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# Absolute value doubts

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Absolute value doubts [#permalink]  26 Sep 2013, 03:32
If its given that a=|b|,
then can I square both sides and say $$a^2 = b^2$$ ?
In this case, if I take square root of the above equation, then it becomes |a| = |b| right?
Why is this different from the original equation?
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Re: Absolute value doubts [#permalink]  26 Sep 2013, 04:06
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resolehtmai1 wrote:
If its given that a=|b|,
then can I square both sides and say $$a^2 = b^2$$ ?
In this case, if I take square root of the above equation, then it becomes |a| = |b| right?
Why is this different from the original equation?

That is a very good question!
Lets go back to basics: You know:
|x|>=0
|x|=sqrt(x^2)
|-x|=x

But what do these mean? Lets take an example:
In the example you gave, if a = |b|
To answer your question: then can I square both sides and say $$a^2 = b^2$$ ?
Yes, but you must remember the condition that this is only possible under 2 conditions:
a = |b| = b .....when b>0
a = |b| =-b ....when b<0
In this case, if I take square root of the above equation,
Right, but again with the condition that,
sqrt(b^2) = b....when b>0 (Example: b =5, sqrt(25) = 5)
sqrt (b^2) = -b....when b<0 (Example: b =-5, sqrt(25) = 5)

[b]then it becomes |a| = |b| right?
[/b] Why is this different from the original equation?[
Umm.. here you are putting the +/- condition on a as well. That changes the original equation.

Hope this helps and its not tooo confusing!
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Re: Absolute value doubts [#permalink]  27 Sep 2013, 00:15
thanx for the reply igotthis, but idintgetthis.
first of all:
igotthis wrote:
then it becomes |a| = |b| right?[/b] Why is this different from the original equation?[
Umm.. here you are putting the +/- condition on a as well. That changes the original equation.

ofcourse i applied +/- condition on a as well, coz according to your basics quote
igotthis wrote:
|x|=sqrt(x^2)

so $$a^2=b^2$$ becomes |a|=|b|.
but isn't something wrong with this? how can we derive |a|=|b| from just a=|b|?
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Re: Absolute value doubts [#permalink]  27 Sep 2013, 09:20
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Re: Absolute value doubts [#permalink]  27 Sep 2013, 14:26
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Quote:
but isn't something wrong with this? how can we derive |a|=|b| from just a=|b|?

there is nothing wrong. we can derive |a| = |b| from a = |b| i.e.
a = |b| => |a| = |b| (but not the other way around).

you just need to consider the range of numbers that the first equation implies for 'a'. in the first equation (a = |b|) the right side ( |b| ) is 0 or greater than 0, therefore the left side (a) is also 0 or greater, so "a" can't be a negative number
(a >= 0).

let me explain in a different way:

1) original equation:
a = |b| implies that:
I. 'b' can be anything
II. a is 0 or greater than 0 i.e. 'a' cant be negative (a>=0).
for example:
b= -0.6 and a is 0.6
b = 0 and a is 0
b = 10 and a is 10

2) the new equation:
|a| = |b| says that absolute value of 'a' is equal to absolute value of 'b'.
why is this ok? because from the first equation we already know that 'a' is not negative, and since absolute value of non-negative numbers are the same as the number, the new equation true also.

remember always pay attention to range of values a variable can take. for example in X = 1/Y you should beware that y can never be 0.
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Re: Absolute value doubts [#permalink]  27 Sep 2013, 20:15
amostofi1999 wrote:
Quote:
but isn't something wrong with this? how can we derive |a|=|b| from just a=|b|?

there is nothing wrong. we can derive |a| = |b| from a = |b| i.e.
a = |b| => |a| = |b| (but not the other way around).

you just need to consider the range of numbers that the first equation implies for 'a'. in the first equation (a = |b|) the right side ( |b| ) is 0 or greater than 0, therefore the left side (a) is also 0 or greater, so "a" can't be a negative number
(a >= 0).

let me explain in a different way:

1) original equation:
a = |b| implies that:
I. 'b' can be anything
II. a is 0 or greater than 0 i.e. 'a' cant be negative (a>=0).
for example:
b= -0.6 and a is 0.6
b = 0 and a is 0
b = 10 and a is 10

2) the new equation:
|a| = |b| says that absolute value of 'a' is equal to absolute value of 'b'.
why is this ok? because from the first equation we already know that 'a' is not negative, and since absolute value of non-negative numbers are the same as the number, the new equation true also.

remember always pay attention to range of values a variable can take. for example in X = 1/Y you should beware that y can never be 0.

Thanks amostofi1999. i get it now. After reading your explanation, i also figured out why the other way round (that is a=|b| derived from |a|=|b|) is not correct.
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Re: Absolute value doubts [#permalink]  27 Sep 2013, 22:40
Can anyone help me with fundamentals of absolute numbers??

if you have difficulty understanding, i have to recommendations for you regarding absolute value function. 1- try to really understand and learn the definitions deeply; dont just memorize a bunch of formulas 2- at first, learn the concept in your language. best source is probably official high school math or other books that are teaching the subject to first time learners.
Re: Absolute value doubts   [#permalink] 27 Sep 2013, 22:40
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