# Clarification of Divide by Zero [closed]

I'm confused by something that I think has caused an issue my entire life and all mathematicians will likely destroy me for. But divide by zero seems trivial in the regards that it should just be the number above.

We have so many rules for how math works, why can we not add one where if the product of the divisor is 0 it is removed as if it does not exist there. This would happen before any other factoring or cancellations would occur.

When we look at 0/x on a graph it is equal to the x-axis.

This would mean that the graph 1/x would still be this, but it would also include the y-axis. Obviously this doesn't break any concepts that already exist because the two lines on the graph don't touch already therefore a theoretical third line could exist.

This concept shouldn't even have a major impact in slowing computers realistically. As to allow divide by 0 it would just be an ignored state and removed from the calculation. We would then remove the ability for computers to crash when divided by zero because we would have an agreed upon way of moving forward.

A semi-viral TikTok from user @chunkyging has one of their friends demonstrating an absurd math solution to 2+2=5.

The gist of it is.

2+2 = 5

if 4 - 4 = 10 - 10

and 4-4 = (2^2 - 2^2)

and 10 - 10 = 5(2) - 5(2)

and (2^2 - 2^2) can be written (2 + 2)(2 - 2)

and 5(2) - 5(2) can be written 5(2-2)

then you can divide (2-2) from both sides to leave

2 + 2 = 5

Obviously this is bad math and not allowed because 0 cannot be used to divide. Technically this individual doesn't break other math rules.

If you try this in a computer it would crash. Our brains melt in a way. Because we say "No we can't". If we just make a rule - if you try to divide by zero it is just ignored and cannot be used to cancel from the equation - we would have a definitive rule that provides the same outcome as not doing it any way.

This just seems like the obvious answer. I know it is hated when people say that and somehow I might be missing a complex concept that creating this rule would break, but that is what I'm kind of asking for. I don't see anything that would cause an issue with allowing this concept to exist in our world.

Edit::

Some questions have been asked and I don't think those who are looking at this are understanding the differences.

• 0 as the divisor is not 1 ... it is nothing. if you don't divide 8 what are you left with? 8. If you don't divide 1 what are you left with... 1. While dividing those by 1 you are also left with 1 this still doesn't break the concepts that exist in other places.
• Someone says "division is the opposite of multiplication" which isn't exactly correct either. Multiplication is what happens when you take this group and add more to it equally multiple times. so a group of 2 multiplied 8 times is 16 total for that group. Division then is if you take a group of 16 and split it 8 times what is each group left with and the answer is 2. But if you don't split a group of 16 what are you left with 16.
• But then we say 16 times 0 is 0. Anything times 0 is 0. Which is still fine. This is the same as saying you have a group of 16 and you give all of them to a friend. You have 0, but that does not mean the 16 are destroyed.

We know matter cannot be added or destroyed. This concept is the same...

Further...

when saying a/b = c/d we assume for every a/b there exists some c/d that is equal. That is all we are saying.

So... 1/0 could equal 9999999999/10000000000 or in other math 1=1. We can easily see other scenarios where this would work where 2/0 would equal 6/3 or 12/6. The curve for this would be exponential. 3/0 = 18/6,15/5 etc...

If this rubs people the wrong way we are already OK with undefined numbers. So I don't care in the end, but it seems so strange we just write off this as common knowledge when we definitely seem to just not understand the reality.

• "if the product of the divisor is 0 it is removed as if it does not exist there" Does that mean to pretend the divisor is $1$? Commented Sep 5 at 3:38
• This question is similar to: Why not to extend the set of natural numbers to make it closed under division by zero?. If you believe it’s different, please edit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. Commented Sep 5 at 3:39
• Possibly relevant: Re-invention of the wheel. Commented Sep 5 at 4:22

The best way of verifying that is thinking of division as reverse multiplication. 8 $$\div$$ 4 is just asking "what number $$\times$$ 4 = 8?". So 8 $$\div$$ 0 is just asking "what number $$\times$$ 0 = 8?". There is no such number since anything times 0 is 0, so any nonzero number divided by 0 is undefined.
However, $$0 \div 0$$ is asking "what number times 0 equals 0"? Since any number times 0 is 0, all numbers are a correct answer, and we cannot "determine" a specific correct answer, so $$0 \div 0$$ is indeterminate.