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>.