Question 1:
By definition absolute value gives just no of units and does not indicate any direction neither positive nor negative then why in practice we use +ve direction like $\left|4\right|=+4$ it should be just 4 not +4
Question 2:
We know that $$\sqrt{{x}^2} = \pm x = \left|x\right| $$ Then why $\left|4\right|=+4$ and not $\left|4\right| = \pm 4$
Also $\sqrt{{4}^2} = +4$ and not $\sqrt{{4}^2} = \pm 4$
Is this all have do with involvement of a variable only (when variable involved use $\pm$ and when constants involved don't use $\pm$) ………… or there are some other rules !
Best Answer
The plus does nothing, it's $+4=4$. It's there to draw attention to the fact that the absolute value left a positive value unchanged -- they wanted to stress the difference between $-$ and $+$ case.
Also, $\sqrt{x^2}=\pm x$ is wrong. It's $\sqrt{x^2}=|x|$. The result of square root is always positive by definition. You can't get two values from a function. If you want to solve a quadratic equation, for instance, $y^2=x$, then you have to explicitly put $\pm$ in front of the square root to specify both solutions: $y=\pm \sqrt{x}$.