Prove that |-a| = |a|

Question

I'm struggling to understand the basic proof for the following theorem:

$\lvert -a\rvert = \lvert a \rvert$

The provided solution goes on to describe the following:

$(i)$ For $a \ge 0 $, since $-a \le 0$

        $\lvert -a \rvert = -(-a)=a, \lvert a \rvert = a$

My question: Why does $\lvert -a \rvert = -(-a)$ hold true?

Answer 1

Provided we are working the a subring $A$, of the reals (i.e. $\mathbb{Z, Q, R}$), then the absolute value function is defined for $a \in A$ as:

\begin{equation} |a|= \begin{cases} -a, & \text{if}\ a<0\\ 0, & \text{if}\ a=0 \\ a, & \text{if}\ a>0 \end{cases} \end{equation}

So to answer your question, in the case $a \geq 0$, $|-a|=-(-a)$ comes straight from the definition, as $-a \leq 0$ (obviously noting $0=-0$).

Answer 2

The definition of $|\cdot|$ is: $$|x| := \begin{cases}x & x \ge 0\\-x & x<0\end{cases}$$

So, for $a \ge 0$, since $-a\le 0$, we have $|-a| = -(-a)$, by the definition of $|\cdot|$.

Answer 3

The definition of $|a|$ is:

$|a| := \begin{cases} a \quad a \geq 0\\ -a \quad a\leq0 \end{cases}$

So, suppose $a \geq 0$

Then, $|a| = a$

$|-a| = -(-a) = a$. Therefore, $|-a| = |a|$

Now, suppose $a \leq 0$

Then, $|a| = -a$

$|-a| = -a$

Hence, $|a| = |-a|$

So, for all $a \in \mathbb{R}$, we find:

$$|a| = |-a|$$

Answer 4

Let's take -a≥0 then |-a|=-a ........(1)

        0<a then |a |=a  ........(2)

by (1) and (2) |-a|=|a |

Let's take -a<0 then |-a|=-(-a)=a .........(3)

        0≤a then |a |=a ...............(4)

by (3) and (4) |-a|=|a |

conclusion= |-a|=|a |

NOTE:The absolute value function always gives the positive value of some number.That is why the negative number -a gives the positive number -(-a) as the absolute value.