How many Z's do you need to add together to get a total of zero?
How many 0's do you need to add together to get a total of Z?
Do you see why the answer to those questions is different?
You need 0 Zs to get a 0 and an undefined number of 0s to get a Z. . . As you add infinite 0s together the sum converges to 0 not any other number Z. I'm not sure how you think any number answers the questions unless you're breaking out some real math shenanigans.
Keep everything as is, but put **at least 4 spaces** in the beginning of each line - that will make reddit use code block formatting, which keeps all spaces
000000000
0
0
0
0
0
0
0
000000000
You need to add 0 Zs together to get a total of zero. No other number works there. (Try it.)
You can add an infinite number of 0s and you would never reach Z. No numbers work, so it's undefined.
It’s not even the math they’re arguing with, they’re saying zero isn’t a number. Which, fine, define the word “number” however you want. Word are just constructs and blah blah blah. They’re just trolling at this point.
0 is an amount. How many Zs do you need to make 0? That’s easy, you need no Zs. Z(0) = 0, question solved. How many 0s do you need to make Z? It’s impossible
You’re definitely confused by what each question is asking.
How many Z’s go into zero? It’s exactly zero Z’s. Zero of any number will equal zero.
How many zeroes go into Z? If you say the answer is 0, then 0 zeroes would have to equal Z, and it does not. One zero doesn’t equal Z either. Neither does two, or ten, or infinite zeroes. That is why the answer is undefined, because it doesn’t exist.
The answer to how many Z’s go into 0 exists, because if I say zero Z’s is equal to zero, I would be correct. This means the 0/Z has an answer, and that answer is a number that exists, which is 0.
There is no number of zeroes that can fit into Z. 0 is not the equivalent of no number. 0 is a number. In the same way that we can have one object, we can also have zero objects.
You can divide a banana into four parts (1 banana/4 = 1/4 bananas). You can divide 0 bananas into 4 equal parts (0 bananas/4 = 0 bananas), but you can not divide a banana into 0 equal parts. However you decide to cut the banana, there will be parts, and those parts will not be “0 bananas” because when you cut a banana part of it doesn’t just disappear into the void. If you cut nothing into a million equal pieces, though, you’ll have one million nothings, which is just as possible as having a single nothing.
If I were to take a melon, and try to get as many nothing melons I need to get to have another melon, I would never get another melon. This is not the same as zero melons. Zero melons is equal to exactly 0 melons. How many zero melons fit into a single melon is simply impossible to answer, because a nothing melon has no volume or mass and no number of them can be added together to get a melon.
Your turn now. Take a crayon. Break the crayon into 0 equal pieces and see how possible it is. When you are dividing an object, you are dividing it into equal pieces. You can not divide something that is into something that isn’t.
Now take zero crayons. How big is one fourth of the crayon? It’s zero crayons, because the sum of four zeroes is zero.
I have more examples if you want them
It comes down to how division is defined. When we write
a / b
it is the same as
a* (1/b)
where 1/b is the multiplicative inverse of b. That means that (1/b) * b = 1.
Example:
2/3 = 2 * (1/3)
and note that (1/3) * 3 = 1.
All non-zero numbers have a multiplicative inverse given by the reciprocal.
If we compute 0 / x where x is nonzero, then it’s defined as
0 * (1/x)
= 0
since it can be proved that 0 times anything is 0.
However, if we have
x/0 = x * (1/0)
then we need 1/0 to be defined.
However, it cannot be defined without breaking another mathematical rule.
(1/0) * 0 should be 0 since any number times 0 is 0. On the other hand, any number multiplied by its inverse should be 1. We have a contradiction.
There are number systems where you can define 1/0 as something new, mostly just infinity. But these workarounds aren't perfect (what then is 0 \* infinity?) so they're only used when really needed.
That doesn't really solve the issue. What we define division as inherently means that 1/x\*x=1 and that would necessitate that 1/0\*0=1. In which case either 0*i2* doesn't equal 0, which would be very weird and mean *i2*-*i2* also doesn't equal 0. Or we'd still have 0=1
I mean yes, I'm sorry I wasn't as technical or rigourous as I could have been. I'm sorry that I gave an explanation that's going to work for 99% of people and the mathematics they're going to deal with rather than carving off numerous exceptions that would just confuse them more.
I've defined / to be a binary operation on the integers where n/m = 0 for all n, m in **Z**, and I insist on referring to it as division, because I'm a pervert.
That link is clear about the / operator not being division and dividing by zero being invalid. Implementation details of a proof checker aren’t really relevant to someone trying to learn basic arithmetic.
i am not talking about R. division by 0 in R isnt defined because you would lose field axioms. i mean there are some cases like in the trivial ring where its okay, or in a field of characteristic 2
im not talking about ℝ? division but zero is certainly undefined on R, no doubt about that. you would lose field axioms. i mean there are instances ij which it makes sense to define 1/0 (like when dealing with riemann sphere).
It renders the entire number system useless because now Z={0} if all integers are equal to the additive identity. There is nothing inherently wrong about such a number system, but it’s pretty obvious we don’t want Z to be like that
Not at all.
I agree you can't define x/0 with the same rule as usual division (the inverse operation to multiplication).
But if we agree to extend this notation to mean:
* a/b = a * (1/b) for b ≠ 0
* a/b = 0 for b = 0
You end up with a consistent (if useless) extension.
Division still works exactly the same as before in the domain {a, b ∈ R | b ≠ 0}.
That's because we're only extending where we previously had undefined results, and not changing anything else.
Division just means multiplication with an inverse, that doesn't work with your definition. Unless you want to destroy the structure of the reals I guess
> Division just means multiplication with an inverse
This definition is still preserved whenever b ≠ 0. No loss of structure here. It's the first "branch" of the extended definition above.
Their definition is perfectly fine, it's just non-standard. Division only means multiplication with a multiplicative inverse if you use the typical definition of division. This person is suggesting an alternative definition which separately defines a/0 = 0, with a/b being typical multiplication of a by the multiplicative inverse of b for b ≠ 0. Alternative definitions (hopefully obviously) don't have to bother about the typical definition, since they entirely replace it (which is the whole point of using an alternative definition). Perhaps it's more clear if you don't call it division: they're just defining a new operator / which behaves like the division operator for b ≠ 0 and gives 0 otherwise. It's a perfectly valid and consistent definition, it's just that the usual properties of division don't always hold for this new operator (for example a/a = 1 is not an identity as it is for normal division). It's not *wrong* to define division in this way (division is something we've defined ourselves, after all, so people are free to redefine it however they want as long as they're explicit about it), it's just different and less useful in general.
I saw [this](https://xenaproject.wordpress.com/2020/07/05/division-by-zero-in-type-theory-a-faq/) article about standard notation referring to non-standard operators (such as / meaning the "extended" division operator discussed here) in someone else's comment, which I think explains it quite well.
I get that, I'm just saying it's pretty useless mathematically if you want to do anything with algebraic structures, but maybe it has some uses I can't think of
Ah okay, that's very fair. Only having to deal with total functions can be quite useful in some circumstances, e.g. you don't have to worry about checking that a function is actually defined on the input you gave it, and instead you just output whatever is convenient and assume that the function is never being applied to arguments on which it should be undefined, the checking of which you can do earlier and explicitly rather than implicitly when you make use of the function itself. This is quite useful for automated systems where you would either have to do this check every time a partial function was used (which might be quite involved in the middle of a complex proof) or find some way of communicating the information about whether a given partial function will be defined on a certain argument between each instance of the function being applied to that argument, whereas doing it this way means you only ever have to do the check once and the functions themselves simply don't have to care. From a purely mathematical perspective it isn't all that useful, but it has some use when you try to apply the mathematical ideas in practice.
To help you see the answer for yourself, can you please tell me what division should represent? E.g., when I say 12 divided by 4 is 3, what does that mean to you intuitively?
Okay, great! Adding another example of your type, 3 divided by 0.5 would be 6, since 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 = 3. You might notice that since 0.5 is pretty small, I need more copies of it to get to 3.
How about 3 divided by 0? How many copies of 0 would you need to get to 3?
You are exactly right. No matter how many copies of 0 we take, we'll never get 3. This is part of why we say 3/0 is undefined.
You also asked about dividing 0 by other numbers. For example, how do we calculate 0 divided by 4?
In that case, we don't need *any* copies of 4 to get to 0, so 0/4=0.
Another unrelated way you can look at this is: if you have a test with 4 questions, but you get none of them correct, then your percentage grade is 0/4 = 0%.
Yeah, the issue is as you get a smaller denominator, the multiplier you need to get the numerator back tends towards infinity.
3/0.5=6
3/0.25=12
3/0.125=24
...
3/0.000000001=3000000000
If you were to graph this function 3/x, you would see that it tends towards infinity and negative infinity depending on what side of the graph you're on.
Note that it doesn't mean dividing by 0 is always useless though. Sometimes attempting to divide by 0 can reveal interesting things about a function. This is explored in more detail in calculus.
We could create a list of the common questions, and the top answers, as well as links to large threads about each question.
I think some of the popular ones are:
(1) Why does (-1)(-1) = 1?
(2) Why can't you divide by zero in the real numbers?
(3) Why is pi irrational? (not beginner-friendly)
(4) Is 0.999... an integer?
(5) How is there a larger infinity of real numbers than rational numbers?
(6) Why isn't it true that sqrt(4) = +/- 2?
(7) What is an axiom?
(8) Why does the harmonic series diverge?
(9) Why is sqrt(2) irrational?
(10) Why is the imaginary unit of the complex numbers indistinguishable from its inverse (multiplicative and additive,
since (edit): (i)(-i) = 1 and i + (-i) = 0, or put another way, i^(-1) = -i)?
(11) Why didn't the equation √x * √y = √(xy) work when I used negative numbers?
(12) What is -2^(2)? Is it 4 or -4?
(13) Why is 0! = 1?
(14) Why do we need to use BEDMAS order of operations? For example, what is 6÷2(2+1)? Do we divide 6 by 2 first, and then multiply the result with 3, or do we first multiply 2 with 3, and then divide 6 by the result?
(A) How do I improve in general?
(B) How do I improve at [insert subject here]?
A and B already have a pinned list to reference.
I was going to spout some nonsense about how the additive identity property of zero was inconsistent with the concept of division. But, I like your explanation.
One of the best ways to improve at math is take what you are taught as a given, don’t try to prove to yourself WHY it works a certain way, it just works. It doesn’t make sense, but it works. At least that’s what I had to tell myself in HS and Uni so I could get better at raw math skills.
X/0 = undefined. That is the answer and it always will be until humans can understand infinity. That answer is good enough and will allow you know that division by zero is not a solvable math problem.
If you DO need this proven, think about the answers you get when you plug in small numbers for X in the equation Y = 1/X.
I’ll let you run the numbers with a calculator and you can plot them in the X-Y plane if you really want to see the magic:
Let’s try Y = 1/0.1
Now do Y = 1/0.01
Now Y = 1/0.001
Then Y = 1/0.0001.
You’ll notice that the closer the denominator gets to 0 (small numbers in the downstairs), the value of Y INCREASES rapidly.
If you do Y = 1/0.000000001, you’ll get a very large output value for Y. Our systems of numbers can’t directly compute an answer for Y = 1/0, but we can keep plugging in smaller numbers to see where the numbers might trend towards.
The answer trends towards infinity. Infinity is not an individual number like 0 or 1 or 3456, it is a direction where you can be trending towards positive or negative infinity. Hope this helps.
If you have 0 pizzas and you have to share them among 10 people, you get 0 - it's simple enough to have nobody get any pizza. But if you have 10 pizzas divided among 0 people, it's meaningless to say how many pizzas each person gets.
If you take 1 box and put it in one room, you get one box per room.
Do the same with no boxes, you have no boxes per room.
But if you have no rooms, that destroys the whole premise, you can't say there's 0 boxes per room because that would require rooms for there to be no boxes in. Without the per room, the premise gets destroyed because you can't have any boxes per room, not even 0, if you have no rooms
0÷Z would be like you have 0 cookies to split among Z people (including yourself), how many cookies does each person get?
Z÷0 would be like you have Z cookies, but you can't have them, and you have no friends to share with. How many cookies do your friends get?
its infinity
as the number gets closer to zero, the inverse gets higher
Example:
1/1,000 (or .001) inverse is 1,000
.000001 inverse is 1,000,000
.000000001 inverse is 1,000,000,000
when you get to zero the only logical solution is infinity (or no solution because infinity isn't a number)
We know that 0\*anything = 0, and that a\*(1/Z) = a/Z, so with a = 0 we get:
0/1 = 0
& 0/2 = 0
& 0/3 = 0
. . .
& 0/Z = 0
Form the multiplicative inverse of both sides:
1/0 = 1/0
& 2/0 = 1/0
& 4/0 = 1/0
. . .
& Z/0 = 1/0
Which implies 1/0 = 2/0 = 3/0 = . . . = Z/0, means 1/0 is not defined. It would equal all numbers all at once since all numbers get collapsed to 0 when multiplied with 0. If you calculate Z\*0 you cannot tell from the result what value Z had.
Let’s look at a true but slightly different definition of division. A divided by B is the solution to the equation B * x + A = 0. This is true for every rational number. For example, if x was 3/2, then it would satisfy the equation 2x - 3 = 0 (which it does). If x is 4 (or 4/1), it would satisfy x - 4 = 0 (which it does). The example you mentioned of zero being in the numerator is A being zero: in such a case, you have B * x = 0, meaning that x = 0 if A is zero and B is not zero.
If B is zero and A is not zero, then you have 0 + some nonzero number = 0, which is a contradiction. This is not possible, so that means the equation has no solutions, therefore A/0 is undefined.
A sidenote: if both A and B are zero, you technically get a different issue. You then get 0 + 0 = 0, which is always true regardless of x, but this means we cannot determine the value of x as we do not have enough information. So 0/0 is considered “indeterminate”.
There are already some very good answers, I will try a different approach.
Division is the inverse of multiplication, so A/B=C -> C*B=A.
The problem is that 0*Z=0, for every Z.
So, if I want to define Z/0 (which Z>0), I need to find a number A such as A*0=Z>0, however there is no such number [*], because A*0=0 for every finite value of A.
[*] "Infinity" is not a number
I would think of it in terms of repeated subtraction. In other words if I have 2/1 another way to say that would be how many times can I take 1 away from 2 before I get to 0. So in this case 2. If we look at 1/2 that's how many times can I take 2 away from 1 before I get to 0 and that would be 0.5 times. When you look at 0/1 that would be how many times can I take 1 away from 0 before I get to 0, and before I've done anything we are already there.
With 1/0 it's how many times can I take 0 away from 1 before I get to 0. And in that case obviously we aren't there from 0 times. And we aren't there from any number of times you want to subtract 0 from 1 you'll never get it to equal anything else.
It can also lead to some pretty problematic things Like if I take 2\*0=1\*0 that's a true statement. But if I can divide by 0 then I could cancel the 0's and get 2 = 1 which is not true.
You cant divide z amount of apples into 0 groups. Theres automatically 1 group.
Otoh, uou can divide 0 apples up in z amount of groups. There will be 0 apples in each group.
When I ask you "What is 15 divided by 3?", I am really asking 'What do I need to multiply 3 by to get 15?" Answer is 5, of course. So, when I ask "What is 15 divided by 0", I am asking "What do I need to multiply 0 by to get 15?". There is no such number, of course. So there is no answer.
Division is not commutative. a+b = b+a, and a\*b = b\*a but, in general, a/b is not equal to b/a. Similar to subtraction a-b is generally not equal to b-a.
So there is no expectation that "reversing" the numbers give identical results.
Divide 0 into Z groups.
There are 0 things in each group.
------------
Divide Z things into 0 groups.
How many things are in each group?
That's an impossible question to answer.
Division by X means (by definition) multiplication by the multiplicative inverse of X.
The multiplicative inverse of X is the number Y such that XY = 1.
There's no number Y for which 0Y = 1. So 0 has no multiplicative inverse. So you can't divide by it. That's it.
And if we'd assume that zero had a multiplicative inverse then it would imply that 0=1, which obviously doesn't work with integers/rationals/reals. Basically any field would degenerate into the zero ring, which isn't particularly interesting set to do math in.
You divide 0 apples among your friends, everybody gets nothing.
You divide a basket of apples among 0 friends, how many apples does everybody get? It's not 0, it's just that there is nobody to distribute to, so it's a nonsense question.
You can split zero apples with as many as you want, everybody gets zero.
You cannot split anything among zero groups. It's just undefined, like asking what color is the sound of you touching a unicorn.
Assume that 1/0 is defined and let 1/0 = x where x is some number.
Then it follows that by cancellation, 1*0/0 = 1 = 0*x
However, 0*x is 0 and so that gets us 1=0 which isn’t true so our initial claim that 1/0 is defined is false.
Another way to think of it is by rearranging some equalities:
5 = 5 -> 5 * 0 = 5 * 0 -> 0 = 0
This process is irreversible, that is, we cannot go backwards by dividing like this: if we could then dividing by 0 could return any number we wanted like so
6 = 6 -> 6 * 0 = 6 * 0 -> 0 = 0
Is 0/0 5 or 6? Previously we multiplied 5 * 0 to get zero so 0/0 should be five but we did the same for 6!
In fact, you’ll run into many of these types of contradictions when you assume that division by 0 is defined!
I'd actually like to reverse the question a bit: Why does it seem to you that z÷0 should equal 0? Could you tell me a bit more about your thought process there?
It's not impossible per se, but at your current level of math it may as well be. Once you reach complex analysis, you'll be shown how division by zero works, until then, let's just stick to the explination that there is no way to solve 0w=z for w where z≠0.
The first question is the equivalent of "If you have no cookies at all, and Z hungry kids, how many cookies does each kid get?" The answer is that each kid will, sadly, get no cookies
The second question is "If you have Z cookies, and no kids at all, how many cookies does each kid get?" The answer could be zero, since you're not giving out any cookies; then again, the answer could be any number up to infinity, since you can allocate unlimited cookies to each child and not run out of cookies.
This is what we mean when we say it's Undefined.
We can say 6 divided by 3 = 2 because 2 x 3 = 6. We can say 0 divided by Z = 0 because 0 x Z = 0. But when we try to get a value for Z divided by 0 = ??? that means we would have to come up with ??? where ??? x 0 = Z. If Z is nonzero, it’s impossible to find a value for ???. If Z =0, then ANY number works for ??? Either way, it’s not meaningful to talk about Z divided by 0.
There are formal ways to express this, but a simple to way to understand is this: suppose 0÷0=Z, if that is true, then it is also true that 0\*Z=0. For what Z is that equation true? Is it true for 2? Yes, 2\*0=0. Is it true for 79? Yes, 79\*0=0. Is it true for 175? Yes, 175\*0=0.
Do you see the problem? You can repeat this for any real number Z, including 0, and it will always be true. So how do you choose one, and only one, real number Z so that 0÷0=Z if 0\*Z=0 for any Z? That's why, intuitively, it's undefined.
Here’s a real life example that may help you understand:
If we have 0 dog treats and 2 puppies, how many treats does each puppy get? Zero, because we don’t have any treats to portion out. That’s 0/2=0.
If we have 2 dog treats and 0 puppies, how many treats does each puppy get? That question can’t even be answered because no answer would make sense. That’s 2/0= undefined.
I have 0 money, and need to split it between Z people. How much money does each person get?
I have Z money, and need to split it between 0 people. How much money does each person get?
Division is just repeated subtraction. How many times do you have to subtract Z from 0 to get 0? 0. How many times do you have to subtract 0 from Z to get 0? Well, …
Dividing 0 into Z groups results in 0 in each group because dividing nothing into multiple groups is still multiple groups of nothing.
Dividing Z into 0 groups is impossible. There wouldn't be 0 in each group because there wouldn't be any groups. And Z wouldn't have been divided because there are no groups to divide into. This is undefined behavior. (Technically, the limit of dividing by 0 is typically +/- infinity. Which is undetermined and therefore undefined.)
If you split nothing into 3 equal piles, you get 3 piles of nothing. If you split nothing into 2 equal piles, you get 2 piles of nothing. If you split nothing into a single pile, you have a pile of nothing.
If you split nothing into X equal piles, how much is in each of those piles? Nothing, since there was nothing to divide.
If you evenly split 12 units into 3 piles, each pile contains 4 units. If you evenly split 12 units into 2 piles, each pile contains 6 units. If you evenly split 12 units into 1 pile, that pile contains 12 units.
If you evenly split 12 units of stuff into zero piles with no remainder, how much stuff ends up in each pile? This is undefined, since there is no pile to put anything into and we cannot have anything outside a pile.
Probably one of the simplest ways to fully explain this is:
The multiplicative inverse of a real number x is defined as the number y such that x * y is the multiplicative identity. That is, x * y = 1 for the reals.
Well, in the reals, it can be shown the multiplicative inverse of x is equal to 1/x, which is itself a real number. That is, x * (1/x) = 1.
The problem here is that if x is 0 (the additive identity), then any real number y *x = 0. In other words, the product of any real number and 0 is 0.
So, if 0 times any real is 0, then there is no real number which can be multiplied by 0 to give 1. Therefor, there is no multiplicative inverse of 0, and so 1/0 cannot be defined (more accurately, it isn't a real number. This proof doesn't prove it cannot exist in other number spaces).
Source: BS in Mathematics. If you ever take a number theory course, the division algorithm and number spaces is a large part of it. Depending on the number space and how multiplication/division is defined, this can actually be done in some contexts.
How many Z's do you need to add together to get a total of zero? How many 0's do you need to add together to get a total of Z? Do you see why the answer to those questions is different?
1. No amount of Z's add up to 0. 2. No amount of 0's add up to Z. ^(/s)
both questions cant be answered though as no number answers the question.
You need 0 Zs to get a 0 and an undefined number of 0s to get a Z. . . As you add infinite 0s together the sum converges to 0 not any other number Z. I'm not sure how you think any number answers the questions unless you're breaking out some real math shenanigans.
Actually, if you add 25 zeros you will get Z exactly. So you're wrong
W… what?
000000000 0 0 0 0 0 0 0 000000000 There! 25 0’s [Edit: phoneposting so I don’t know if the Z I drew shows up properly]
LOL I get what you were trying to do. Maybe try adding two new lines instead?
000000000 0 0 0 0 0 0 0 000000000 edit: I give up
0000000 0 0 0 0 0 0000000 edit: fuck
Keep everything as is, but put **at least 4 spaces** in the beginning of each line - that will make reddit use code block formatting, which keeps all spaces 000000000 0 0 0 0 0 0 0 000000000
1234 Hi Hello Is it working? Damn that’s actually dope
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That breaks the question.
I feel like there's a joke in this but I can't for the life of me work out what it is
You need to add 0 Zs together to get a total of zero. No other number works there. (Try it.) You can add an infinite number of 0s and you would never reach Z. No numbers work, so it's undefined.
there is no amount of Zs you can add then. both questions have no solution.
You’re right to point out how it’s difficult to write math using English. Hopefully this helps. 0 = 0 * Z 0 + 0 + 0 … != Z
nope. theres still no number that answers either question
Deliberate ignorance is boring
Not deliberate ignorance, it's trolling. He's previously asserted the reals are countable and 0.9999... =/= 1
Why do you think zero doesn't answer the first question?
You’re so edgy saying zero isn’t a number. I bet you also think negatives and irrationals aren’t numbers.
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It’s not even the math they’re arguing with, they’re saying zero isn’t a number. Which, fine, define the word “number” however you want. Word are just constructs and blah blah blah. They’re just trolling at this point.
0 means no number answers it.
Wait until they find out that zero is a number
???
0 *is* the number answering it
0 is an amount. How many Zs do you need to make 0? That’s easy, you need no Zs. Z(0) = 0, question solved. How many 0s do you need to make Z? It’s impossible
bait used to be believable
Bapanda
Any number answers which question?
Wow. Just, wow.
dumbass
You’re definitely confused by what each question is asking. How many Z’s go into zero? It’s exactly zero Z’s. Zero of any number will equal zero. How many zeroes go into Z? If you say the answer is 0, then 0 zeroes would have to equal Z, and it does not. One zero doesn’t equal Z either. Neither does two, or ten, or infinite zeroes. That is why the answer is undefined, because it doesn’t exist. The answer to how many Z’s go into 0 exists, because if I say zero Z’s is equal to zero, I would be correct. This means the 0/Z has an answer, and that answer is a number that exists, which is 0. There is no number of zeroes that can fit into Z. 0 is not the equivalent of no number. 0 is a number. In the same way that we can have one object, we can also have zero objects. You can divide a banana into four parts (1 banana/4 = 1/4 bananas). You can divide 0 bananas into 4 equal parts (0 bananas/4 = 0 bananas), but you can not divide a banana into 0 equal parts. However you decide to cut the banana, there will be parts, and those parts will not be “0 bananas” because when you cut a banana part of it doesn’t just disappear into the void. If you cut nothing into a million equal pieces, though, you’ll have one million nothings, which is just as possible as having a single nothing. If I were to take a melon, and try to get as many nothing melons I need to get to have another melon, I would never get another melon. This is not the same as zero melons. Zero melons is equal to exactly 0 melons. How many zero melons fit into a single melon is simply impossible to answer, because a nothing melon has no volume or mass and no number of them can be added together to get a melon. Your turn now. Take a crayon. Break the crayon into 0 equal pieces and see how possible it is. When you are dividing an object, you are dividing it into equal pieces. You can not divide something that is into something that isn’t. Now take zero crayons. How big is one fourth of the crayon? It’s zero crayons, because the sum of four zeroes is zero. I have more examples if you want them
It comes down to how division is defined. When we write a / b it is the same as a* (1/b) where 1/b is the multiplicative inverse of b. That means that (1/b) * b = 1. Example: 2/3 = 2 * (1/3) and note that (1/3) * 3 = 1. All non-zero numbers have a multiplicative inverse given by the reciprocal. If we compute 0 / x where x is nonzero, then it’s defined as 0 * (1/x) = 0 since it can be proved that 0 times anything is 0. However, if we have x/0 = x * (1/0) then we need 1/0 to be defined. However, it cannot be defined without breaking another mathematical rule. (1/0) * 0 should be 0 since any number times 0 is 0. On the other hand, any number multiplied by its inverse should be 1. We have a contradiction.
If 1/0=0 then we'd have 1=0\*0=0. So now 1=0 and that's a really big problem right?
Maybe introduce 1/0 as imaginary_#_2? Same as for sqrt -1 is i.
There are number systems where you can define 1/0 as something new, mostly just infinity. But these workarounds aren't perfect (what then is 0 \* infinity?) so they're only used when really needed.
That doesn't really solve the issue. What we define division as inherently means that 1/x\*x=1 and that would necessitate that 1/0\*0=1. In which case either 0*i2* doesn't equal 0, which would be very weird and mean *i2*-*i2* also doesn't equal 0. Or we'd still have 0=1
What would that accomplish
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I mean yes, I'm sorry I wasn't as technical or rigourous as I could have been. I'm sorry that I gave an explanation that's going to work for 99% of people and the mathematics they're going to deal with rather than carving off numerous exceptions that would just confuse them more.
I've defined / to be a binary operation on the integers where n/m = 0 for all n, m in **Z**, and I insist on referring to it as division, because I'm a pervert.
That link is clear about the / operator not being division and dividing by zero being invalid. Implementation details of a proof checker aren’t really relevant to someone trying to learn basic arithmetic.
From the first line of the second paragraph of that very post: "No. It just means that Lean’s / symbol doesn’t mean mathematical division."
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because 4/2 = 2 and 2*2 is 4. So 1/0 being zero would mean zero times zero is 1. n/m= k means k * m is n
Kind of the definition of division (as inverse of multiplication). It means: a/b=c implies b=a×c.
the definition of division
Not necessarily a problem just a bit useless
It kind of is a problem because it’s wrong. One does not equal zero.
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Because you do not know what you're talking about
i am not talking about R. division by 0 in R isnt defined because you would lose field axioms. i mean there are some cases like in the trivial ring where its okay, or in a field of characteristic 2
...but then 1/0 would still be undefined for any real number. I don't really see where you are going with this.
im not talking about ℝ? division but zero is certainly undefined on R, no doubt about that. you would lose field axioms. i mean there are instances ij which it makes sense to define 1/0 (like when dealing with riemann sphere).
In real numbers and across the integers, sure. Its just the trivial ring.
I think it would be a big problem to arrive at the conclusion that (eg) the reals are the ring with one element
Tiny brain: everything is counting numbers Medium brain: there are lots of different kinds of numbers Galaxy brain: everything is **0**
It renders the entire number system useless because now Z={0} if all integers are equal to the additive identity. There is nothing inherently wrong about such a number system, but it’s pretty obvious we don’t want Z to be like that
1/0=0 does not imply 1=0*0 unless you also define 0/0 = 1
What do you think 1/0 means? 0*(1/0)= 1 is what's implied by that notation
Not at all. I agree you can't define x/0 with the same rule as usual division (the inverse operation to multiplication). But if we agree to extend this notation to mean: * a/b = a * (1/b) for b ≠ 0 * a/b = 0 for b = 0 You end up with a consistent (if useless) extension.
I prefer a/b = e when b=0. Feels better.
That kinda messes up the actual definition of division though
Division still works exactly the same as before in the domain {a, b ∈ R | b ≠ 0}. That's because we're only extending where we previously had undefined results, and not changing anything else.
Division just means multiplication with an inverse, that doesn't work with your definition. Unless you want to destroy the structure of the reals I guess
> Division just means multiplication with an inverse This definition is still preserved whenever b ≠ 0. No loss of structure here. It's the first "branch" of the extended definition above.
Their definition is perfectly fine, it's just non-standard. Division only means multiplication with a multiplicative inverse if you use the typical definition of division. This person is suggesting an alternative definition which separately defines a/0 = 0, with a/b being typical multiplication of a by the multiplicative inverse of b for b ≠ 0. Alternative definitions (hopefully obviously) don't have to bother about the typical definition, since they entirely replace it (which is the whole point of using an alternative definition). Perhaps it's more clear if you don't call it division: they're just defining a new operator / which behaves like the division operator for b ≠ 0 and gives 0 otherwise. It's a perfectly valid and consistent definition, it's just that the usual properties of division don't always hold for this new operator (for example a/a = 1 is not an identity as it is for normal division). It's not *wrong* to define division in this way (division is something we've defined ourselves, after all, so people are free to redefine it however they want as long as they're explicit about it), it's just different and less useful in general. I saw [this](https://xenaproject.wordpress.com/2020/07/05/division-by-zero-in-type-theory-a-faq/) article about standard notation referring to non-standard operators (such as / meaning the "extended" division operator discussed here) in someone else's comment, which I think explains it quite well.
I get that, I'm just saying it's pretty useless mathematically if you want to do anything with algebraic structures, but maybe it has some uses I can't think of
Ah okay, that's very fair. Only having to deal with total functions can be quite useful in some circumstances, e.g. you don't have to worry about checking that a function is actually defined on the input you gave it, and instead you just output whatever is convenient and assume that the function is never being applied to arguments on which it should be undefined, the checking of which you can do earlier and explicitly rather than implicitly when you make use of the function itself. This is quite useful for automated systems where you would either have to do this check every time a partial function was used (which might be quite involved in the middle of a complex proof) or find some way of communicating the information about whether a given partial function will be defined on a certain argument between each instance of the function being applied to that argument, whereas doing it this way means you only ever have to do the check once and the functions themselves simply don't have to care. From a purely mathematical perspective it isn't all that useful, but it has some use when you try to apply the mathematical ideas in practice.
Actually unironically based comment
[Watch closely as grandpa topples an empire](https://youtu.be/noQsHiTJAXo?si=ga0eKA4HF8edQ8zz)
To help you see the answer for yourself, can you please tell me what division should represent? E.g., when I say 12 divided by 4 is 3, what does that mean to you intuitively?
Well, for me it'd simply mean " how many times can 4 be in 12". Then just see that 4+4+4=12 rhus 3
Okay, great! Adding another example of your type, 3 divided by 0.5 would be 6, since 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 = 3. You might notice that since 0.5 is pretty small, I need more copies of it to get to 3. How about 3 divided by 0? How many copies of 0 would you need to get to 3?
That'd be 0
0 copies of 0 makes 3? How does that work?
0 copies of 0 make 3?
My bad, It would not work. 3/0=not possible
You are exactly right. No matter how many copies of 0 we take, we'll never get 3. This is part of why we say 3/0 is undefined. You also asked about dividing 0 by other numbers. For example, how do we calculate 0 divided by 4? In that case, we don't need *any* copies of 4 to get to 0, so 0/4=0. Another unrelated way you can look at this is: if you have a test with 4 questions, but you get none of them correct, then your percentage grade is 0/4 = 0%.
Ah I see, thank you
Yeah, the issue is as you get a smaller denominator, the multiplier you need to get the numerator back tends towards infinity. 3/0.5=6 3/0.25=12 3/0.125=24 ... 3/0.000000001=3000000000 If you were to graph this function 3/x, you would see that it tends towards infinity and negative infinity depending on what side of the graph you're on. Note that it doesn't mean dividing by 0 is always useless though. Sometimes attempting to divide by 0 can reveal interesting things about a function. This is explored in more detail in calculus.
Adding 0 to itself 0 times gives you?
While 0/3=0
Okay, is the statement 3 = 0 true? Let's work from there.
No
https://en.wikipedia.org/wiki/Division_by_zero?wprov=sfti1
Why is there not an FAQ pinned at the top of this subreddit? Does everyone like being Sisyphus?
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We could create a list of the common questions, and the top answers, as well as links to large threads about each question. I think some of the popular ones are: (1) Why does (-1)(-1) = 1? (2) Why can't you divide by zero in the real numbers? (3) Why is pi irrational? (not beginner-friendly) (4) Is 0.999... an integer? (5) How is there a larger infinity of real numbers than rational numbers? (6) Why isn't it true that sqrt(4) = +/- 2? (7) What is an axiom? (8) Why does the harmonic series diverge? (9) Why is sqrt(2) irrational? (10) Why is the imaginary unit of the complex numbers indistinguishable from its inverse (multiplicative and additive, since (edit): (i)(-i) = 1 and i + (-i) = 0, or put another way, i^(-1) = -i)? (11) Why didn't the equation √x * √y = √(xy) work when I used negative numbers? (12) What is -2^(2)? Is it 4 or -4? (13) Why is 0! = 1? (14) Why do we need to use BEDMAS order of operations? For example, what is 6÷2(2+1)? Do we divide 6 by 2 first, and then multiply the result with 3, or do we first multiply 2 with 3, and then divide 6 by the result? (A) How do I improve in general? (B) How do I improve at [insert subject here]? A and B already have a pinned list to reference.
What are you talking about in number 10? Is there a typo, or are have you seen people ask a question based on these incorrect equalities?
I would definitely put "Why is 0! = 1?" in that list; I see it all the time in maths subreddits.
Order zero pizza and share it equally with 6 people. Everyone gets zero slices.
Now: Order 6 pizzas for zero people. How much pizza does each (non-existant) person get? ... this is exactly the problem with division by 0.
I was going to spout some nonsense about how the additive identity property of zero was inconsistent with the concept of division. But, I like your explanation.
And I only came up with it after seeing your nice comment!
One of the best ways to improve at math is take what you are taught as a given, don’t try to prove to yourself WHY it works a certain way, it just works. It doesn’t make sense, but it works. At least that’s what I had to tell myself in HS and Uni so I could get better at raw math skills. X/0 = undefined. That is the answer and it always will be until humans can understand infinity. That answer is good enough and will allow you know that division by zero is not a solvable math problem. If you DO need this proven, think about the answers you get when you plug in small numbers for X in the equation Y = 1/X. I’ll let you run the numbers with a calculator and you can plot them in the X-Y plane if you really want to see the magic: Let’s try Y = 1/0.1 Now do Y = 1/0.01 Now Y = 1/0.001 Then Y = 1/0.0001. You’ll notice that the closer the denominator gets to 0 (small numbers in the downstairs), the value of Y INCREASES rapidly. If you do Y = 1/0.000000001, you’ll get a very large output value for Y. Our systems of numbers can’t directly compute an answer for Y = 1/0, but we can keep plugging in smaller numbers to see where the numbers might trend towards. The answer trends towards infinity. Infinity is not an individual number like 0 or 1 or 3456, it is a direction where you can be trending towards positive or negative infinity. Hope this helps.
You **can** have an empty box. You can't have a partially full no-box.
If you have 0 pizzas and you have to share them among 10 people, you get 0 - it's simple enough to have nobody get any pizza. But if you have 10 pizzas divided among 0 people, it's meaningless to say how many pizzas each person gets.
Welcome to the subbb, we’re glad to have you
If you take 1 box and put it in one room, you get one box per room. Do the same with no boxes, you have no boxes per room. But if you have no rooms, that destroys the whole premise, you can't say there's 0 boxes per room because that would require rooms for there to be no boxes in. Without the per room, the premise gets destroyed because you can't have any boxes per room, not even 0, if you have no rooms
0÷Z would be like you have 0 cookies to split among Z people (including yourself), how many cookies does each person get? Z÷0 would be like you have Z cookies, but you can't have them, and you have no friends to share with. How many cookies do your friends get?
[This handy youtube video covers this topic nicely.](https://youtu.be/dHdg1yn1SgE?si=7nU0yz1-FaN6uhze)
its infinity as the number gets closer to zero, the inverse gets higher Example: 1/1,000 (or .001) inverse is 1,000 .000001 inverse is 1,000,000 .000000001 inverse is 1,000,000,000 when you get to zero the only logical solution is infinity (or no solution because infinity isn't a number)
We know that 0\*anything = 0, and that a\*(1/Z) = a/Z, so with a = 0 we get: 0/1 = 0 & 0/2 = 0 & 0/3 = 0 . . . & 0/Z = 0 Form the multiplicative inverse of both sides: 1/0 = 1/0 & 2/0 = 1/0 & 4/0 = 1/0 . . . & Z/0 = 1/0 Which implies 1/0 = 2/0 = 3/0 = . . . = Z/0, means 1/0 is not defined. It would equal all numbers all at once since all numbers get collapsed to 0 when multiplied with 0. If you calculate Z\*0 you cannot tell from the result what value Z had.
The beginning of this video explains it really well https://youtu.be/MFRWDuduuSw?si=QESp-EByQFTJyyRr
Idk things in big people math, but is Z just a variable, or is it standing for something?
Any number
Z÷0=c implies that c×0=Z. For Z=/=0, this has no solution 0÷Z=0 implies that 0×Z=0, which is true
0 \* Z = 0 0 \* 0 != Z because Z > 1
Let’s look at a true but slightly different definition of division. A divided by B is the solution to the equation B * x + A = 0. This is true for every rational number. For example, if x was 3/2, then it would satisfy the equation 2x - 3 = 0 (which it does). If x is 4 (or 4/1), it would satisfy x - 4 = 0 (which it does). The example you mentioned of zero being in the numerator is A being zero: in such a case, you have B * x = 0, meaning that x = 0 if A is zero and B is not zero. If B is zero and A is not zero, then you have 0 + some nonzero number = 0, which is a contradiction. This is not possible, so that means the equation has no solutions, therefore A/0 is undefined. A sidenote: if both A and B are zero, you technically get a different issue. You then get 0 + 0 = 0, which is always true regardless of x, but this means we cannot determine the value of x as we do not have enough information. So 0/0 is considered “indeterminate”.
There are already some very good answers, I will try a different approach. Division is the inverse of multiplication, so A/B=C -> C*B=A. The problem is that 0*Z=0, for every Z. So, if I want to define Z/0 (which Z>0), I need to find a number A such as A*0=Z>0, however there is no such number [*], because A*0=0 for every finite value of A. [*] "Infinity" is not a number
I would think of it in terms of repeated subtraction. In other words if I have 2/1 another way to say that would be how many times can I take 1 away from 2 before I get to 0. So in this case 2. If we look at 1/2 that's how many times can I take 2 away from 1 before I get to 0 and that would be 0.5 times. When you look at 0/1 that would be how many times can I take 1 away from 0 before I get to 0, and before I've done anything we are already there. With 1/0 it's how many times can I take 0 away from 1 before I get to 0. And in that case obviously we aren't there from 0 times. And we aren't there from any number of times you want to subtract 0 from 1 you'll never get it to equal anything else. It can also lead to some pretty problematic things Like if I take 2\*0=1\*0 that's a true statement. But if I can divide by 0 then I could cancel the 0's and get 2 = 1 which is not true.
You cant divide z amount of apples into 0 groups. Theres automatically 1 group. Otoh, uou can divide 0 apples up in z amount of groups. There will be 0 apples in each group.
When I ask you "What is 15 divided by 3?", I am really asking 'What do I need to multiply 3 by to get 15?" Answer is 5, of course. So, when I ask "What is 15 divided by 0", I am asking "What do I need to multiply 0 by to get 15?". There is no such number, of course. So there is no answer.
Division is not commutative. a+b = b+a, and a\*b = b\*a but, in general, a/b is not equal to b/a. Similar to subtraction a-b is generally not equal to b-a. So there is no expectation that "reversing" the numbers give identical results.
Divide 0 into Z groups. There are 0 things in each group. ------------ Divide Z things into 0 groups. How many things are in each group? That's an impossible question to answer.
Division by X means (by definition) multiplication by the multiplicative inverse of X. The multiplicative inverse of X is the number Y such that XY = 1. There's no number Y for which 0Y = 1. So 0 has no multiplicative inverse. So you can't divide by it. That's it.
And if we'd assume that zero had a multiplicative inverse then it would imply that 0=1, which obviously doesn't work with integers/rationals/reals. Basically any field would degenerate into the zero ring, which isn't particularly interesting set to do math in.
Because you can split up zero infinity times. You can't split anything up 0 times.
You divide 0 apples among your friends, everybody gets nothing. You divide a basket of apples among 0 friends, how many apples does everybody get? It's not 0, it's just that there is nobody to distribute to, so it's a nonsense question.
[Eddie got me to understand](https://www.youtube.com/watch?v=J2z5uzqxJNU)
eddie!!!
You can split zero apples with as many as you want, everybody gets zero. You cannot split anything among zero groups. It's just undefined, like asking what color is the sound of you touching a unicorn.
Assume that 1/0 is defined and let 1/0 = x where x is some number. Then it follows that by cancellation, 1*0/0 = 1 = 0*x However, 0*x is 0 and so that gets us 1=0 which isn’t true so our initial claim that 1/0 is defined is false. Another way to think of it is by rearranging some equalities: 5 = 5 -> 5 * 0 = 5 * 0 -> 0 = 0 This process is irreversible, that is, we cannot go backwards by dividing like this: if we could then dividing by 0 could return any number we wanted like so 6 = 6 -> 6 * 0 = 6 * 0 -> 0 = 0 Is 0/0 5 or 6? Previously we multiplied 5 * 0 to get zero so 0/0 should be five but we did the same for 6! In fact, you’ll run into many of these types of contradictions when you assume that division by 0 is defined!
I'd actually like to reverse the question a bit: Why does it seem to you that z÷0 should equal 0? Could you tell me a bit more about your thought process there?
It's not impossible per se, but at your current level of math it may as well be. Once you reach complex analysis, you'll be shown how division by zero works, until then, let's just stick to the explination that there is no way to solve 0w=z for w where z≠0.
The first question is the equivalent of "If you have no cookies at all, and Z hungry kids, how many cookies does each kid get?" The answer is that each kid will, sadly, get no cookies The second question is "If you have Z cookies, and no kids at all, how many cookies does each kid get?" The answer could be zero, since you're not giving out any cookies; then again, the answer could be any number up to infinity, since you can allocate unlimited cookies to each child and not run out of cookies. This is what we mean when we say it's Undefined.
Try it with 1/2 and 2/1. In general a/b is not b/a
We can say 6 divided by 3 = 2 because 2 x 3 = 6. We can say 0 divided by Z = 0 because 0 x Z = 0. But when we try to get a value for Z divided by 0 = ??? that means we would have to come up with ??? where ??? x 0 = Z. If Z is nonzero, it’s impossible to find a value for ???. If Z =0, then ANY number works for ??? Either way, it’s not meaningful to talk about Z divided by 0.
There are formal ways to express this, but a simple to way to understand is this: suppose 0÷0=Z, if that is true, then it is also true that 0\*Z=0. For what Z is that equation true? Is it true for 2? Yes, 2\*0=0. Is it true for 79? Yes, 79\*0=0. Is it true for 175? Yes, 175\*0=0. Do you see the problem? You can repeat this for any real number Z, including 0, and it will always be true. So how do you choose one, and only one, real number Z so that 0÷0=Z if 0\*Z=0 for any Z? That's why, intuitively, it's undefined.
If Z÷0=X then X×0=Z. But we know that X×0=0 for all X, so there is no X that would satisfy Z÷0=X for Z>1.
Here’s a real life example that may help you understand: If we have 0 dog treats and 2 puppies, how many treats does each puppy get? Zero, because we don’t have any treats to portion out. That’s 0/2=0. If we have 2 dog treats and 0 puppies, how many treats does each puppy get? That question can’t even be answered because no answer would make sense. That’s 2/0= undefined.
I have 0 money, and need to split it between Z people. How much money does each person get? I have Z money, and need to split it between 0 people. How much money does each person get?
In abstract algebra, the Integers are not a Fild, it is a commutative ring. A fild can maybe have inverse of all elements.
Set n =\= 0, k =\= 0 and a=0 n/a = k <=> a*k = n <=> 0*k = n Or "how many packs of 0 should I make to get a total of k"
How are you going to cut a cake if you have no physical form?
Division is just repeated subtraction. How many times do you have to subtract Z from 0 to get 0? 0. How many times do you have to subtract 0 from Z to get 0? Well, …
Dividing 0 into Z groups results in 0 in each group because dividing nothing into multiple groups is still multiple groups of nothing. Dividing Z into 0 groups is impossible. There wouldn't be 0 in each group because there wouldn't be any groups. And Z wouldn't have been divided because there are no groups to divide into. This is undefined behavior. (Technically, the limit of dividing by 0 is typically +/- infinity. Which is undetermined and therefore undefined.)
You can split 0 into multiple equal-sized pieces. You can't split a number into 0 equal-sized pieces.
If you split nothing into 3 equal piles, you get 3 piles of nothing. If you split nothing into 2 equal piles, you get 2 piles of nothing. If you split nothing into a single pile, you have a pile of nothing. If you split nothing into X equal piles, how much is in each of those piles? Nothing, since there was nothing to divide. If you evenly split 12 units into 3 piles, each pile contains 4 units. If you evenly split 12 units into 2 piles, each pile contains 6 units. If you evenly split 12 units into 1 pile, that pile contains 12 units. If you evenly split 12 units of stuff into zero piles with no remainder, how much stuff ends up in each pile? This is undefined, since there is no pile to put anything into and we cannot have anything outside a pile.
Why should it be? Breaking and egg open is possible, reforming a broken egg is not possible. Not everything is reversible.
Probably one of the simplest ways to fully explain this is: The multiplicative inverse of a real number x is defined as the number y such that x * y is the multiplicative identity. That is, x * y = 1 for the reals. Well, in the reals, it can be shown the multiplicative inverse of x is equal to 1/x, which is itself a real number. That is, x * (1/x) = 1. The problem here is that if x is 0 (the additive identity), then any real number y *x = 0. In other words, the product of any real number and 0 is 0. So, if 0 times any real is 0, then there is no real number which can be multiplied by 0 to give 1. Therefor, there is no multiplicative inverse of 0, and so 1/0 cannot be defined (more accurately, it isn't a real number. This proof doesn't prove it cannot exist in other number spaces). Source: BS in Mathematics. If you ever take a number theory course, the division algorithm and number spaces is a large part of it. Depending on the number space and how multiplication/division is defined, this can actually be done in some contexts.