# Divide a number by 0

## Main Question or Discussion Point

Hey just wondering

if u divide any real number by 0 is it + infinite or - infinite

is it : +infinite when the number if positive, and -infinite when the number is negative ?

• 442nick

benorin
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Technically, division by zero is undefined; but, essentially yes, with a few modifications: dividing a positive real number by a number which approaches zero through positive values approaches $$+\infty$$. Consider the graph of $$y=\frac{3}{x}$$ below: note that the function value (or y value) approaches $$+\infty$$ as x approaches zero through positive values (from the right side of zero); but the function value (or y value) approaches $$-\infty$$ as x approaches zero through negative values (from the left side of zero).

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yeah but i mean , is 0 positive or negative ?
or is it just undefined as u said ?

dextercioby
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0 is neither positive, nor negative and division by 0 in arithmetics over reals is undefined.

DaveC426913
Gold Member
Dividing by zero does not result in infinity, it results in undefined. Here's why:

The division operation, by definition, is the inverse of multiplication. 6/3 is equal to 2 because 2*3=6.

But if we try that with zero, we start with 0*x=6. This equation has no solution at all - there is no value of x (including infinity) that can make that equation true.

Since the multiplicative has no solution, the inverse has no solution (not even infinity).

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Dividing by zero is undefined. In fact when someone says a number A (positive) divided by zero, he means it is divided by an epsilon and then find the lim of the result when epsilon approaches to zero. So depending the sign of epsilon, the result can be + infinitive or - infinitive. I mean epsilon can approache zero origine either from the left or from the right then we have -0 or +0.

arildno
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Dearly Missed
Dividing by zero is undefined. In fact when someone says a number A (positive) divided by zero, he means it is divided by an epsilon and then find the lim of the result when epsilon approaches to zero.
Does he mean that? I don't think so..
SHOULD he mean that? He certainly should not.

I don't understand why some people are obstinate about undefined operations. Mathematics is not nature made, it's man made, and man dictates its functioning. How we define our numbers and their arithmetic makes the expression x / 0 irrelevant. It does not mean anything.

INFINITY IS NOT A NUMBER! Thus we cannot define dividion with zero. To prove that infinity is not a number
let suppose
inf = 1/0;
then 0* inf = 1,
but 0 = 1 * 0 = 0 * (0 * inf ) = (0 * 0 ) * inf = inf * 0 = 1
Thus division with zero is still remain undefined.

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dextercioby: 0 is neither positive, nor negative and division by 0 in arithmetics over reals is undefined.

The standard proceedure with historians is to go from 1BC to 1AD, thus 1000AD is 999 years after 1 BC. If you consider BC to be negative and AD to be positive, well then, as you see here, there is no such thing as year 0, so it can not be positive or negative in this system.

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Hurkyl
Staff Emeritus
Gold Member
The standard proceedure with historians is to go from 1BC to 1AD, thus 1000AD is 999 years after 1 BC. If you consider BC to be negative and AD to be positive, well then, as you see here, there is no such thing as year 0, so it can not be positive or negative in this system.
What does that have to do with anything? Hurkyl: What does that have to do with anything?

i am sorry if this is troublesome, i thought it was a joke. I could have added, it does illustrate the fact that 0 was not accepted by early historians, and so the tradition just continued. It is the reason why we are now in the 21st Century, but in the year 2007.

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HallsofIvy
Homework Helper
I don't understand why some people are obstinate about undefined operations. Mathematics is not nature made, it's man made, and man dictates its functioning. How we define our numbers and their arithmetic makes the expression x / 0 irrelevant. It does not mean anything.
Yes- it does not mean anything because it is not defined to be anything- it is undefined. When you refer to people being "obstinate" about undefined operations, exactly what do you mean?

DaveC426913
Gold Member
I don't understand why some people are obstinate about undefined operations. Mathematics is not nature made, it's man made, and man dictates its functioning. How we define our numbers and their arithmetic makes the expression x / 0 irrelevant. It does not mean anything.
But simply declaring that "it is undefined because we say so" is not a satisfactory answer.

There IS a good, logical reason why it is undefined, and it can be shown, as posts 5 and 9.

Hurkyl
Staff Emeritus
Gold Member
But simply declaring that "it is undefined because we say so" is not a satisfactory answer.

There IS a good, logical reason why it is undefined, and it can be shown, as posts 5 and 9.
There are two separate issues here.

"It is (un)defined because we say so" is essentially the only correct answer to the question "Is it defined?" There is no higher reason for it -- either an expression syntactically satisfies the requirements of a definition, or it does not.

1/0 is undefined (for real division) precisely because this expression fails to satisfy the requirement that the dividend is a real number and the divisor is a nonzero real number. Similarly, if x is a variable denoting a real number, then x/x is undefined. (y/y would be defined if y is a variable denoting a nonzero real number, though)

You are talking about the reason why we would ever have chosen to define division this way. There is a good reason for that. (I think "practical" is more accurate than "logical", though) But the reasons for choosing the definition are entirely irrelevant to the question of what is or is not defined.

DaveC426913
Gold Member
Why would you insist that we defined it a certain way before there was a reason to need it to be that way? Isn't that putting the cart before the horse? Or a fabulously lucky premonition?

Surely, the historical order of occurence is:
1] We "invent" division (as the inverse of multiplcation)
2] We realize that dividing by zero is problematic
3] We put in place a rule so as not to cause problems

Hurkyl
Staff Emeritus
Gold Member
Why would you insist that we defined it a certain way before there was a reason to need it to be that way?
I'm not. There is a difference between the question

"Why is 1/0 undefined?"

and the question

"Why did we define division so that 1/0 is undefined?"

If we say y(a number) divided by x as x approched infity equals 0. Can we say 0 times x as x approches infity equals Y.
Does division by 0 is undefined for only real numbers or all complex numbers.

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Hurkyl
Staff Emeritus
Gold Member
y divided by x as x approched infity equals 0
This doesn't make sense. I will assume you meant to say:

"The limit of y divided by x, as x approaches infinity, equals 0"

or equivalently

"y divided by x approaches 0 as x approaches infinity"

Can we say 0 times x as x approches infity equals Y.
Similarly, I will assume you meant to say

"The limit of 0 times x, as x approaches infinity, equals y"

or equivalently

"0 times x approaches y as x approaches infinity"

The second statement does not follow (directly or indirectly) from the first. The second statement is true iff y = 0.

Can you use the same rule for limits as an ordinary equation y/x=0, 0*x=Y.

HallsofIvy
Homework Helper
"If we say y(a number) divided by x as x approched infity equals 0. Can we say 0 times x as x approches infity equals Y."
"limit of y/x, as x goes to infinity, y fixed, equals 0" is only true if y is not 0.

No, we cannot say "0*x, as x approaches infinity equals y". 0*x for any real x is 0 so the limit is 0.

When we did integration isn't it the same thing what we were doing when adding up all the rectangles. The sum of all the rectangles as dx approches 0. Sorry i am a bit confused. Math is not my strongest subject.

HallsofIvy
Homework Helper
No, it's not at all the same thing.

HallsofIvy
Homework Helper
Don't you guys know? 1/0 = infinity and 0^0=0/0 = golden ratio. How poetic!

http://www.bbc.co.uk/berkshire/content/articles/2006/12/06/divide_zero_feature.shtml

I shed a tear for those kiddies learning such trash.
No, we don't know that because it isn't true. "infinity" is not a number so 1/0= infinity is non-sense. The golden ratio is a number but neither 00 nor 0/0 is so that last statement is false.