Proving 0 = -0: Axioms & Solutions

In summary, the conversation discusses how to prove the statement 0 = -0. The use of the distributive property and axioms of multiplication by zero are suggested as possible approaches. The definition of -0 and the importance of working in a group are also mentioned.
  • #1
rsaad2
2
0

Homework Statement



prove : 0 = -0


Homework Equations





The Attempt at a Solution


 
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  • #2
What have you done so far?
 
  • #3
i do not know how to go about it so that is why i posted it here.
 
  • #4
If you multiplied both sides of the inequality by -0 and used the distributive property what would you arrive at ? Can you arrive at the same result by using -1 instead ?
 
  • #5
The crucial question is, what axioms are you allowed to use? The "distributive law" has been suggested but that assumes that you are working in the real numbers or at least a ring in which the distributive law is true. But "0= -0" only requires the "0" element and additive inverse- you should be able to prove this in any group. What is the definition of "-0"? Is it (-1)(0) or "the additive inverse of the multiplicative identity time the additive identity" or just "the additive inverse of the additive identity"?
 
  • #6
hint:

-0 = -1 * 0

what do you know as axioms about multiplication by zero? :)
 

FAQ: Proving 0 = -0: Axioms & Solutions

What does it mean for 0 to equal -0?

For a real number to equal another real number, they must have the same value. In this case, both 0 and -0 have a value of 0, so they are considered equal.

What are axioms and how do they relate to proving 0 = -0?

Axioms are self-evident truths that serve as the basis for a mathematical system. They are used to prove mathematical statements, such as 0 = -0, by following logical steps.

Can you prove 0 = -0 using axioms?

Yes, it is possible to prove 0 = -0 using axioms. One way to do so is by using the axiom of identity, which states that any number is equal to itself. Since 0 and -0 both have a value of 0, they are equal to themselves.

Why is it important to prove 0 = -0?

Proving mathematical statements, such as 0 = -0, is important because it helps to establish the validity of a mathematical system. It also allows us to better understand the properties and relationships of numbers.

Are there any other ways to prove 0 = -0 besides using axioms?

Yes, there are other ways to prove 0 = -0, such as using the definition of equality, which states that two numbers are equal if they have the same value and sign. Since 0 and -0 both have a value of 0 and a sign of positive, they are considered equal by definition.

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