# Integers involving absolute values

• Sep 9th 2008, 12:28 PM
hmwin
Integers involving absolute values
Assuming a is a non zero integer, which of the following statements is never true...
1. a > -a
2. a < -a
3. [a] = -a
4. [a] = - [a]

any explanation is welcome!
• Sep 9th 2008, 01:46 PM
hmwin
So, are you saying that all four answers are false? Could there be more than one false answer?
• Sep 11th 2008, 07:17 AM
bkarpuz
Quote:

Originally Posted by Moo
Hello,

In general, negative numbers are great counter examples.
For 1. let a be a negative integer (-1,-2,etc... you can choose).
For 2. let a be a positive integer.

False too...

Recall that $|x|=\left\{\begin{array}{ll} -x \quad \text{ if } x<0 \\ x \quad \text{ if } x \ge 0 \end{array} \right.$

Once again, you can tackle it with counterexamples.

Assuming a is a non zero integer, which of the following statements is never true...

1 is true with $a=1$ and 2 is true with $a=-1$, also 3 is true with $a=-1$, I think the answer is 4
Because if $a$ is non-zero then $|a|\neq0$ too.

Question. Does $[a]$ denotes the lowest or greatest integer function? If so it is not meaningful since $[a]=a$ holds ( $a$ is an integer).
But it would be nice to consider this question with real $a\neq0$.
In this case, the solution is 3.
• Sep 11th 2008, 09:09 AM
masters
Quote:

Originally Posted by hmwin
Assuming a is a non zero integer, which of the following statements is never true...
1. a > -a
2. a < -a
3. [a] = -a
4. [a] = - [a]

any explanation is welcome!

Judging from the title "Integers involving absolute value", I assume that [a] means "the absolute value of a".

So, in #4 [a] = -[a] ==> |a| = - |a|

This would never be true since the absolute value will always produce a positive result.
• Oct 2nd 2008, 06:36 AM
nioton
Non of the answers are true
if a is not 0 ,it can be positive or nagative ,
check it with one positive for example 3,
then check it with negative for example -3
if these to numbers are not correct for any of the answer,any non ziro integer will act the same
• Oct 19th 2008, 04:35 AM
HallsofIvy
Quote:

Originally Posted by nioton
Non of the answers are true
if a is not 0 ,it can be positive or nagative ,
check it with one positive for example 3,
then check it with negative for example -3
if these to numbers are not correct for any of the answer,any non ziro integer will act the same

The first, a> -a, is true whenever a is positive and so is not "never true".

The second, a< -a, is true whenever a is negative.

The third and fourth, |a|= -a and |a|= -|a| are true for a= 0.

NONE of the statments is "NEVER true".