Absolute value doubts

Question

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?

igotthis · Answer

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) then it becomes |a| = |b| right? 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!

resolehtmai1 · Answer

thanx for the reply igotthis, but idintgetthis.

first of all: ofcourse i applied +/- condition on a as well, coz according to your basics quote 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|?

Devon · Answer

Nevermind, totally misread OP

amostofi1999 · Answer

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.

resolehtmai1 · Answer

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.

amostofi1999 · Answer

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.

Absolute value doubts

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