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Shoot. I lost sight of the goal. Sorry. a·0 = 0 for all numbers a Thus it is true that –1·0 = 0 Replace 0 on left hand side with (–1 + 1) to get –1·(–1+1

Shoot. I lost sight of the goal. Sorry. a·0 = 0 for all numbers a Thus it is true that –1·0 = 0 Replace 0 on left hand side with (–1 + 1) to get –1·(–1+1) = 0 Now use the distributive property –1·(–1) + (–1)·1 = 0 Keeping in mind that 1·a = a for all numbers gives us –1(–1) + (–1) = 0 Thus –1(–1) is a number such that when added to –1 gives 0. Hence –1(–1) must be the opposite of –1 which is 1.

esrauch 14y ago
You are assuming that additive inverse is unique. Also you are assuming that -1(-1) is some value at all; you could write an extremely similar proof that leaves you with 0/0 = 1 and it would be faulty to conclude that 0/0 actually is 1.
- ColinWright 14y ago
  > You are assuming that additive inverse is unique Suppose that some number a has two additive inverses,x and y. a + x = 0 = x + a a + y = 0 = y + a Consider x + a + y = (x + a) + y = 0 + y = y x + a + y = x + (a + y) = x + 0 = x Thus x=y, and so the additive inverse is unique. > you are assuming that -1(-1) is some value at all; We can define it to have a value and then derive the properties of the value. We can show it to be consistent by creating an explicit model. Such things are relatively easy to do, but require a level of detail inappropriate for this context. > you could write an extremely similar proof that > leaves you with 0/0 = 1 and it would be faulty to > conclude that 0/0 actually is 1. Could you present such a proof for us?
- yequalsx 14y ago
  Additive inverses are unique in any ring. The subtle things I skipped over because of the technicalities involved is how do I know the distributive property holds for negative integers. Indeed, what is a negative integer? How does one get them from the natural numbers? Suffice it to say that this can all be defined in a consistent, precise way and everything works out.
  - gizmo686 14y ago
    Actually, we only think that it can be described in a consistent way. We have proven it consistent using set theory , but Godwell's theorems tell us that we cannot prove a system to be consistent without using something outside of said system (unless the system is inconsistent). This means that there is some level in our chain of proofs that cannot be proven consistent.
    - deleted 14y ago
      [deleted]

esrauch 14y ago

You are assuming that additive inverse is unique. Also you are assuming that -1(-1) is some value at all; you could write an extremely similar proof that leaves you with 0/0 = 1 and it would be faulty to conclude that 0/0 actually is 1.

ColinWright 14y ago
> You are assuming that additive inverse is unique Suppose that some number a has two additive inverses,x and y. a + x = 0 = x + a a + y = 0 = y + a Consider x + a + y = (x + a) + y = 0 + y = y x + a + y = x + (a + y) = x + 0 = x Thus x=y, and so the additive inverse is unique. > you are assuming that -1(-1) is some value at all; We can define it to have a value and then derive the properties of the value. We can show it to be consistent by creating an explicit model. Such things are relatively easy to do, but require a level of detail inappropriate for this context. > you could write an extremely similar proof that > leaves you with 0/0 = 1 and it would be faulty to > conclude that 0/0 actually is 1. Could you present such a proof for us?
yequalsx 14y ago
Additive inverses are unique in any ring. The subtle things I skipped over because of the technicalities involved is how do I know the distributive property holds for negative integers. Indeed, what is a negative integer? How does one get them from the natural numbers? Suffice it to say that this can all be defined in a consistent, precise way and everything works out.
- gizmo686 14y ago
  Actually, we only think that it can be described in a consistent way. We have proven it consistent using set theory , but Godwell's theorems tell us that we cannot prove a system to be consistent without using something outside of said system (unless the system is inconsistent). This means that there is some level in our chain of proofs that cannot be proven consistent.
  - deleted 14y ago
    [deleted]