Qiaochu_Yuan already answered your question, but because he was pretty technical with his answer, I thought I should try to simplify the point here a bit. The problem with division by zero is that division is essentially defined through multiplication and existence of certain inverse elements. It’s an axiom in itself in group theory that there are inverse elements, that is, for each a, there is x such that ax = 1. Our notation for x here would be 1/a, and it’s easy to see why a 1/a = 1. Division is defined by these inverse elements: a/b is calculated by a * (1/b), where (1/b) is the inverse of b.
But, if you have both multiplication and addition, there is one interesting thing. If we assume addition is the group operation for all numbers(and we use “0” to signify additive neutral element you get from adding together an element and its additive inverse, that is, “a + (-a) = 0″), and we want multiplication to work the way we like it to work(so that a(x + y) = (ax) + (a*y), that is, distributivity hold, something interesting happens.
Now, neutral element 0 is such that x + 0 = x, this is by definition of neutral element. Now watch the magic happen:
0x
= (0 + 0)x = 0x + 0x
So 0x = 0x + 0x.
We subtract 0x from both sides, leaving us with 0x = 0.
Doesn’t matter what you are multiplying 0 with, you always end up with zero. So, assuming 1 and 0 are not the same number(in zero ring, that’s the case, also, 0 = 1 is the only number in the entire zero ring), you can’t get a number such that 0*x = 1. Lacking inverse elements, there’s no obvious way to define what it would mean to divide by zero. There are special situations where there is a natural way to interpret what it means to divide by zero, in which cases, go for it. However, it’s separate from the division defined for other numbers.
And, if you end up dividing by zero because you somewhere assumed that there actually was such a number x that 0*x = 1, well, that’s just your own clumsiness.
Also, you can prove 1=2 if you multiply both sides by zero. 1 = 2. Proof: 10 = 20 ⇒ 0 = 0. Division and multiplication work in opposite directions, multiplication gets you from not equals to equals, division gets you from equals to not equals.
Excellent explanation, thank you. I’ve been telling everyone I know about your resolution to my worry. I believe in math again.
Maybe you can solve my similarly dumb worry about ethics: If the best life is the life of ethical action (insofar as we do or ought to prefer to do the ethically right thing over any other comforts or pleasures), and if ethical action consists at least largely in providing and preserving the goods of life for our fellow human beings, then if someone inhabited the limit case of the best possible life (by permanently providing immortality, freedom, and happiness for all human beings), wouldn’t they at the same time cut everyone else off from the best kind of life?
Ethical action is defined by situations. The best life in the scenario where we don’t have immortality freedom and happiness is to try to bring them about, but the best life in the scenario where we already have them is something different.
Good! That would solve the problem, if true. Do you have a ready argument for this thesis (I mean “but the best life in the scenario where we already have them is something different.”)?
“If true” is a tough thing here because I’m not a moral realist. I can argue by analogy for the best moral life in different scenarios being a different life but I don’t have a deductive proof of anything.
By analogy: the best ethical life in 1850 is probably not identical to the best ethical life in 1950 or in 2050, simply because people have different capacities and there exist different problems in the world. This means the theoretical most ethical life is actually divorced from the real most ethical life, because no one in 1850 could’ve given humanity all those things and working toward would’ve taken away ethical effort from eg, abolishing slavery. Ethics under uncertainty means that more than one person can be living the subjectively ethically perfect life even if only one of them will achieve what their goal is because no one knows who that is ahead of time.
Qiaochu_Yuan already answered your question, but because he was pretty technical with his answer, I thought I should try to simplify the point here a bit. The problem with division by zero is that division is essentially defined through multiplication and existence of certain inverse elements. It’s an axiom in itself in group theory that there are inverse elements, that is, for each a, there is x such that ax = 1. Our notation for x here would be 1/a, and it’s easy to see why a 1/a = 1. Division is defined by these inverse elements: a/b is calculated by a * (1/b), where (1/b) is the inverse of b.
But, if you have both multiplication and addition, there is one interesting thing. If we assume addition is the group operation for all numbers(and we use “0” to signify additive neutral element you get from adding together an element and its additive inverse, that is, “a + (-a) = 0″), and we want multiplication to work the way we like it to work(so that a(x + y) = (ax) + (a*y), that is, distributivity hold, something interesting happens.
Now, neutral element 0 is such that x + 0 = x, this is by definition of neutral element. Now watch the magic happen: 0x = (0 + 0)x
= 0x + 0x So 0x = 0x + 0x.
We subtract 0x from both sides, leaving us with 0x = 0.
Doesn’t matter what you are multiplying 0 with, you always end up with zero. So, assuming 1 and 0 are not the same number(in zero ring, that’s the case, also, 0 = 1 is the only number in the entire zero ring), you can’t get a number such that 0*x = 1. Lacking inverse elements, there’s no obvious way to define what it would mean to divide by zero. There are special situations where there is a natural way to interpret what it means to divide by zero, in which cases, go for it. However, it’s separate from the division defined for other numbers.
And, if you end up dividing by zero because you somewhere assumed that there actually was such a number x that 0*x = 1, well, that’s just your own clumsiness.
Also, you can prove 1=2 if you multiply both sides by zero. 1 = 2. Proof: 10 = 20 ⇒ 0 = 0. Division and multiplication work in opposite directions, multiplication gets you from not equals to equals, division gets you from equals to not equals.
Excellent explanation, thank you. I’ve been telling everyone I know about your resolution to my worry. I believe in math again.
Maybe you can solve my similarly dumb worry about ethics: If the best life is the life of ethical action (insofar as we do or ought to prefer to do the ethically right thing over any other comforts or pleasures), and if ethical action consists at least largely in providing and preserving the goods of life for our fellow human beings, then if someone inhabited the limit case of the best possible life (by permanently providing immortality, freedom, and happiness for all human beings), wouldn’t they at the same time cut everyone else off from the best kind of life?
Ethical action is defined by situations. The best life in the scenario where we don’t have immortality freedom and happiness is to try to bring them about, but the best life in the scenario where we already have them is something different.
Good! That would solve the problem, if true. Do you have a ready argument for this thesis (I mean “but the best life in the scenario where we already have them is something different.”)?
“If true” is a tough thing here because I’m not a moral realist. I can argue by analogy for the best moral life in different scenarios being a different life but I don’t have a deductive proof of anything.
By analogy: the best ethical life in 1850 is probably not identical to the best ethical life in 1950 or in 2050, simply because people have different capacities and there exist different problems in the world. This means the theoretical most ethical life is actually divorced from the real most ethical life, because no one in 1850 could’ve given humanity all those things and working toward would’ve taken away ethical effort from eg, abolishing slavery. Ethics under uncertainty means that more than one person can be living the subjectively ethically perfect life even if only one of them will achieve what their goal is because no one knows who that is ahead of time.
I think you mean x + 0 = x
yes. yes. i remember thinking “x + 0 =”. after that it gets a bit fuzzy.