Question

What is the most convincing argument you have found for why can you never divide by zero?

Answer 1

given by my teacher xD

Answer 2

I think it is because u can't divide something in nothing, u are going to do nothing, there if nothing to do!

Answer 3

there is*

Answer 4

But isn't that the same as dividing by 1?

Because you have done nothing and left the number unchanged, right?

x/1 = x

Answer 5

you add add as many 0's as you like - ad infinitum - but you'll always get a total of zero

Answer 6

* you can add

Answer 7

also if dividing by 0 is valid you can prove things like 1 = 2

Answer 8

if we let x = 1

then x -1 = 2(x -1)

divide both sides by x-1 :-

1 = 2

Answer 9

\(\mathbb{R}\) is not a group with multiplication. So the question does not even make sense. :)

Answer 10

For the same reason I can't divide house by car

Answer 11

What does "a group with multiplication" mean?

Answer 12

For a set to form a group it needs to have the property that every element has an inverse. Multiplication on R does not pass this test. The identity is 1, but there is no number you can multiply by 0 to get 1. So that R with multiplication is not a group. R\{0} is however

Answer 13

so in short, since there is no number we can multiply 0 by to get 1, we cant divide by 0

Answer 14

that video is nice: https://www.khanacademy.org/math/algebra2/functions_and_graphs/undefined_indeterminate/v/why-dividing-by-zero-is-undefined

Answer 15

Yes, that is convincing if you define division as multiplication by the reciprocal.

But what about a/b = c iff a = b•c?

Answer 16

that implies that a/b is defined, which is not for any additive identity on a ring, when we are calling the two operations additive and multiplicative.

Answer 17

there are two operations on the real numbers, additive and multiplication

Answer 18

division is not an operation for a group or ring because it is NOT associative

Answer 19

To get a very basic understanding of why this must be, go look up groups, and rings, and then fields if you get bored.

Answer 20

If b = 0, the above definition is still valid if a = 0, correct? But the c is no longer unique.

That^ is the second most convincing argument I've seen after yours :-)

Answer 21

what that really says is \(ab^{-1} = c \iff a=cb\)

Answer 22

I think its best not to think of division as a thing, its really the inverse of a thing ;)

Answer 23

Indeed, it then fits perfectly with subtraction.

Answer 24

but that is not defined for 0 by properties of being a group and you need those properties to make a nice structure.

Answer 25

there is no rule that says you can never divide by zero.

Answer 26

some courses define 1/0 as infinity

Answer 27

unless otherwise defined, dividing a real number by 0 remains 'undefined'.

Answer 28

how would you know if 1/0 is infinity or negative infinity ?

Answer 29

when defining the slope of a line

Ax + 0y = C

0y = -Ax + C

y = -A/0 x + C/0

the slope of a vertical line is .... infinity for some.

Answer 30

again, depends on the course ... i leave it to them to define it the way they see fit.

Answer 31

\[\lim_{x \rightarrow 0^+}\frac{1}{-x}\]
if you plug in 0 you get 1/0
but the limit is negative infinity not positive infinity

if i take that negative sign off though
you would your 1/0 where the limit goes to positive infinity

Answer 32

oh are you defining undefined as infinity ?

Answer 33

infinity is not a real number :)

Answer 34

oh wait no nevermind kinda confused

I always seen vertical lines with undefined slopes

Answer 35

i know infinity isn't a number
but just because something isn't defined doesn't mean it is infinity

Answer 36

1/0 is not a real number, and x+1 = x has no real solution

Answer 37

or even negative infinity
it could just not be defined at all

Answer 38

Some books say we can't define a/0 because if a/0 = k, then a = 0*k and this leads to a contradiction. for example if 3/0 = k, then 3 = 0*k , but then 3 = 0.
this is supported by the first post here
http://mathforum.org/dr.math/faq/faq.divideby0.html
But the problem with this line of reasoning is, 0/0= 0 seems to be valid, since 0 = 0*0.
so you really need a more sophisticated answer than that.

Answer 39

if we define it, then it is defined. I havent taken the courses so I wouldnt be all that knowledgable about the specfic.

Answer 40

dividing by zero is undefined, until it is defined. if it comports itself to a noncontradictory philosophy, then its valid.

Answer 41

oh so I guess you are saying it is up to the instructor to define a/0

in my opinion a/0 is indeterminate
you cannot determine it as infinity or -negative infinity
I prefer to say a/0 is undefined because of that
until we get to limits of course when we actually consider what something is approaching or whatever but even then sides might not agree and you can't say anything but it is undefined
like \[\lim_{x \rightarrow 0}\frac{1}{-x} \text{ is undefined since both sides don't agree on the same infinity }\]

Answer 42

so it just seems kinda confusing to define 1/0 as infinity when students get to calculus they will see different stories to the 1/0 thing

Answer 43

but anyways I'm done about division by 0
this topic is scary

Answer 44

in calculus we say expressions 0/0 (in limits) are indeterminate, but a/0 with a not equal to zero diverge

Answer 45

indeterminate has a special meaning in the context of limits, it is not the same as undefined

Answer 46

I'm sorry what i meant by what I said a/0 can't be determined as infinity or negative infinity
should have just said that even though that still seems not determined :p

Answer 47

no need to apologize :)

Answer 48

i agree with that , there might be some special number system that has only one infinity

Answer 49

I apologize all the time when I don't mean it
it is hard to tell when I really mean it and when I don't
My husband or wife (that is indeterminate also) says I say sorry too much
and I tell him I don't mean it half the time :p

Answer 50

If you are pressed to, you can define a/0 with a ≠ 0 as |∞|

Answer 51

I think I would be more comfortable defining |a/0| where a not equal 0
as infinity

Answer 52

But using absolute value bars around infinity is treating it like a real number?

Answer 53

yes that makes sense

Answer 54

you can treat infinity as a real number in something called the extended Real life

Answer 55

http://en.wikipedia.org/wiki/Extended_real_number_line

Answer 56

life?

Answer 57

lol line

Answer 58

line*

Answer 59

:D great discussion, thanks.

Answer 60

I disagree. Dividing my zero is just something you have to handle carefully and that's what calculus is.

Even in projective geometry we can safely define infinity as a point that you approach from all directions in the plane for instance and it makes perfect geometric sense that -infinity and +infinity are the same thing! I'll draw a picture if anyone's curious.

Answer 61

I'm curious.
When I think about the opposite infinities, I think about two totally different directions.

Answer 62

Are you saying it is kinda circular or something?

Answer 63

\(\color{blue}{\text{Originally Posted by}}\) @zzr0ck3r
so in short, since there is no number we can multiply 0 by to get 1, we cant divide by 0
\(\color{blue}{\text{End of Quote}}\)
\[\huge\rm (0)^0 = 1 \]
lel :D ;P

Answer 64

all X^0=1 except for 0^0=0

Answer 65

Yeah, so I'll start with the 2D version and the 3D or higher version is not too crazy. In projective geometry we're just saying we're taking every point on the line and projecting it to a point on the circle: When the projected line is parallel to the top of the circle, that point is called infinity and if we tilt it either way it will either give us a negative or positive number.

The plane version is essentially the same thing, but still only a singular infinite point at the top of the sphere. so if you pick any point infinitely far away it will always point to the exact same point at the top, and it's nice to have for doing computer graphics apparently to have a point at infinity or so I hear, I was more interested in it for other reasons so I didn't get into any applications of it.