Why?

[niviene]

Alright, I realize that this is a super simple question compared to the rest of the stuff I have seen here, but I have no one else to ask...

 

Can anyone explain to me why the square root of 25 is only 5, and not plus or minus 5? I do not understand this, but I am told that this is true. I was always taught otherwise, but this fellow student was a math major, and insists that this is right, although can't show me any reason - she says this is just a law (of course, I asked her to prove it, and she drew a graph, which she said the function of the square root of 25 (as shown in your calculator) only shows positive y values, and I argued that limiting Y is ignoring the whole answer and that calculators only graph "functions"... ). Is it just a law that I don't know about? I assume I am wrong, because I didn't nearly take as much math as she did, but I thought this was something I understood well.... sigh....

[Rincewind]

I would be asking this math major what, in their opinion, is the square of -5, given that they are claiming that it isn't 25.

[niviene]

I would be asking this math major what, in their opinion, is the square of -5, given that they are claiming that it isn't 25.

 

She says that the square of -5 is also 25... but she kept writing that 5 != -5, I don't understand what she meant by that.

 

Her explanation was that if the problem said "Evaluate: Square root of 25" it is not the same as "Solve for x: x^2=25"

 

In my opinion, that's the same thing, unless the problem said "the function of (square root of 25)" which would allow restricting answers... but she insisted that the anytime you have a square root of a number, the positive answer is the only answer, unless you ask for + or - in front of the square root sign (because she argues this is what is required in the calculator, too)... which leads to my previous argument...

[UncleAl]

Every number has n nth roots by de Moivre's theorem. The square root of a positive number is generally taken as |[sqrt(n)]|. However, there are more than n nth roots if you allow non-commutative algebras wherein ab=-ba. This is a big deal, both in math and in physics

 

http://math.ucr.edu/home/baez/week82.html

[Rincewind]

She says that the square of -5 is also 25...

Then the square root of 25 must be -5 (as well as 5 of course).

 

... but she kept writing that 5 != -5, I don't understand what she meant by that...

As far as I'm aware, 5! means 5 factorial (or factorial 5), which is 5 × 4 × 3 × 2 × 1, which equals 120. Well, that's what it means here in Australia, anyway. :hihi:

 

Her explanation was that if the problem said "Evaluate: Square root of 25" it is not the same as "Solve for x: x^2=25"

Is she, by any chance using a graphing calculator to make these assertions? Your OP seemed to imply that. Maybe she needs a better calculator.

 

... but she insisted that the anytime you have a square root of a number, the positive answer is the only answer, unless you ask for + or - in front of the square root sign (because she argues this is what is required in the calculator, too)... which leads to my previous argument...

The limitations of her calculator, and the requirement to specify in it a general case for a machine that assumes a specific case, does not constitute a mathematical law. She needs to learn not to put her unwavering faith in a machine -- it's not always right just because it's a machine.

[Buffy]

As far as I'm aware, 5! means 5 factorial (or factorial 5), which is 5 × 4 × 3 × 2 × 1, which equals 120. Well, that's what it means here in Australia, anyway. :hihi:

To be perfectly clear:

     5! != -5
and
    5 != -5
because
    != = <>

Or, since you're from Australia:

     S¡ ¡= -S
and
    S ¡= -S
because
    ¡= = <>

All of this is perfectly clear to anyone who programs in C/C++/Java/Javascript....

 

Cheers,

Buffy

[Rincewind]

...All of this is perfectly clear to anyone who programs in C/C++/Java/Javascript...

And therein lies my downfall (if that's not too mixed a metaphor :hihi: ).

So what does != mean in words?

[Buffy]

! means both arithmetical and logical "NOT", therefore "!=" (or "¡=" downunder) translates to "NOT EQUALS" which most mathematicians write as "<>"....

 

Lexically yours,

Buffy

[Rincewind]

Thanks for that, Buffy.

 

(or "¡=" downunder)

LOL -- but it's actually you guys that are down under in my FoR. :hihi:

[niviene]

! means both arithmetical and logical "NOT", therefore "!=" (or "¡=" downunder) translates to "NOT EQUALS" which most mathematicians write as "<>"....

 

..Which is your first clue that I have done more coding than math, and why I have this question at all, I suppose. Sorry, I assumed everyone here would just know that second nature. Silly assumption. I am going to look up some of this non-commutative stuff UA posted, because I don't understand that at all... If there are n nth roots, then are you saying that there are n possibilities for the combination of the roots? I have to think more about this... that's the only thing I can figure out from that. My brain hurts. I'll think and come back when I have exhausted myself.. shouldn't be long.

[C1ay]

See square root at MathWorld....

[niviene]

I thought about it on the way to the store, and I think my previous post is pretty dumb. I don't know why I thought nth root would mean n combinations... that makes absolutely no sense.

 

See square root at MathWorld....

 

Thanks. :hihi:

[Qfwfq]

She says that the square of -5 is also 25... but she kept writing that 5 != -5, I don't understand what she meant by that.

 

Her explanation was that if the problem said "Evaluate: Square root of 25" it is not the same as "Solve for x: x^2=25"

Confusing character, this math major is.

 

I think that what she means is that, although the actual definition of square root of a given, single number a is a solution of the equation x^2 = a, it's a different thing to say what is meant by the function square root of x. By definition, a function f(x) should not have more than one value (in the co-domain) for a given value of x in the domain. The usual thing is to define the function root of x as being the positive real value, for positive real arguments. There are other possibilities for the function too, of course.

 

Sez hoo, that a given number doesn't have n nth roots? They might be complex, but they'll be there, somewhere. Actually they will be at equal angles around a circumference in the complex plane.

[GAHD]

Maby they were simply trying to state that -5 isn't a "real" number. you't really hvae -5 rocks (though you can be missing 5 rocks).

[Qfwfq]

Can the number of rocks be the root of 2, or the root of 3, Gahd? Can it be 317/54 or something like that?

 

These are called real numbers, by most mathematicians.

[GAHD]

yes it can, but again, you can't have a negative number of rocks, negative numbers are technically imaginary. Shure math works out that negatives cancel out to a positive upon multiplication, but I'd challenge you to put any negative number of rocks in a box, thus negative numbers are imaginary.

 

 

Here's one for you; what's the square root of -25?

[Qfwfq]

yes it can, but again, you can't have a negative number of rocks, negative numbers are technically imaginary. Shure math works out that negatives cancel out to a positive upon multiplication, but I'd challenge you to put any negative number of rocks in a box, thus negative numbers are imaginary.

Are you quite sure of all this?

 

Here's one for you; what's the square root of -25?

5 i