# 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?

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### 3 Answers

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.**

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` . **

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.