# Question related to Absolute Value

Hi everyone;
I've got this question concerning absolute value, although I understand the concept of absolute value, I can't quite understand why option B is bigger than option A.
I would really appreciate some explanation on the following question:
I am trying to post it in Latex, but in case that the comparison is not properly viewed, here is how it goes:
Option A
(everything inside absolute value symbol, the two vertical lines), sqrt2 - 1
Option B
(everything inside absolute value symbol, the two vertical lines)
1/2 - 1

I am getting very confused with something that appears to be simple and basic enough, do I need to review some other concepts? arithmetic and/or number properties? if so can someone pin point me to some web based explanations.
Thanks
And cheers to everyone

abs(sqrt(2) - 1) < abs(1/2 - 1)

Isn't it obvious?

If you can accept that sqrt(2) is a real between 1.41 and 1.42, you can see that

sqrt(2) - 1 < 1/2
sqrt(2) - 1 < abs(1/2 - 1)
abs(sqrt(2) - 1) < abs(1/2 - 1)

abs(sqrt(2) - 1) < abs(1/2 - 1)

Isn't it obvious?

If you can accept that sqrt(2) is a real between 1.41 and 1.42, you can see that

sqrt(2) - 1 < 1/2
sqrt(2) - 1 < abs(1/2 - 1)
abs(sqrt(2) - 1) < abs(1/2 - 1)

Excuse me....
First I signed to this forum to get help in a proper and educated way.
Secondly I ask that question because I have a doubt related to the concept.
I already know that 1.41 or 1.42 - 1 equals 0.41 or 0.42 I also know that 0.5 - 1 equals - 0.5
and I also know that the absolute value of a number is the value of a real number regarding the sign. that's why I ask that question, there's no need to be petulant, arrogant and/or try to belittle people here Tic-Tac...
Is been 20 years ago that I saw basic Algebra and I am trying to go back to school, so please have a little more of respect.
Plus your answer, doesn't help me at all from the conceptual point.

The absolute value is a function it's defined like this:

abs(x) = x, when x >= 0
abs(x) = -x, otherwise

You can calculate the right hand side in a straightforward way:

1/2 -1 = -1/2, so that means
abs(1/2 - 1) = -(-1/2) = 1/2

The right hand side is tricky because of the square root. But since we're dealing with inequalities and not interested in the exact number, we can abuse the fact that 1.41 < sqrt(2) < 1.42:

1.41 < sqrt(2) < 1.42
0.41 < sqrt(2) - 1 < 0.42

Since sqrt(2) - 1 is positive, it is equal to its own absolute value:

abs(sqrt(2) - 1) = sqrt(2) - 1, so
0.41 < abs(sqrt(2) - 1) < 0.42

And finally, since 0.42 < 1/2, we have

abs(sqrt(2) - 1) < 0.42 < 1/2, and
abs(sqrt(2) - 1) < abs(1/2 - 1)

What steps do you get stuck on? What conceptual difficulties do you have over the absolute value function?

tiny-tim
Homework Helper
I've got this question concerning absolute value, although I understand the concept of absolute value, I can't quite understand why option B is bigger than option A.

Option A
(everything inside absolute value symbol, the two vertical lines), sqrt2 - 1
Option B
(everything inside absolute value symbol, the two vertical lines)
1/2 - 1
I already know that 1.41 or 1.42 - 1 equals 0.41 or 0.42 I also know that 0.5 - 1 equals - 0.5
and I also know that the absolute value of a number is the value of a real number regarding the sign. that's why I ask that question

You seem to be asking about why |√2 - 1| < |1/2 - 1|.

You accept that it obviously is, but you're not telling us why the obvious answer is bothering you.

As Tac-Tics says, what steps do you get stuck on? What conceptual difficulties do you have over the absolute value function?

Mark44
Mentor
Hi everyone;
I've got this question concerning absolute value, although I understand the concept of absolute value, I can't quite understand why option B is bigger than option A.
I would really appreciate some explanation on the following question:
I am trying to post it in Latex, but in case that the comparison is not properly viewed, here is how it goes:
Option A
(everything inside absolute value symbol, the two vertical lines), sqrt2 - 1
Option B
(everything inside absolute value symbol, the two vertical lines)
1/2 - 1

I am getting very confused with something that appears to be simple and basic enough, do I need to review some other concepts? arithmetic and/or number properties? if so can someone pin point me to some web based explanations.
Thanks
And cheers to everyone

You seem to be asking whether |sqrt(2) - 1| < |1/2 - 1|. (Note that the | character is on the same key as the \ character.)
The left side is about 0.414. The right side is exactly 0.5.

The only thing the absolute value function does is return the magnitude (distance from 0) of the expression inside the vertical bars. There's really nothing very mysterious here.

HallsofIvy
He assumed you know, or could use a calculator to find that $\sqrt{2}$ is about 1.414 and so $\sqrt{2}- 1$ is about .414. Then $$\displaystyle |\sqrt{2}-1|$$ is about .414.