Prove that |a|=0 if and only if a=0.

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Lesson Parent: 
Rules of Absolute Values
Problem: 

Prove that |a|=0 if and only if a=0.

Answer: 

|a|=0 if and only if a=0

SAMPLE SOLUTION

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Solution: 

Recall that when we say "if and only if" we have to prove two things. For this particular problem we have to prove the following

  • If |a|=0 then a=0.
  • If a=0 then |a|=0.

Recall that the absolute value is always greater than or equal to zero. We can see this from the definition <Sign in to see all the formulas>.

First to prove that if |a|=0 then a=0. Since <Sign in to see all the formulas> then <Sign in to see all the formulas>. Let's suppose <Sign in to see all the formulas>. Then <Sign in to see all the formulas>. We have just shown that <Sign in to see all the formulas>. By contraposition, <Sign in to see all the formulas>. But since <Sign in to see all the formulas>, then <Sign in to see all the formulas>.

Now let's prove that if a=0 then |a|=0. By definition <Sign in to see all the formulas>. Since 0=−0 then <Sign in to see all the formulas>.