason why the commutative law is important.)

From the point of view of everyday life, this, for example, means that you can find not only the width of a rectangle if you know its area and length but also find the length if you know the area and the width.

However, the idea of division only makes sense if, when knowing A and C in the formulas above, you can find a unique value of B (the stress is on both words "can" and "unique"). If you cannot find a value of B, or you cannot find it in a unique way, division just makes no sense.

If any number A gets multiplied by 0, the result is known to be 0. Therefore, if I tell you that A=0, then for any B A*B=0. In other words, C cannot be anything other than 0. So, if I give you a non-zero value of C and ask you to find B from a multiplication relationship A*B=C with A=0 (in other words, 0*B=C with a non-zero C, which is impossible), no B can be found.

If, however, C is also equal to 0 (so we have 0*B=0, and there is no contradiction like in the previous paragraph), then there are lots of values of B that would fit in - remember, any number multiplied by 0 renders 0!

So, the result of division by zero defined as an inversion of the multiplication relationship either does not exist or is not unique. For this reason we say that in elementary math division by zero is not defined - or, in simpler words, you can't divide by zero.

In more advanced branches of math, where people deal with multiplication and division of objects other than numbers, division by zero (which is usually called a neutral element rather than zero) can be defined and performed.

dividing by zero simply means you are not dividing by any number and therefore you get no answer, it is the same as multiplying by zero. "literally" nothing equals nothing .

dividing by zero simply means you are not dividing by any number and therefore you get no answer, it is the same as multiplying by zero. "literally" nothing equals nothing .

Because (the y such that: x=y*0) does not exist, ie.there is no unique y such that y=x/0 for all x.

Ok, it's very simple to understand why you can't divide by zero. Lets say we make our numerator 12 and lets make the denominator smaller.

12/12=1
12/4=3
12/3=4
12/2=6
12/1=12 Now at this point, we know our next smaller denominator into our answer will be bigger.
12/0.5=24
12/0.25=48
12/0.125=96

And on and on and on. Until it gets to the smallest thing ever until nothing known as ZERO
12/0=infinity
So far we can see this, but now heres the problem, the number one reason n/0 is undefined. The mathimathical rule says that 0 is between -1 and 1.
12/-0.5=-24
12/-1=-12
We can't just go from all the way to positive infinity to negative infinity. You would have to start back at 0 then go to negative infinity. So how in the world could 12/0 be two answers at the same time (+infinity and -infinity) So that's why the rule is n/0=undefined

Because there would be nobody to share items between.

Because the answer would be OVER 9000!?!

Any number can be divided by some other number to produce a unique result. If division by 0 cannot then 0 is not a number, just like infinity. It's a concept, nothing more. Declaring it to be a number does not make it one.
Be wary of long, complicated answers to simple questions.

Think about it - there's no possible solution...