Question

i) If $x$ is a real number, then $|x|=\{\begin{array}{c}x,ifx\ge 0\\ -x,ifx<0\end{array}$
ii) $|x|\ge 0$ for all real numbers $x$

Answer 1

Modulus $\left|x\right|$ of a real number $x$ is the non-negative value of $x$ without regard to its sign .Namely,

$\left|x\right|=x$ for a positive $x$,

$\left|x\right|=-x$ for a negative $x$ (Here $(-x)$ is positive)

For any real number $x$, the absolute value or modulus of $x$ is denoted by $\left|x\right|$.

From an analytic geometry point of view,

The absolute value of a real number is that number's distance from zero along the real number line, and more generally the absolute value of the difference of two real numbers is the distance between them. That’s why absolute value of $x$ is thus always either positive or zero, but never negative, since $x<0$ implies $-x>0$ Since , Distance can’t be negative .

$|a-b|$ indicates distance from $a$ to $b$ along the real number line i.e. $|a-b|=a-b$

This is also defined for complex numbers as well as real numbers. If $z=x+iy$, then $\left|z\right|=|x+iy|=\sqrt{x^2+y^2}$. It is the distance of $z$ in the complex plane from the origin $O$.

For example :

$\left|5\right|$ indicates distance from $0$ to $5$ along the real number line i.e. $\left|5\right|=(5-0)=5$

$|-3|$ indicates distance from $0$ to $-3$ along the real number line i.e. $|-3|=\{0-(–3)\}=–(–3)=3$