Decoding the Mystery: The Truth Behind (-1)^2^(1/2) and Its Value

[Nikitin]

Some little detail has been giving me a head-ache for 1 of my problems.. And here is why.

Does ((-1)²)^1/2 equal 1 or -1? You decide...

 

[CompuChip]

((-1)²)^1/2 = (1)^1/2 = 1

Because of your bracketing, there is nothing to decide. It is just 1, end of story ;)

 

[humanino]

(-1)² = 1
(1)^1/2 = 1
The function "square root" is not uniquely defined and requires a convention. I did not decide, this is the usual convention, I find it natural however.

If you intend to continue in mathematics, it will be best that you keep this example in mind : it will most probably be one of the first examples in the Riemann surface chapter.
edit
I have never been as fast as CompuChip !

 

[Nikitin]

Isn't the better answer +-1??

Seriously, a problem like this thing ruined my entire evening.. I couldn't figure out what was wrong when I was simplifying a ridiculously long expression during my math homework...

 

[Nikitin]

Also why is the other way around (eliminating the exponents first) wrong?

and yeh, I'l continue studying math when I go engineering.. I don't think they will teach us (imaginary numbers and unwanted stuff like that) such stuff until the final year tho..

 

[humanino]

I can guarantee that there are integrals which you could calculate in a few minutes, by hand, using complex numbers (Cauchy residue theorem), and that you could never figure out, even using softwares, using only real numbers. In fact, I believe Feynman took precisely this bet, and lost. So the short form of the story is : please do not expect imaginary numbers to be "unwanted".

 

[Hurkyl]

Isn't the better answer +-1??

No. But if you asked the question

What are the square roots of 1​

the answer would be

+1 and -1​

The square root function simply isn't an inverse to the squaring function.

Seriously, a problem like this thing ruined my entire evening.. I couldn't figure out what was wrong when I was simplifying a ridiculously long expression during my math homework...

Incorrect simplification is a pervasive problem. People make this mistake with the trigonometric functions and their "inverses" a lot too.

One of the more notorious mistakes is something like:

Oh, I have the equation

x(3+x) = x(5-x)​

so I'll divide both sides by x

3+x = 5-x​

and so the answer is x = 1​

Can you spot the error?





Many (most?) real-life problems break up into cases, and you have to solve all of the different cases (or, at least, admit to solving only a particular case). Unfortunately, many people seem to have great difficulty accepting this fact, and habitually ignore all except for the one -- e.g. they often ignore things like the fact that each positive real number has a negative square root or that variable expressions could be zero.

 

[BobG]

You need to read the wording of the responses and be sure not to gloss over the word "function". Functions have a specific definition.

As Hurkyl mentioned, that can cause problems with such things as trigonometry problems (quadrant resolution issues) since you often need all the possible answers. (Your calculator or a spreadsheet is only going to give you the answer to the function, not all possible answers.)

The most fun example of the problem with functions (and imaginary numbers) is graphing something such as $$y= -2^x$$.

The answer is:
-2 when x=1
4 when x=2
-8 when x =3

When does the graph cross zero?

There's a reason the exponential functions, $$f(x) = n^x$$, have to have a positive base other than 1.

Hmm, for some reason, my edits to latex seem to exist in some other dimension than visible posts. They show up in preview, but not in the post.

 

[Hurkyl]

Try reloading the page, or right clicking on the image to reload it specifically (or to just view the image in its own page, and possibly refreshing).

 

[Nikitin]

yeh, obviously x=0 is an answer as well in your question. I am no newb (not a master tho, obviously :p). I know that stuff can have multiple answers, but this problem I brought is kind of different.

Bob, hurkly; I am still not understanding, why is the correct way from bottom-to-top and not from top-to-bottom? This is just some simple curiosity.. It just seems that the only answer is, though,: "it just is so, so accept that!"

And trust me hurkly, that thing I was simplifying (the 1st derivative of a function of the type f(x)=u^(0.5) + p^(0.5).. and it didn't even look simple in its base-form) was not a "simple expression", it was a beast from hell.
This entire (1^2)^(1/2) problem was too much for mathcad, so I couldn't check if I had done any mistakes or not. That caused me great problems and finally brought me to this forum :)

 

[BobG]

The mathematical concept of a function expresses the intuitive idea that one quantity (the argument of the function, also known as the input) completely determines another quantity (the value, or the output). A function assigns exactly one value to each input of a specified type.

So, some constraints have to be set since some equations can have two (or more) values that correspond to a given input.

 

[Nikitin]

So one can have different functions that have the same input? Like, -1 and 1 being an output in my question depending on how you read it?

 

[BobG]

So one can have different functions that have the same input? Like, -1 and 1 being an output in my question depending on how you read it?

No.

I assume you're asking if you could write two functions which happen to look identical to each other and say one equals -1 and the other 1?

Not unless you're redefining your constraints, in which case you're comparing apples in Washington to oranges in Finland. (And, yes, I know Washington is a state and Finland is a country - that's half the point.)

 

[Rasalhague]

So one can have different functions that have the same input? Like, -1 and 1 being an output in my question depending on how you read it?

If you mean, "Can a function have two different outputs for the same input?", then no. It's part of the definintion of a function that it has only one output for a given input. So $f(x) = \sqrt{x}$ defines a function $f$ which gives the positive square root of a real number $x$. But this function isn't the inverse of the function $g$ defined by $g(x) = x^2$. In fact, if $g$ is a function from the real numbers to the real numbers, it has no inverse function.

Something else to watch out for: sometimes the term "square root of x" refers to the positive square root, that is, $\sqrt{x} \equiv x^{1/2}$, while other times, the "square root of x" means "a real number, y, such that y² = x", where y can be positive or negative.

http://en.wikipedia.org/wiki/Square_root

 

[Rasalhague]

Some little detail has been giving me a head-ache for 1 of my problems.. And here is why.

Does ((-1)²)^1/2 equal 1 or -1? You decide...

Let f map the set of nonnegative real numbers to itself, such that f(x) = x^1/2, which means the positive square root of x. Let g map the set of real numbers to the set of nonnegative real numbers, such that g(x) = x². Then the composite function $f \circ g$ maps the set of real numbers to the set of nonnegative real numbers, such that

$$f \circ g \, (x) = (x^2)^{1/2} = \sqrt{x^2} = |x|.$$

The value of this function on x is called the "absolute value of x". An equivalent way of defining the absolute value of x is to say:

If x < 0, |x| = -x,
otherwise |x| = x.

The absolute value of 1 is 1, since 1 is not less than 0.

 

[Nikitin]

If you mean, "Can a function have two different outputs for the same input?", then no. It's part of the definintion of a function that it has only one output for a given input. So LaTeX Code: f(x) = \\sqrt{x} defines a function LaTeX Code: f which gives the positive square root of a real number LaTeX Code: x . But this function isn't the inverse of the function LaTeX Code: g defined by LaTeX Code: g(x) = x^2 . In fact, if LaTeX Code: g is a function from the real numbers to the real numbers, it has no inverse function.

Something else to watch out for: sometimes the term "square root of x" refers to the positive square root, that is, LaTeX Code: \\sqrt{x} \\equiv x^{1/2} , while other times, the "square root of x" means "a real number, y, such that y2 = x", where y can be positive or negative.

http://en.wikipedia.org/wiki/Square_root

I meant, are there functions that have a different way of processing input? I am basically wondering if the "+1" answer to my question is ALWAYS right, i.e. none consider anything else correct.

Like 1+1=2.

Sorry for my English...

Rasal, heh don't worry.. I obviously know stuff like that! I always provide the +- y answer unless it would be illogical (like in a practical assignment).

Heh, I am starting to feel slightly stupid after asking this question.. :P

Edit: Yeah, thinking of it as an absolute-value makes sense though this is different from things like sqrt X = +-y (where +-y is the correct answer, only +y is wrong).. I mean, you can get either 1 or -1 depending on how you read it..

 

[Mark44]

Assuming that x is a real number that is >= 0, $\sqrt{x}$ has a single answer.

For example, $\sqrt{4}$ = 2, not +/- 2.

It is true that 4 has two square roots, 2 and -2, but 2 is the principal square root, and that is what is intended by the notation $\sqrt{4}$.

 

[Rasalhague]

I meant, are there functions that have a different way of processing input?

No, there is no function which has more than one different way of processing a particular input. By definition, a function associates only one element each of its output set (codomain) with every element of its input set (domain). If a relation associates more than one output with a single input, then that relation is not a function.

I am basically wondering if the "+1" answer to my question is ALWAYS right, i.e. none consider anything else correct.

Like 1+1=2.

Sorry for my English...

Yes, ((-1)²)^1/2 is always equal to 1 (never -1), just as 1+1 is always equal 2. That's how the notation has been defined.

Rasal, heh don't worry.. I obviously know stuff like that! I always provide the +- y answer unless it would be illogical (like in a practical assignment).

Heh, I am starting to feel slightly stupid after asking this question.. :P

Edit: Yeah, thinking of it as an absolute-value makes sense though this is different from things like sqrt X = +-y (where +-y is the correct answer, only +y is wrong).. I mean, you can get either 1 or -1 depending on how you read it..

More stupid not to ask ;-)

Compare the questions:

(1) If x² = 4, what is x? (What is the square root of 4?)

(2) What is 4^1/2? (What is the positive square root of 4?)

The answer to the first question is 2 or -2. The answer to the second question is 2.

g(x) = y such that y² = x doesn't defined a function; it would have more than one output each for every input except x = 0.

f(x) = x^1/2 does define a function; it has only one output each for every input.

(As Mark44 mentioned, another name for the positive square root is "the principle square root".)

 

[Studiot]

Also why is the other way around (eliminating the exponents first) wrong?

Just to take up this point, since others have done the main question to death.

Perhaps you are thinking of differentiation rules for 'function of a function' which does indeed proceed from the outside to the inside.

 

[HallsofIvy]

A "relation" can assign two different outputs to the same input. The "relation" $x= y^2$ assigns both y= 1 and y= -1 to the same input, 1. But as "function", by definition, assigns only one value of y to each value of x. The "inverse" of $x= y^2$ (which, exactly because two values of x correspond to one value of y doesn't really have an true "inverse") is either $y= \sqrt{1}= 1$ or $y= -\sqrt{x}$. In order to make it a "function", we have to choose one or the other, the conventional choice is $\sqrt{y}$ that is the positive number whose square if y. You are welcome to change that if you like, but you have to say so!

 

[yuiop]

Some little detail has been giving me a head-ache for 1 of my problems.. And here is why.

Does ((-1)²)^1/2 equal 1 or -1? You decide...

I am no expert so this is just for discussion.

Let us say we we have the equation x² = (-1)² then there are two solutions:

x = + ((-1)²)^1/2 = +1 ... Soln 1

x = - ((-1)²)^1/2 = -1 ... Soln 2

Your question is in the form of solution 1 so the positive solution has already been chosen.

When we say 1^1/2 we are saying perform the function of raising 1 by the power of (1/2) which is subtly different to asking what are the possible roots of the equation x²=1.

 

[Studiot]

When we say 11/2 we are saying perform the function of raising 1 by the power of (1/2) which is subtly different to asking what are the possible roots of the equation x2=1.

Instead of function use operation.

 

[Nikitin]

well obviously a single function can't read one input in more than 1 ways.. Nevermind the entire function talk. What I was trying to ask, is if there -1 could be considered a correct answer.

Which it can't, so question solved, thanks :)

 

[Rasalhague]

Instead of function use operation.

What a cryptic thing to say.