Explain why if the absolute value of a number is always nonnegative, | a | can equal -a.My textbooks says the answer is: when a is a negative number and the negative of a negative number is...

Explain why if the absolute value of a number is always nonnegative, | a | can equal -a.

My textbooks says the answer is: when a is a negative number and the negative of a negative number is positive.

I don't really understand this. Are they trying to say that something like | 3 | = -3? Because that's definately  not true because 3 does not equal -3.

Or, are they trying to say that a is a negative number and the negation of a negative number is positive, but is negative when in variables/symbols?

 

 

Asked on by willneeded

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sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to remember what absolute value represents, hence you should know by now that the absolute value expresses the distance the number lies on x axis with respect to origin.

Hence, since the aboslute value is a distance, it needs to be positive.

You need to remember the definition of absolute value such that: 

`|a| = a,agt=0 `

`|a| = -a,alt0 `

Hence the result the textbook provides is trying to say that if the number a is negative then the absolute value is the negative of negative number, thus a positive number.

embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

I believe your confusion lies in the meaning of `-a` . This confusion is understandable since the symbol - represents so many things.

The key is to realize that `-a` is not negative a, it is the opposite of a. -3 is the opposite of 3, etc... The quadratic formula should say that if `ax^2+bx+c=0` then x equals the opposite of b plus/minus etc...

Thus ```|a|` is a if a>0, and the opposite of a if a<0. Or |a|=a if a>0, |a|=-a if a<0. Then |-3|=-(-3)=3 (The opposite of -3; -(-3) )

** This comes up again when you take square roots (or any even root); `sqrt(a^2)=|a|` since `sqrt((-3)^2)=3` . **

Wiggin42's profile pic

Wiggin42 | Student, Undergraduate | (Level 2) Valedictorian

Posted on

Absolute value refers to the distance to the origin (in this case, zero). 

If you take a look at the number line, you'll see that both 3 and -3 are both 3 spaces away from zero. 

That is why we the absolute value of any number; negative or otherwise; will always be positive. Distance is positive. 

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