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I just want to know if there is a mistake in my text book.

The proof from my textbook is as follows:

For any real number $a \in \mathbb{R} $ we have $-\lvert a \rvert \leq a \leq \lvert a \rvert$.

Proof. [...] When $a < 0$ then $a < 0 < \lvert a \rvert $, showing that $a \leq \lvert a \rvert$, while $-\lvert a \rvert = -(-a) = a$ [...]

I don't understand the last part regarding $-\lvert a \rvert = -(-a) = a $.

If $ a = -4 $, for example, would it not be $-\lvert -4 \rvert = -\lvert 4 \rvert = -4$?

lone student
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Curulian
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1 Answers1

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The example you gave does not contradict the book and the book is correct.

If $a<0$, then

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

This is correct by definition of absolute value. Because,

$$|a|=a, \thinspace a≥0\iff -|a|=-a$$

For $ a = -4 $, you wrote

$$-\lvert -4 \rvert = -\lvert 4 \rvert = -4$$

That is correct. But, note that here we don't need to write $|4|$ here. Because, $|-4|=4$ by definition of absolute value. Thus, we can write

$$-|-4|=-4$$

By definition of absolute value we can also write,

$$|a|≥a ~~\text{and}~~ |a|≥-a$$

This immediately implies, we have

$$-|a|≤a≤|a|.$$

This comes from,

$$\begin{align}&a≥0\\ \iff &-a≤0≤a\\ \iff &|a|=a≥a≥-a\end{align}$$

and

$$\begin{align}&a<0\\ \iff&-a>0>a\\ \iff& |a|=-a≥-a>a.\end{align} $$

lone student
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  • I'll update my answer if I know the reason for the downvote. – lone student Oct 05 '21 at 06:34
  • Not sure why the downvote, but minor possible improvements: (1) Maybe say "true" instead of "correct", since "correct" has a slight connotation of being an assertion about the proof (a meta notion) rather than of being an assertion about a formal mathematical statement within the proof. (2) "If $a = -4 $, then $-\lvert -4 \rvert = -\lvert 4 \rvert = -4$" Note that $-\lvert -4 \rvert = -\lvert 4 \rvert = -4$ is true regardless of the value of $a,$ so you probably don't want to write this as an "if ... then" statement. (continued) – Dave L. Renfro Oct 05 '21 at 08:14
  • Instead, you probably intended to say something like "For example, substituting $-4$ for $a$ in the previous equality gives ..." (3) You should probably avoid notation such as $a \geq \pm a$ in an explanation like this in which part of the OPs problem might be interpreting symbolic statements. (4) Overall, I would probably start with the definition of absolute value, noting that removal of absolute value bars involves two sign cases, and thus the subsequent analysis breaks down into two sign cases, $\ldots$ (I'll return several hours later $\ldots$ need to leave now.) – Dave L. Renfro Oct 05 '21 at 08:15
  • @DaveL.Renfro I added points. – lone student Oct 05 '21 at 10:27