# real numbers: which is true? I. m2⎯⎯⎯⎯⎯√ II. ||−m|| III. −||m||

#### jamie bean

##### New member
real numbers: which is true? I. m2⎯⎯⎯⎯⎯√ II. ||−m|| III. −||m||

Which of the following expressions, if any, are equal for all real numbers m ?
I. [FONT=STIXMathJax_Main-italic]m[FONT=STIXMathJax_Main]2[/FONT][FONT=STIXMathJax_Symbols]⎯[/FONT][FONT=STIXMathJax_Symbols]⎯[/FONT][FONT=STIXMathJax_Symbols]⎯[/FONT][FONT=STIXMathJax_Symbols]⎯[/FONT][FONT=STIXMathJax_Symbols]⎯[/FONT][FONT=STIXMathJax_Main]√[/FONT]
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II. [FONT=STIXMathJax_Main]|[FONT=STIXMathJax_Main]|[/FONT][FONT=STIXMathJax_Main]−[/FONT][FONT=STIXMathJax_Main-italic]m[/FONT][FONT=STIXMathJax_Main]|[/FONT][FONT=STIXMathJax_Main]|[/FONT]
[/FONT]

III. [FONT=STIXMathJax_Main]−[FONT=STIXMathJax_Main]|[/FONT][FONT=STIXMathJax_Main]|[/FONT][FONT=STIXMathJax_Main-italic]m[/FONT][FONT=STIXMathJax_Main]|[/FONT][FONT=STIXMathJax_Main]|[/FONT]
[/FONT]

1. I and II only
2. I and III only
3. II and III only
4. I, II, and III
5. None of the expressions are equivalent.

#### tkhunny

##### Moderator
Staff member
Which of the following expressions, if any, are equal for all real numbers m ?
I. [FONT=STIXMathJax_Main-italic]m[FONT=STIXMathJax_Main]2[/FONT][FONT=STIXMathJax_Symbols]⎯[/FONT][FONT=STIXMathJax_Symbols]⎯[/FONT][FONT=STIXMathJax_Symbols]⎯[/FONT][FONT=STIXMathJax_Symbols]⎯[/FONT][FONT=STIXMathJax_Symbols]⎯[/FONT][FONT=STIXMathJax_Main]√[/FONT]
[/FONT]

II. [FONT=STIXMathJax_Main]|[FONT=STIXMathJax_Main]|[/FONT][FONT=STIXMathJax_Main]−[/FONT][FONT=STIXMathJax_Main-italic]m[/FONT][FONT=STIXMathJax_Main]|[/FONT][FONT=STIXMathJax_Main]|[/FONT]
[/FONT]

III. [FONT=STIXMathJax_Main]−[FONT=STIXMathJax_Main]|[/FONT][FONT=STIXMathJax_Main]|[/FONT][FONT=STIXMathJax_Main-italic]m[/FONT][FONT=STIXMathJax_Main]|[/FONT][FONT=STIXMathJax_Main]|[/FONT]
[/FONT]

1. I and II only
2. I and III only
3. II and III only
4. I, II, and III
5. None of the expressions are equivalent.
I'm guessing that's supposed to be $$\displaystyle \sqrt{m^{2}}$$?

I Positive value, Magnitude of m.
II Positive value, Magnitude of m.
III Negative value.

You're not thinking $$\displaystyle \sqrt{m^{2}} = \pm m$$ , are you? That would be incorrect. You tell me why.

#### jamie bean

##### New member
I'm guessing that's supposed to be $$\displaystyle \sqrt{m^{2}}$$?

I Positive value, Magnitude of m.
II Positive value, Magnitude of m.
III Negative value.

You're not thinking $$\displaystyle \sqrt{m^{2}} = \pm m$$ , are you? That would be incorrect. You tell me why.

What is magnitude of m?
i understand why I=+m and -m
but i dont understand why is ii equal for all real numbers of m?

#### jamie bean

##### New member
i understand i=+m and -m
but i dont understand why l-ml is equal to all real numbers of m???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
isnt l-ml=+m?

#### HallsofIvy

##### Elite Member
Do you know the basic definitions of |x| and $$\displaystyle \sqrt{x}$$? |x| is defined as: x if $$\displaystyle x\ge 0$$, -x if x< 0. So |x| is never negative. $$\displaystyle \sqrt{x}$$ is defined as "the positive number m such that $$\displaystyle m^2= x$$. So $$\displaystyle \sqrt{x}$$ is never negative.

#### Dr.Peterson

##### Elite Member
Which of the following expressions, if any, are equal for all real numbers m ?
I. [FONT=STIXMathJax_Main-italic]m[FONT=STIXMathJax_Main]2[/FONT][FONT=STIXMathJax_Symbols]⎯[/FONT][FONT=STIXMathJax_Symbols]⎯[/FONT][FONT=STIXMathJax_Symbols]⎯[/FONT][FONT=STIXMathJax_Symbols]⎯[/FONT][FONT=STIXMathJax_Symbols]⎯[/FONT][FONT=STIXMathJax_Main]√[/FONT]
[/FONT]

II. [FONT=STIXMathJax_Main]|[FONT=STIXMathJax_Main]|[/FONT][FONT=STIXMathJax_Main]−[/FONT][FONT=STIXMathJax_Main-italic]m[/FONT][FONT=STIXMathJax_Main]|[/FONT][FONT=STIXMathJax_Main]|[/FONT]
[/FONT]

III. [FONT=STIXMathJax_Main]−[FONT=STIXMathJax_Main]|[/FONT][FONT=STIXMathJax_Main]|[/FONT][FONT=STIXMathJax_Main-italic]m[/FONT][FONT=STIXMathJax_Main]|[/FONT][FONT=STIXMathJax_Main]|[/FONT]
[/FONT]

1. I and II only
2. I and III only
3. II and III only
4. I, II, and III
5. None of the expressions are equivalent.
To help you understand, try a couple examples. Suppose that m = 5; what are the values of I, II, and III? Then repeat if m = -5.

What do you find?

#### jamie bean

##### New member
To help you understand, try a couple examples. Suppose that m = 5; what are the values of I, II, and III? Then repeat if m = -5.

What do you find?
when m=5,

i=+5

ii=+5

iii=-5

when m= -5

i= -5

ii=+5

iii=-5

is this correct? im still confused TT sorry for my dumbness.
TO clarify my understandings, all real numbers = numbers other than complex numbers and imaginary numbers, right?
That means +ve and -ve numbers are included as all real numbers, isnt it?
then why arent all 3 options correct?REally thank you all for your help TT

#### JeffM

##### Elite Member
when m=5,

i=+5

ii=+5

iii=-5

when m= -5

i= -5

ii=+5

iii=-5

is this correct? im still confused TT sorry for my dumbness.
TO clarify my understandings, all real numbers = numbers other than complex numbers and imaginary numbers, right?
That means +ve and -ve numbers are included as all real numbers, isnt it?
then why arent all 3 options correct?REally thank you all for your help TT
One problem is that you do not use function notation.

$$\displaystyle f(m) = \sqrt{m^2},\ g(m) = |-\ m|,\ \text { and } h(m) = -\ |m| \text { for any } m \in \mathbb R.$$

$$\displaystyle m = 0 \implies f(m) = g(m) = h(m) = 0.$$

Do you see why?

$$\displaystyle m \ne 0 \implies h(m) < 0.$$

Do you see why?

$$\displaystyle m < 0 \implies f(m) = -\ m > 0 \text { and } g(m) = -\ m > 0 \implies f(m) = g(m) > 0.$$

Do you why?

$$\displaystyle m > 0 \implies f(m) = m > 0 \text { and } g(m) = m > 0 \implies f(m) = g(m) > 0.$$

Do you see why?

$$\displaystyle \therefore m \ne 0 \implies f(m) = g(m) > 0 > h(m).$$

$$\displaystyle \text {So, to sum up, } f(0) = g(0) = h(0) \text { and } f(m) = g(m) \ne h(m) \text { if } m \ne 0.$$

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#### Dr.Peterson

##### Elite Member
when m=5,

i=+5

ii=+5

iii=-5

when m= -5

i= -5

ii=+5

iii=-5

is this correct? im still confused TT sorry for my dumbness.
TO clarify my understandings, all real numbers = numbers other than complex numbers and imaginary numbers, right?
That means +ve and -ve numbers are included as all real numbers, isnt it?
then why arent all 3 options correct?REally thank you all for your help TT
Real numbers are numbers on the number line, positive, negative, and zero.

You were given 5 options, not 3. Do you mean, why aren't all three expressions equal?

But if your answers above are correct, none of the three expressions is always equal to any of the others (that is, equal for all real numbers, which would imply they are equal for both 5 and -5).

On the other hand, you got one of your values above wrong. Expression I is $$\displaystyle \sqrt{x^2}$$; what is $$\displaystyle \sqrt{(-5)^2}$$ ? Think carefully; don't jump to conclusions, but do the actual calculation.