Absolute Value Definition

Status
Not open for further replies.
mathdad

mathdad

Full Member
Joined
Apr 24, 2015
Messages
737
Please, explain the definition of absolute value as defined in the book.

"The absolute value of a real number a, denoted by the symbol |a|, is defined by |a| = a is a is >or= 0 and |a| = -a if a < 0."

Can you provide an example when we get -a?
 
mathdad said:
Please, explain the definition of absolute value as defined in the book.

"The absolute value of a real number a, denoted by the symbol |a|, is defined by |a| = a is a is >or= 0 and |a| = -a if a < 0."

Can you provide an example when we get -a?
Click to expand...
|-2| = 2
 
mathdad said:
Please, explain the definition of absolute value as defined in the book.

"The absolute value of a real number a, denoted by the symbol |a|, is defined by |a| = a is a is >or= 0 and |a| = -a if a < 0."

Can you provide an example when we get -a?
Click to expand...
mathdad said:
No. Khan, provide an example in terms of - a.
Click to expand...
Did you think through it?

It seems that you are posting super-elementary thoughts??!!
 
Correct me please if I'm wrong but absolute value is the distance between the number and 0 on the number line as far as I know. So for example, the distance between 4 and 0 is the same as -4 and 0 on the number line. They are both 4 away from 0.

So because of that,
|-4|=4 and |4|=4.
 
Last edited:
mathdad said:
So, give me a super-elementary answer, Mr. Khan.
Click to expand...
The problem here is that you have been given a definition in terms of a variable. The only pertinent examples will be numeric ones (otherwise you are just substituting one variable for another).

DEFINITION

[MATH]a < 0 \iff |a| = -\ a \text { and } a \ge 0 \iff |a| = a.[/MATH]
Think of the absolute value as how far away from zero a number is. So - 3 is 3 units distant away from zero.

Here is an alternate but equivalent definition that requires you to remember that the square root function is always non-negative.

[MATH]|a| \equiv \sqrt{a^2}.[/MATH]
I like the alternative definition because now it is obvious that

[MATH]|-\ 3| = \sqrt{(-\ 3)^2} = \sqrt{9} = 3.[/MATH]
But the two definitions mean exactly the same thing.
 
Subhotosh Khan said:
|-2| = 2
Click to expand...
mathdad said:
No. Khan, provide an example in terms of - a.
Click to expand...
Here is his example in terms of -a, as well as can be done, I think:

Suppose a = -2. Then

|a| = |-2| = 2, and​
-a = -(-2) = 2.​
So |a| = -a.​

If that isn't clear, perhaps you can elaborate on your own thoughts about the definition. It's very common for students at first to be confused about negative numbers vs. negatives of numbers, as in "If a is negative, then -a is positive."
 
Dr.Peterson said:
Here is his example in terms of -a, as well as can be done, I think:

Suppose a = -2. Then

|a| = |-2| = 2, and​
-a = -(-2) = 2.​
So |a| = -a.​

If that isn't clear, perhaps you can elaborate on your own thoughts about the definition. It's very common for students at first to be confused about negative numbers vs. negatives of numbers, as in "If a is negative, then -a is positive."
Click to expand...

More great notes.
 
Dr.Peterson said:
Here is his example in terms of -a, as well as can be done, I think:

Suppose a = -2. Then

|a| = |-2| = 2, and​
-a = -(-2) = 2.​
So |a| = -a.​

If that isn't clear, perhaps you can elaborate on your own thoughts about the definition. It's very common for students at first to be confused about negative numbers vs. negatives of numbers, as in "If a is negative, then -a is positive."
Click to expand...

What about -|-7|?

- | -7 | = -7
 
The negative of any non-negative is a non-positive. Major premise. All A is B.

The absolute value of any number is a non-negative. Minor premise. C is an A.

Therefore, the negative of any absolute value is a non-positive. Conclusion. Thus, C is B.

Basic syllogism.
 
mathdad said:
Please, explain the definition of absolute value as defined in the book.

"The absolute value of a real number a, denoted by the symbol |a|, is defined by |a| = a is a is >or= 0 and |a| = -a if a < 0."

Can you provide an example when we get -a?
Click to expand...
|-2| = - (-2)
There is your example. Just note that -(-2)=2
 
Dr.Peterson said:
What is the meaning of the "but"? Are you implying you see a contradiction with what had been said?
Click to expand...

The "but" should clearly indicate that there are questions in terms of absolute values that end up as negative answers..
 
Please elaborate. Do you have a specific question about it? Clarity trumps brevity.

Yes, you can end up with a negative value when an absolute value is used in an expression. That has nothing to do with the fact that the absolute value itself is never negative.

In "- |-2| = -( 2) = - 2", we are first taking an absolute value, yielding the positive number 2; then we take the negative of that. The result is not an absolute value, so it doesn't have to be positive.

The trouble here is that because you were so brief, I can't tell whether you were merely giving an illustration of this fact as an addition to the conversation, or making some kind of objection to what had been said. (In fact, I'm still not at all sure, because I can't tell whether you are wrongly using the word "question" to mean "expression", or referring to a question you still have.)
 
Status
Not open for further replies.
Top