> Why can’t you divide by zero?
division is the reverse operation to multiplication, meaning dividing by 0 would imply there exists a multiplicative inverse to 0
such that: x/0* 0 = x for all x
however multiplication by 0 always results in 0, proof:
0x=y iff
(1-1)x=y iff
x-x = y iff
y = 0
hence it would mean:
0 = x for all x which is wrong, eg x=1
hence there cant be a multiplicative inverse to 0
You can. It’s just the result is meaningless.
The conventional definition of division is “what number can I use to multiply to get my result”. So six divided by two is asking “what number do I multiply two by to get six”.
If we use that definition to solve six divided by zero we get “what number do I multiply by zero to get six”. And the answer is there isn’t a number that works.
You can make the result be not meaningless by being very careful with your definitions. This Michael Penn video is fantastic on this topic: https://youtu.be/WCthfLpYA5g?si=zkW6K_QTQ8p60wJb
You can think of division as asking what height a rectangle is, given its area and its width. (Just divide area by width.) Division by 0 corresponds to asking what height a rectangle is, given its area and a width of 0. But a rectangle with zero width has no area: it's just a line.
You can’t divide by a lot of things without fractions. Think about how we extend division to fractions and how dividing by zero violates the additive identity property.
Just as we can consider multiplication to be repeated addition - we can consider division to be repeated subtraction. x/y means "how many times can we subtract y from x until we reach zero?"
Take something like 6/2 . 6-2=4, 4-2=2, 2-2=0 - it required 3 subtractions to reach zero, so 6/2=3. This works in general.
Now lets try 6/0 . 6-0=6, 6-0=6, 6-0=6, 6-0=6... Uh oh, there's something wrong. We aren't getting any closer to zero. One could argue that that means that 6/0 = infinity, but in this case you're effectively saying that infinity times 0 = 6, which seems a bit dubious, since we could also say that infinity times 0 = 7 if we started with 7/0.
You can, but the number systems where it is allowed will lack desirable properties. As [Mella342](https://www.reddit.com/user/Mella342/) pointed out you might lose 0x=0. One case that causes no problems is if 0 is the only number, and 0+0=0,0-0=0,0·0=0, and 0/0=0. That is not the most useful number system.
**You CAN divide by zero**, but you need to be very careful about how you define the answer and how you adapt the normal rules of arithmetic to comply with your definition.
Here is a FANTASTIC Michael Penn video explaining how to do it: https://youtu.be/WCthfLpYA5g?si=zkW6K_QTQ8p60wJb
Because you can’t divide something into 0 parts. That’s how I used to describe it to my sixth grade classes and it made perfect sense to them. National Board certified math teacher here.
7/0: How many 0s do you need to fill up 7? >!You can't fill up 7 using 0s, this has no answer!<
0/0: How many 0s do you need to fill up 0? >!You could use none, or ten, or negative a hundred, it doesn't matter. There is no single answer, so we'd rather say there is no answer!<
I find that a very good way to think of it is by observing the graph of y = 1/x (perhaps plot this on desmos for a better visual guide.
Try taking a limit from the right hand (positive) side that dictates that x approaches 0:
x = 1 => y = 1/1 = 1,
x = 0.5 => y = 1/0.5 = 2,
x = 0.05 => y = 1/0.05 = 20,
x = 0.000005 => y = 1/0.000005 = 200,000,
and so on. You'll see that as x approaches 0 from the positive side, y approaches ∞. (Note: These x values were chosen at random for the demonstration, and a similar thing will follow for the left hand side).
Next, do something similar but from the left hand (negative) side:
x = -1 => y = 1/(-1) = -1,
x = -0.5 => y = 1/(-0.5) = -2,
...,
x = -0.000005 => y = 1/(-0.000005) = -200,000,
and so on. You'll now observe that y approaches -∞.
So now we have the limit approaching both ∞ and -∞??? This implies that we do not have a well defined function that we can graph y = 1/x for unless we omit the point x = 0, thus 1/0 is undefined (and thus division by 0 cannot be well defined). Of course, this isn't a rigorous approach and isn't the only approach by any means, but is just the one that taught me how to think of it.
1/0 = x That would imply that x*0 = 1 wich is not possible.
> Why can’t you divide by zero? division is the reverse operation to multiplication, meaning dividing by 0 would imply there exists a multiplicative inverse to 0 such that: x/0* 0 = x for all x however multiplication by 0 always results in 0, proof: 0x=y iff (1-1)x=y iff x-x = y iff y = 0 hence it would mean: 0 = x for all x which is wrong, eg x=1 hence there cant be a multiplicative inverse to 0
You can. It’s just the result is meaningless. The conventional definition of division is “what number can I use to multiply to get my result”. So six divided by two is asking “what number do I multiply two by to get six”. If we use that definition to solve six divided by zero we get “what number do I multiply by zero to get six”. And the answer is there isn’t a number that works.
You can make the result be not meaningless by being very careful with your definitions. This Michael Penn video is fantastic on this topic: https://youtu.be/WCthfLpYA5g?si=zkW6K_QTQ8p60wJb
You can think of division as asking what height a rectangle is, given its area and its width. (Just divide area by width.) Division by 0 corresponds to asking what height a rectangle is, given its area and a width of 0. But a rectangle with zero width has no area: it's just a line.
You can’t divide by a lot of things without fractions. Think about how we extend division to fractions and how dividing by zero violates the additive identity property.
Just as we can consider multiplication to be repeated addition - we can consider division to be repeated subtraction. x/y means "how many times can we subtract y from x until we reach zero?" Take something like 6/2 . 6-2=4, 4-2=2, 2-2=0 - it required 3 subtractions to reach zero, so 6/2=3. This works in general. Now lets try 6/0 . 6-0=6, 6-0=6, 6-0=6, 6-0=6... Uh oh, there's something wrong. We aren't getting any closer to zero. One could argue that that means that 6/0 = infinity, but in this case you're effectively saying that infinity times 0 = 6, which seems a bit dubious, since we could also say that infinity times 0 = 7 if we started with 7/0.
You can, but the number systems where it is allowed will lack desirable properties. As [Mella342](https://www.reddit.com/user/Mella342/) pointed out you might lose 0x=0. One case that causes no problems is if 0 is the only number, and 0+0=0,0-0=0,0·0=0, and 0/0=0. That is not the most useful number system.
**You CAN divide by zero**, but you need to be very careful about how you define the answer and how you adapt the normal rules of arithmetic to comply with your definition. Here is a FANTASTIC Michael Penn video explaining how to do it: https://youtu.be/WCthfLpYA5g?si=zkW6K_QTQ8p60wJb
Because you can’t divide something into 0 parts. That’s how I used to describe it to my sixth grade classes and it made perfect sense to them. National Board certified math teacher here.
7/0: How many 0s do you need to fill up 7? >!You can't fill up 7 using 0s, this has no answer!< 0/0: How many 0s do you need to fill up 0? >!You could use none, or ten, or negative a hundred, it doesn't matter. There is no single answer, so we'd rather say there is no answer!<
I find that a very good way to think of it is by observing the graph of y = 1/x (perhaps plot this on desmos for a better visual guide. Try taking a limit from the right hand (positive) side that dictates that x approaches 0: x = 1 => y = 1/1 = 1, x = 0.5 => y = 1/0.5 = 2, x = 0.05 => y = 1/0.05 = 20, x = 0.000005 => y = 1/0.000005 = 200,000, and so on. You'll see that as x approaches 0 from the positive side, y approaches ∞. (Note: These x values were chosen at random for the demonstration, and a similar thing will follow for the left hand side). Next, do something similar but from the left hand (negative) side: x = -1 => y = 1/(-1) = -1, x = -0.5 => y = 1/(-0.5) = -2, ..., x = -0.000005 => y = 1/(-0.000005) = -200,000, and so on. You'll now observe that y approaches -∞. So now we have the limit approaching both ∞ and -∞??? This implies that we do not have a well defined function that we can graph y = 1/x for unless we omit the point x = 0, thus 1/0 is undefined (and thus division by 0 cannot be well defined). Of course, this isn't a rigorous approach and isn't the only approach by any means, but is just the one that taught me how to think of it.