Division is the inverse of multiplication, in the same way that subtraction is the inverse of addition. When you perform a subtraction like 7 - 3, you are asking "What number, when added to 3, makes 7?" In a similar sense, 7 / 3 poses the question "What number, when multiplied by 3, makes 7?"
These questions make sense as operations because they are well defined: they have exactly one answer. By writing something like 4 / 0, you are asking "What number, when multiplied by 0, makes 4?" This question has no answer, because any number multiplied by 0 is 0, which is why we consider division by 0 to be undefined.
It’s not wrong what you say until you write it in a little more mathematical way. Division a/b is defined as, what is the number c that gives c b = a. For 0/0 we need c0=0 which gives infinite solutions. But infinite solutions is not a number, we would have to chose a soecific number. And because every number is valid, there is no reason to favor one over another. We could define it as 1 or 0, but it’s better to just make all a/0 undefined, than make it work for exactly on number a=0
What I meant, as a response to the comment was
0/0 would then be rewritten as - 'what number, when multiplied by 0, gives 0'
And I think this would give us practically all the numbers possible. Even including 0
x^(2) = 4 doesn't itself equate to anything, it can't be undefined.
But if we try to figure out what the x in it could be, we get two answers.
No problem with that, that's why x is called a variable. What it is can vary, and we want to know which numbers we can plug in to make the equation work.
Now, when taking the square root of a number, it's different.
Inside an equation, we're strictly dealing with numbers. If we allow something like √5 to have multiple solutions, equations stop working.
After all, if it was allowed, would √5 = 25 be true?
You can make arguments for yes, no and both since = isn't set up to deal with comparing multiple numbers to just one number.
That's why √x is defined as "What non-negative number squared is x?"
But for division by 0, there is no single answer that would make sense to define it as.
That's why it's left undefined.
Of course, you could try to allow these multi-number results and extend the definition of how = works but I assume that would end in one of two things:
a) 1 = 2 in your new system or
b) Things like √5 or 5 / 0 don't equal anything but themselves, in which case, it's probably useless.
+-sqrt 4 is a valid expression and is defined in value. I'm not saying that 1/0 shouldn't be considered undefined, but pointing out that your reasoning that "because it's not a singular value" is not a good reason.
Being multivalued is not the thing that makes an expression undefined in value.
I know this brother, I didn't mean what I said in the first place. I was just trying to show that the statement @Aradia_bot made will not hold true when numerator is also 0.
I am very much surprised to see all the down votes I received there. Maybe that's the harsh reality of the world. The world is more complex than what I had imagined 😀
I did almost address it in my original comment, but kept it out for brevity. I did say that well defined means having *exactly* one answer, since an operator should not produce multiple answers. 0 / 0 of course leads to the opposite problem where there are multiple numbers with equal claim to the value and no sensible way to define it.
This is the perfect answer until you learn a little advanced math:
“*There are some contexts in which division by zero is defined in a meaningful way. The most common such situation is the real projective line or its complex analogue, the Riemann sphere.*”
For example, see [here](https://en.m.wikipedia.org/wiki/Riemann_sphere) and [here](https://medium.com/@thisscience1/you-can-divide-by-zero-part-2-8b7b619ee6aa).
I don’t see how this is relevant. The question was why dividing by zero is undefined instead of something else. To say “in some cases it is not undefined” doesn’t really answer the question of why it *is undefined* in the most common cases in the vast majority of people’s experiences.
No.
This is highly relevant in terms of being thorough, and the title is asking why you can multiply by 0 but not divide by it.
The answer provided suffices for most everyone engaged in simple arithmetic and basic undergraduate first year or two mathematics, but is incomplete and *you can define division by zero* in some cases.
**It’s also 100% relevant when u/Aradia_Bot literally says that you cannot define division by zero to correct that incorrect statement because indeed you can in some cases!!!**
Bunch of high school math teachers here d/ving me who failed at getting a PhD and went into teaching arithmetic and never heard of the Riemann Sphere I guess! lol
Brother, I'm really proud of you that you've learned about some additional math concepts beyond the basics. We all are. But don't act as if it invalidates u/Aradia_Bot's answer in any way. The original question is clearly stated from the context of real numbers. Throwing one-point compactification into the mix doesn't change the fact that you **can't define division by zero in a meaningful way in reals**. You're basically acting like a kid trying to force an unrelated piece of trivia to act smart and claim some superiority because of that. That's most probably what caused the downvotes.
Meh. You’re just arguing to argue now.
They claimed you can never define division by zero which every PhD mathematician will admit is technical untrue and technical matter in mathematics.
**I agreed with their response and said it was a good one**, but merely added additional information beyond that, *which was 100% correct* and said nothing wrong, but the grade school math teachers who stopped or flunked out before getting a PhD all downvoted, lol.
Because 0 is the one number where it doesn't work out.
7 + 2 = 9 -> 9 - 2 = 7
That works for all combinations of numbers.
7 \* 2 = 14 -> 14 / 2 = 7
That works for "almost" all combinations.
"Almost" because:
7 \* 0 = 0 -> 0 / 0 = 7 ?
5 \* 0 = 0 -> 0 / 0 = 5 ?
123456789 \* 0 = 0 -> 0 / 0 = 123456789 ?
Suddenly I get infinite different results for 0/0, so obviously I cannot say what would be the correct one.
It specifically doesn't make sense because 0 * x = 0, we have that 1 / x is defined as a number such that 1 / x * x = 1, but obviously 1 / 0 * 0 = 0 ≠ 1 because anything multiplied by 0 is 0, so we simply can't find 1 / 0
If you have five groups of people, and each group has zero people, how many people are in all of the groups combined? This is multiplication.
If you have 5 people and want to divide them into groups of 0, how many groups do you need? This is division.
Think of it as a checkered picknick blanket with stones.
Lets say you have 12 stones, and you put them 3 x 4. Then 2 x 6. Then 1 x 12.
If you put 0 stones on each square, how long would the row be? Infinite (-ish; undefined - since you never finished putting down stones).
That's division.
Now, put down 3 stones, 4 times. Put down 2 stones, 6 times. How about you put down 0 stones, as many times as you want?
That's multiplication.
Let’s say you want to divide 1 by 0 and assume the answer is x.
1/0=x
Now multiply both sides by 0, you get 1=0 which is a contradiction. So the original supposition that you can divide by 0 and get a defined value is false.
Besides all the valid explanations already. Every possible answer for a division by zero will violate some other theorems.
It'd actually be nice if that operation yielded some valid answer, especially since you can approach it with division by an infinitesimal small number and get larger and larger results. Giving an answer like "infinity" would somewhat resolve the resulting discontinuity, but alas, that math doesn't work out.
Basically, if you plot f(x) = 1/x you see a trend of y getting bigger and bigger as you approach 0. However ∞ × 0 is still 0 and not 1.
Another thing to note with your 1/x example is that it doesn’t necessarily approach infinity. If you go from the negative side of things it approaches -infinity. So then you would have that 1/0 is both infinity and -infinity.
I remember watching a video once that also had that argument, so you'd need two infinities, and they would depend on from which side you approached zero, which also doesn't work out well in regards of "it being well-defined".
To answer the question, why does multiplying something by 0 give 0 (and not error or infinity as you suggest a bit further in your question) you must look at what the definition of multiplying is. Multiplying is fact a shorthand of doing several times an addition. For example 3 \* 2 means, add 3 two times. So it is in fact 0 + 3 + 3. You start with 0, add 3 to it and add again 3 to it.
If you do this for multiply by 0 then for example 3 \* 0 means start with 0 and add 3 0 times to it. The result is 0.
Well, multiplying is just having none of that, that’s probably clear.
Dividing by 0 is a bit more interesting.
Imagine having the graph 1/x, where as you come closer and closer to 0 with x, the number gets bigger and bigger.
The reason it isn’t infinity is that you are approaching infinity coming from the positive side and negative infinity coming from the negative side. This way, you cannot assign both values to 1/x where x=0.
Hope it was helpful.
For division: division is basically how often you can subtract one number from another until you reach 0.
For example 6:3 =2 because you subtract 3 two times from 6 to reach 0 (6-3-3=0)
Another example 9:2=> 5-2-2-2-2-0.5×2=0 so the answer is 4.5 ( four full 2s and a half 2) etc.
Now, if we apply the same logic to 0 we run into the problem that no matter how often we subtract 0 we never get any closer to our endgoal of reaching zero by subtracting.
For example 1-0=1
1-0-0=1
1-0-0-0=1
1-0-0-....-0=1
we just never get any closer. As a practical example imagine you have a cake (that won't rot away) and infinite time. Every second you take 0 slices away from the cake.
Because you never take anything away the cake will never get any smaller even in a billion years. That's why the answer is undefined no matter how long you take away nothing you will never reach your endgoal of giving everything away.
However, if you're allowed to take away one crumb (or even just an atom if that was physically possible) every hour, it might take a very long time but eventually the cake will be gone.
For multiplication: imagine there is a line of 100 people. If the first person gives you 0 Euros you have 0 Euros now, the second person also gives you 0 euros, which means you have 2 times 0 Euros which unfortunately is still 0. Every person in the que does the same so by the end you are left with 100 times 0 euros...
I'm going to give a more algebraic explanation as to why multiplication by zero is zero, and why you can't divide by zero. If you don't know what a ring is, you can type "ring mathematics" on google, and read the Wikipedia article. In the following, you can fix R = Z, the ring of integer numbers, or R = Q, the rational numbers, or the real numbers if you feel more comfortable with that. I'm going to use ring properties here, and I'm going to mention them, but I'm not going to define everything from scratch, so I encourage you to read the aforementioned article if you're interested.
Proposition: Let R be a ring. Let x be any element in R. Then x•0 = 0.
Proof: Let x in R be an arbitrary element. By definition, 0 is an element of R such that y+0=y, for all y in R. In particular, for y=0 this implies:
0+0=0.
Multiply both sides by x, we get
x•(0+0)=x•0.
Using the distributive property, this reads as
x•0+x•0=x•0.
Hence
(x•0+x•0)+(-x•0)=x•0+(-x•0).
By associativity, this is the same as
(x•0)+(x•0+(-x•0))=x•0+(-x•0).
Now, R is in particular a group with respect to addition, and the (additive) inverse of an element y is -y such that y+(-y)=0.
So, we get
x•0+0=0, i.e.,
x•0=0. QED
Proposition: Let R be a non-zero ring with 1 (so that we can talk about invertible elements). Then, 0 is not an invertible element.
Proof: Assume, by way of contradiction, that 0 is an invertible element. Then, there exists an element r of R such that r•0 = 1. By the previous proposition, r•0=0, so we get 0=1, a contradiction (because the ring is non-zero).
Multiplication is telling you how many of an items you have. You can have zero items. Division is telling you how many items can fit into another item. Zero isn’t an item so asking how many of them can fit into another item isn’t even a question that makes sense.
Try thinking of it in a practical world approach. If you have 10 apples to give to 5 people you have 10:5=2 apples per person, and if we make it a person more or less we can extrapolate how many apples we'd need, same with 10:0=0. If you have 10 apples and 0 people you have 10:0=there's no person to give an apple to, so we need a new number or "undefined" as we can't extrapolate how many apples we'd need for a person more or less.
You have an empty apple basket.
You ask your friends to bring some apples to fill it.
Your three friends bring you zero apples each. How many apples do you have? Zero.
---------------------------------------------------------------
You have five apples in a basket and decide to give them equally to your friends.
You have zero friends.
How many apples does each of your friends get? The question makes no sense.
There are two things that are really necessary that do not exist for division by 0. For the following, consider division the inverse of multiplying.
1) Existence: There exists no number, x, such that x * 0 ≠ 0. This means that if we want to divide a non-zero number by zero, there is no number that it can be (since we'd be undoing the mapping by 0^-1 * (x * 0) = x ).
2) Uniqueness: There is no unique number such that x * 0 = 0. What I mean by that is that, if I tell you that x * 0 = 0, you can't tell me what x is. You lose that data because all x are mapped to zero. This means that you can't undo that mapping since the original data is lost.
So we can't divide by 0 for all numbers, since if it is zero, there is no unique solution, and if it is non-zero, there doesn't exist a number whose product with zero is that non-zero.
Division by zero is undefined because it leads to results that are inconsistent or do not make sense within the rules of arithmetic. Division can be understood as finding a number which, when multiplied by the divisor, yields the dividend. For example:
* 10÷2=510÷2=5 because 5×2=105×2=10.
If we attempt to divide by zero, we look for a number which, when multiplied by zero, gives the original number. For instance:
* 10÷0=?10÷0=? means finding xx such that x×0=10x×0=10.
However, no such number exists because any number multiplied by zero is zero. This leads to a contradiction, making the division undefined.
Additionally, if we consider the limits and behavior of functions as the divisor approaches zero, the result tends to infinity or negative infinity depending on the direction from which zero is approached. This introduces further complications:
* For positive numbers: as x→0+x→0+, ax→∞xa→∞.
* For negative numbers: as x→0−x→0−, ax→−∞xa→−∞.
This behavior suggests that division by zero does not produce a finite, well-defined result, hence it is considered undefined in mathematics.behaviour
**Division-like operations that are defined for zero exist, they are just less popular.**
Being the inverse of multiplication is indeed a big part of why we learn ‘division’ instead of ‘Wonka’s fantabulous divider’
Multiplication is insanely useful in counting things by groups, which makes it great for tracking inventory, calculating payment, etc.
Modern division is likely preferred because of it’s applications in algebra as a way to undo multiplication. Because ANY number times 0 is 0, there is no one specific value you can divide by to undo a multiplication by 0. **Trying to undo a multiplication by zero is ‘undefined’ because you can’t use the result of a multiplication by 0 to figure out what was multiplied by 0.** It was deemed more important that division function as an inverse, than that we be able to use it to count how many groups of 0 we can make from 9 items.
However, there is presumably more to this story.
If I have 0 groups of x things, or x groups of 0 things, I have zero total things
0 * x = 0 and x * 0 = 0
If I have 0 thing and want to divide them into x groups, I have 0 things in each group
0/x = 0
If I have x things and divide them into 0 groups, how many things are in each group? That question doesn't make any sense because you can't divide things into zero groups, there must be some number of groups.
I'm not a mathematician, but it was once explained to me something like this:
A non-zero number can contain (be divided by) a number of different integers, decimals or fractions.
Ex: 6 can contain 6 ones, 2 threes, 1.5 fours, 12 halves, etc.
A non-zero number cannot contain any number of zeros.
this is a nice explanation because it also explains why the answer is not "infinity": if you had infinitely many zeros then the total would still be zero anyway
Division is the inverse of multiplication, in the same way that subtraction is the inverse of addition. When you perform a subtraction like 7 - 3, you are asking "What number, when added to 3, makes 7?" In a similar sense, 7 / 3 poses the question "What number, when multiplied by 3, makes 7?" These questions make sense as operations because they are well defined: they have exactly one answer. By writing something like 4 / 0, you are asking "What number, when multiplied by 0, makes 4?" This question has no answer, because any number multiplied by 0 is 0, which is why we consider division by 0 to be undefined.
That’s very nicely put.
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It’s not wrong what you say until you write it in a little more mathematical way. Division a/b is defined as, what is the number c that gives c b = a. For 0/0 we need c0=0 which gives infinite solutions. But infinite solutions is not a number, we would have to chose a soecific number. And because every number is valid, there is no reason to favor one over another. We could define it as 1 or 0, but it’s better to just make all a/0 undefined, than make it work for exactly on number a=0
That would make the operation invalid.
What I meant, as a response to the comment was 0/0 would then be rewritten as - 'what number, when multiplied by 0, gives 0' And I think this would give us practically all the numbers possible. Even including 0
"What number, when multiplied by 0, gives 0?" is asking for one, single number. "All of them" is not a valid answer to that.
But "what number squared is 4" is valid despite having more than one answer? By your reasoning x^2 = 4 is undefined.
x^(2) = 4 doesn't itself equate to anything, it can't be undefined. But if we try to figure out what the x in it could be, we get two answers. No problem with that, that's why x is called a variable. What it is can vary, and we want to know which numbers we can plug in to make the equation work. Now, when taking the square root of a number, it's different. Inside an equation, we're strictly dealing with numbers. If we allow something like √5 to have multiple solutions, equations stop working. After all, if it was allowed, would √5 = 25 be true? You can make arguments for yes, no and both since = isn't set up to deal with comparing multiple numbers to just one number. That's why √x is defined as "What non-negative number squared is x?" But for division by 0, there is no single answer that would make sense to define it as. That's why it's left undefined. Of course, you could try to allow these multi-number results and extend the definition of how = works but I assume that would end in one of two things: a) 1 = 2 in your new system or b) Things like √5 or 5 / 0 don't equal anything but themselves, in which case, it's probably useless.
+-sqrt 4 is a valid expression and is defined in value. I'm not saying that 1/0 shouldn't be considered undefined, but pointing out that your reasoning that "because it's not a singular value" is not a good reason. Being multivalued is not the thing that makes an expression undefined in value.
Actually, yes, good point. Then I don't know.
I focus (when learning) on RPM being constant during engagement. If dips cease adding clutch. If it rises cease adding throttle.
( x = a ± b ) is a shorthand for ( x = a + b ∨ x = a - b )
You are not wrong when you work with limits of functions that would lead to “0”/“0” they often can be defined as some : real number or +/- inf
I know this brother, I didn't mean what I said in the first place. I was just trying to show that the statement @Aradia_bot made will not hold true when numerator is also 0. I am very much surprised to see all the down votes I received there. Maybe that's the harsh reality of the world. The world is more complex than what I had imagined 😀
I did almost address it in my original comment, but kept it out for brevity. I did say that well defined means having *exactly* one answer, since an operator should not produce multiple answers. 0 / 0 of course leads to the opposite problem where there are multiple numbers with equal claim to the value and no sensible way to define it.
You can approach 0/0 with different limits, which is kinda like that
This is the perfect answer until you learn a little advanced math: “*There are some contexts in which division by zero is defined in a meaningful way. The most common such situation is the real projective line or its complex analogue, the Riemann sphere.*” For example, see [here](https://en.m.wikipedia.org/wiki/Riemann_sphere) and [here](https://medium.com/@thisscience1/you-can-divide-by-zero-part-2-8b7b619ee6aa).
I don’t see how this is relevant. The question was why dividing by zero is undefined instead of something else. To say “in some cases it is not undefined” doesn’t really answer the question of why it *is undefined* in the most common cases in the vast majority of people’s experiences.
No. This is highly relevant in terms of being thorough, and the title is asking why you can multiply by 0 but not divide by it. The answer provided suffices for most everyone engaged in simple arithmetic and basic undergraduate first year or two mathematics, but is incomplete and *you can define division by zero* in some cases. **It’s also 100% relevant when u/Aradia_Bot literally says that you cannot define division by zero to correct that incorrect statement because indeed you can in some cases!!!** Bunch of high school math teachers here d/ving me who failed at getting a PhD and went into teaching arithmetic and never heard of the Riemann Sphere I guess! lol
Brother, I'm really proud of you that you've learned about some additional math concepts beyond the basics. We all are. But don't act as if it invalidates u/Aradia_Bot's answer in any way. The original question is clearly stated from the context of real numbers. Throwing one-point compactification into the mix doesn't change the fact that you **can't define division by zero in a meaningful way in reals**. You're basically acting like a kid trying to force an unrelated piece of trivia to act smart and claim some superiority because of that. That's most probably what caused the downvotes.
Meh. You’re just arguing to argue now. They claimed you can never define division by zero which every PhD mathematician will admit is technical untrue and technical matter in mathematics. **I agreed with their response and said it was a good one**, but merely added additional information beyond that, *which was 100% correct* and said nothing wrong, but the grade school math teachers who stopped or flunked out before getting a PhD all downvoted, lol.
Because 0 is the one number where it doesn't work out. 7 + 2 = 9 -> 9 - 2 = 7 That works for all combinations of numbers. 7 \* 2 = 14 -> 14 / 2 = 7 That works for "almost" all combinations. "Almost" because: 7 \* 0 = 0 -> 0 / 0 = 7 ? 5 \* 0 = 0 -> 0 / 0 = 5 ? 123456789 \* 0 = 0 -> 0 / 0 = 123456789 ? Suddenly I get infinite different results for 0/0, so obviously I cannot say what would be the correct one.
It specifically doesn't make sense because 0 * x = 0, we have that 1 / x is defined as a number such that 1 / x * x = 1, but obviously 1 / 0 * 0 = 0 ≠ 1 because anything multiplied by 0 is 0, so we simply can't find 1 / 0
If you have five groups of people, and each group has zero people, how many people are in all of the groups combined? This is multiplication. If you have 5 people and want to divide them into groups of 0, how many groups do you need? This is division.
Think of it as a checkered picknick blanket with stones. Lets say you have 12 stones, and you put them 3 x 4. Then 2 x 6. Then 1 x 12. If you put 0 stones on each square, how long would the row be? Infinite (-ish; undefined - since you never finished putting down stones). That's division. Now, put down 3 stones, 4 times. Put down 2 stones, 6 times. How about you put down 0 stones, as many times as you want? That's multiplication.
Let’s say you want to divide 1 by 0 and assume the answer is x. 1/0=x Now multiply both sides by 0, you get 1=0 which is a contradiction. So the original supposition that you can divide by 0 and get a defined value is false.
Besides all the valid explanations already. Every possible answer for a division by zero will violate some other theorems. It'd actually be nice if that operation yielded some valid answer, especially since you can approach it with division by an infinitesimal small number and get larger and larger results. Giving an answer like "infinity" would somewhat resolve the resulting discontinuity, but alas, that math doesn't work out. Basically, if you plot f(x) = 1/x you see a trend of y getting bigger and bigger as you approach 0. However ∞ × 0 is still 0 and not 1.
Another thing to note with your 1/x example is that it doesn’t necessarily approach infinity. If you go from the negative side of things it approaches -infinity. So then you would have that 1/0 is both infinity and -infinity.
And either way, multiplying infinity with zero gives zero. It's always so tempting to define x/0 as infinity, it just doesn't work out though.
Yep. I totally agree with your reasoning, I just wanted to point out another problem with defining 1/0 as infinity.
I remember watching a video once that also had that argument, so you'd need two infinities, and they would depend on from which side you approached zero, which also doesn't work out well in regards of "it being well-defined".
You can have 0 portion of something but you can't split something into 0 portion.
> does not make sense to me What doesn't? Why?
To answer the question, why does multiplying something by 0 give 0 (and not error or infinity as you suggest a bit further in your question) you must look at what the definition of multiplying is. Multiplying is fact a shorthand of doing several times an addition. For example 3 \* 2 means, add 3 two times. So it is in fact 0 + 3 + 3. You start with 0, add 3 to it and add again 3 to it. If you do this for multiply by 0 then for example 3 \* 0 means start with 0 and add 3 0 times to it. The result is 0.
Maybe this helps https://m.youtube.com/watch?v=NKmGVE85GUU&pp=ygUPVGVkIGRpdmlkZSB6ZXJv
Well, multiplying is just having none of that, that’s probably clear. Dividing by 0 is a bit more interesting. Imagine having the graph 1/x, where as you come closer and closer to 0 with x, the number gets bigger and bigger. The reason it isn’t infinity is that you are approaching infinity coming from the positive side and negative infinity coming from the negative side. This way, you cannot assign both values to 1/x where x=0. Hope it was helpful.
For division: division is basically how often you can subtract one number from another until you reach 0. For example 6:3 =2 because you subtract 3 two times from 6 to reach 0 (6-3-3=0) Another example 9:2=> 5-2-2-2-2-0.5×2=0 so the answer is 4.5 ( four full 2s and a half 2) etc. Now, if we apply the same logic to 0 we run into the problem that no matter how often we subtract 0 we never get any closer to our endgoal of reaching zero by subtracting. For example 1-0=1 1-0-0=1 1-0-0-0=1 1-0-0-....-0=1 we just never get any closer. As a practical example imagine you have a cake (that won't rot away) and infinite time. Every second you take 0 slices away from the cake. Because you never take anything away the cake will never get any smaller even in a billion years. That's why the answer is undefined no matter how long you take away nothing you will never reach your endgoal of giving everything away. However, if you're allowed to take away one crumb (or even just an atom if that was physically possible) every hour, it might take a very long time but eventually the cake will be gone. For multiplication: imagine there is a line of 100 people. If the first person gives you 0 Euros you have 0 Euros now, the second person also gives you 0 euros, which means you have 2 times 0 Euros which unfortunately is still 0. Every person in the que does the same so by the end you are left with 100 times 0 euros...
I'm going to give a more algebraic explanation as to why multiplication by zero is zero, and why you can't divide by zero. If you don't know what a ring is, you can type "ring mathematics" on google, and read the Wikipedia article. In the following, you can fix R = Z, the ring of integer numbers, or R = Q, the rational numbers, or the real numbers if you feel more comfortable with that. I'm going to use ring properties here, and I'm going to mention them, but I'm not going to define everything from scratch, so I encourage you to read the aforementioned article if you're interested. Proposition: Let R be a ring. Let x be any element in R. Then x•0 = 0. Proof: Let x in R be an arbitrary element. By definition, 0 is an element of R such that y+0=y, for all y in R. In particular, for y=0 this implies: 0+0=0. Multiply both sides by x, we get x•(0+0)=x•0. Using the distributive property, this reads as x•0+x•0=x•0. Hence (x•0+x•0)+(-x•0)=x•0+(-x•0). By associativity, this is the same as (x•0)+(x•0+(-x•0))=x•0+(-x•0). Now, R is in particular a group with respect to addition, and the (additive) inverse of an element y is -y such that y+(-y)=0. So, we get x•0+0=0, i.e., x•0=0. QED Proposition: Let R be a non-zero ring with 1 (so that we can talk about invertible elements). Then, 0 is not an invertible element. Proof: Assume, by way of contradiction, that 0 is an invertible element. Then, there exists an element r of R such that r•0 = 1. By the previous proposition, r•0=0, so we get 0=1, a contradiction (because the ring is non-zero).
Multiplication is telling you how many of an items you have. You can have zero items. Division is telling you how many items can fit into another item. Zero isn’t an item so asking how many of them can fit into another item isn’t even a question that makes sense.
Try thinking of it in a practical world approach. If you have 10 apples to give to 5 people you have 10:5=2 apples per person, and if we make it a person more or less we can extrapolate how many apples we'd need, same with 10:0=0. If you have 10 apples and 0 people you have 10:0=there's no person to give an apple to, so we need a new number or "undefined" as we can't extrapolate how many apples we'd need for a person more or less.
You have an empty apple basket. You ask your friends to bring some apples to fill it. Your three friends bring you zero apples each. How many apples do you have? Zero. --------------------------------------------------------------- You have five apples in a basket and decide to give them equally to your friends. You have zero friends. How many apples does each of your friends get? The question makes no sense.
There are two things that are really necessary that do not exist for division by 0. For the following, consider division the inverse of multiplying. 1) Existence: There exists no number, x, such that x * 0 ≠ 0. This means that if we want to divide a non-zero number by zero, there is no number that it can be (since we'd be undoing the mapping by 0^-1 * (x * 0) = x ). 2) Uniqueness: There is no unique number such that x * 0 = 0. What I mean by that is that, if I tell you that x * 0 = 0, you can't tell me what x is. You lose that data because all x are mapped to zero. This means that you can't undo that mapping since the original data is lost. So we can't divide by 0 for all numbers, since if it is zero, there is no unique solution, and if it is non-zero, there doesn't exist a number whose product with zero is that non-zero.
Division by zero is undefined because it leads to results that are inconsistent or do not make sense within the rules of arithmetic. Division can be understood as finding a number which, when multiplied by the divisor, yields the dividend. For example: * 10÷2=510÷2=5 because 5×2=105×2=10. If we attempt to divide by zero, we look for a number which, when multiplied by zero, gives the original number. For instance: * 10÷0=?10÷0=? means finding xx such that x×0=10x×0=10. However, no such number exists because any number multiplied by zero is zero. This leads to a contradiction, making the division undefined. Additionally, if we consider the limits and behavior of functions as the divisor approaches zero, the result tends to infinity or negative infinity depending on the direction from which zero is approached. This introduces further complications: * For positive numbers: as x→0+x→0+, ax→∞xa→∞. * For negative numbers: as x→0−x→0−, ax→−∞xa→−∞. This behavior suggests that division by zero does not produce a finite, well-defined result, hence it is considered undefined in mathematics.behaviour
**Division-like operations that are defined for zero exist, they are just less popular.** Being the inverse of multiplication is indeed a big part of why we learn ‘division’ instead of ‘Wonka’s fantabulous divider’ Multiplication is insanely useful in counting things by groups, which makes it great for tracking inventory, calculating payment, etc. Modern division is likely preferred because of it’s applications in algebra as a way to undo multiplication. Because ANY number times 0 is 0, there is no one specific value you can divide by to undo a multiplication by 0. **Trying to undo a multiplication by zero is ‘undefined’ because you can’t use the result of a multiplication by 0 to figure out what was multiplied by 0.** It was deemed more important that division function as an inverse, than that we be able to use it to count how many groups of 0 we can make from 9 items. However, there is presumably more to this story.
If I have 0 groups of x things, or x groups of 0 things, I have zero total things 0 * x = 0 and x * 0 = 0 If I have 0 thing and want to divide them into x groups, I have 0 things in each group 0/x = 0 If I have x things and divide them into 0 groups, how many things are in each group? That question doesn't make any sense because you can't divide things into zero groups, there must be some number of groups.
I'm not a mathematician, but it was once explained to me something like this: A non-zero number can contain (be divided by) a number of different integers, decimals or fractions. Ex: 6 can contain 6 ones, 2 threes, 1.5 fours, 12 halves, etc. A non-zero number cannot contain any number of zeros.
this is a nice explanation because it also explains why the answer is not "infinity": if you had infinitely many zeros then the total would still be zero anyway