3=1?

[Twilight Dirac 721]

I second (or should it be fourth?) Vital's explanation.  The dual value of the square root is how you managed to flip the sign.

Edited November 7, 2014 by Twilight Dirac

[Zekromic 1,054]

A teacher of mine in HS showed me something like that, and he said no one knows why is that possible

[CloudFyre 848]

 

which is:

 

    1/i   or   -1/i   or 1/-i    or    -1/-i

 

which is:

 

    i   or   -i

 

So the same as the first case.

Unless I'm missing something vital here, 1/i does not equal i. There is a difference.

 

1/√(-1) does not equal √(-1), in the same way that 1/√(9) does not equal √(9).

 

 

Edited November 7, 2014 by CloudFyre

[Admiral Regulus 2,769]

The second case is:

 

    √1 / √-1

 

which is:

 

    (1 or -1) / (i or -i)

 

which is:

 

    1/i   or   -1/i   or 1/-i    or    -1/-i

 

which is:

 

    i   or   -i

 

So the same as the first case.

 

The logic holds, as long as you designate the sqrt function to only return a positive or negative number. It's when you use a positive for one and a negative for the other that you have issues, because then you're using two different functions.

Edited November 7, 2014 by Regulus

[CloudFyre 848]

Are you sure about that?

 

I'd say that the operation of taking the square root has two answers, and I'd even make it more precise: that it has two unique answers, because there are sometimes actually infinitely many answers, modulo 2 pi, each of them producing the same square.

 

But square roots, if you mean the answers, which are single numbers, are... well... single numbers, so they don't have two answers – they're the answers themselves (each one of them).

Yep, I'm...positive. (Insert the "AWWWW YEEEEAAAAH" sunglasses here.)

 

To be precise: a square root function that gives ONLY a positive result is known as the principal root, while in practical cases, the square root is assumed to have 2 unique answers. (Your Oxford dictionary will give the principal root definition.)

 

I suppose you said it right: the operation has two answers. But that's exactly what it means: two answers. Until the problem is defined further, you have to assume that either one is valid (perhaps the event does indeed occur twice, in which both of them are acceptable).

 

And I'm not quite sure why you're talking about taking the mean of the answers. The mean is entirely different, and you wouldn't want to take an average when you need exact values.

 

The logic holds, as long as you designate the sqrt function to only return a positive or negative number.

Yes, however we weren't given this, so we can't assume it.

Edited November 7, 2014 by CloudFyre

[Admiral Regulus 2,769]

 

Yes, however we weren't given this, so we can't assume it.

 

Can't we, though?

 

Put sqrt(4) into your calculator. You'll get one answer: 2. Your calculator isn't going to tell you it's -2, because the convention we use is that the square root function returns the positive value.

 

In fact, if I put in sqrt(4) = -2, my nspire CX CAS actually returns "false." Before you say my calculator is being stupid, consider this:

 

If I input sqrt(4) = a, and ask it to solve for a, it returns a = 2. If I input 4 = a², then it returns a = -2 or a = 2. The square root function is always positive—this is the way it's defined. It's only when you solve for a number squared that you convert the answer to ± of the square root.

 

In short, sqrt(4) =/= -2. That is an incorrect expression, because sqrt(4) = 2, and -sqrt(4) = -2.

Edited November 7, 2014 by Regulus

[Twilight Dirac 721]

 

 

I suppose you said it right: the operation has two answers. But that's exactly what it means: two answers. Until the problem is defined further, you have to assume that either one is valid (perhaps the event does indeed occur twice, in which both of them are acceptable).

 

But the problem is further defined.  You started with +1 before using the square root expressions, so when you switch back and get +1 and -1 as the answer, you better darn well pick +1.   This problem is little more than a sleight of hand that tricks the reader into choosing the wrong root, thus giving a nonsense answer.

[CloudFyre 848]

But the problem is further defined.  You started with +1 before using the square root expressions, so when you switch back and get +1 and -1 as the answer, you better darn well pick +1.   This problem is little more than a sleight of hand that tricks the reader into choosing the wrong root, thus giving a nonsense answer.

I was stating that generally, but yes, I suppose that's true in this case.

 

 

Can't we, though?

 

Put sqrt(4) into your calculator. You'll get one answer: 2. Your calculator isn't going to tell you it's -2, because the convention we use is that the square root function returns the positive value.

 

In fact, if I put in sqrt(4) = -2, my nspire CX CAS actually returns "false." Before you say my calculator is being stupid, consider this:

 

If I input sqrt(4) = a, and ask it to solve for a, it returns a = 2. If I input 4 = a², then it returns a = -2 or a = 2. The square root function is always positive—this is the way it's defined. It's only when you solve for a number squared that you convert the answer to ± of the square root.

 

In short, sqrt(4) =/= -2. That is an incorrect expression, because sqrt(4) = 2, and -sqrt(4) = -2.

I see what you're saying. There have been many, many debates on this topic (and I'm just playing along for fun), and thus I'll return to a previous point that had been made: 

 

x^2 (and other higher power functions) have multiple answers, x and -x. And while we have some speculation on the square root function, remember that the square root is merely a function of the power 1/2.

 

So if we're using another power function x^(1/2), there very well could be 2 answers. The square ROOT (singular) has one answer, but the function of taking a square root has 2 answers.

 

Edited November 7, 2014 by CloudFyre

[Twilight Dirac 721]

 

 

If I input sqrt(4) = a, and ask it to solve for a, it returns a = 2. If I input 4 = a2, then it returns a = -2 or a = 2. The square root function is always positive—this is the way it's defined. It's only when you solve for a number squared that you convert the answer to ± of the square root.   In short, sqrt(4) =/= -2. That is an incorrect expression, because sqrt(4) = 2, and -sqrt(4) = -2.

 

I don't know where you got this idea from, but it is wrong.  The square root of n, is by definition, the number r such that r^2 = n.  So if n = 2, then r = -2 is a valid solution because (-2)^2 = 4.  Don't rely on the conventions written into machines, they typically assume that you already understand the mathematical nuances of the square root function.

[SasQ 1,376]

CloudFyre, on 07 Nov 2014 - 02:46 AM, said:

And I'm not quite sure why you're talking about taking the mean of the answers. The mean is entirely different, and you wouldn't want to take an average when you need exact values.

I wasn't talking about taking means. When I said "if you mean the answers", I meant the answers you have in mind. I'll try to put it another way:

 

The operation of taking the square root gives two unique answers.

But if you meant the answers, which are called "square roots", they are single numbers, even if they're written down with the "r" symbol around them. If the "r" symbol would always mean (=designate) the operation of taking the square root, and always gave two answers, then writing "-root(something)" wouldn't make any sense, and there wouldn't be any way to write down just one of these two answers instead of both. E.g how would you write the negative root only if the symbol "r" always gave two answers?

 

That's where root ( ) of confusion is: The same symbol can designate two different things (either the operation or its result), and this introduces lots of confusion in maths. Some people distinguish into arithmetic roots (always positive) and algebraic roots (two answers), but they still use the same symbol for both, so the confusion is still there. Others say that the "r" symbol is a function and it's always positive. Still others that "r" has two answers, because they have the operation in mind (but how to put down just one of its results, then?). And the confusion would never end, and people would be arguing about this as long as each of them has something else in mind when seeing the same symbol; until someone introduces a distinct notation for distinct things.

 

E.g. Regulus above is talking about square root as a function, so he has the "arithmetic root" on his mind. This is the function which returns only the positive answer, because it is used for writing these answers symbolically, as single numbers.

 

Twilight Dirac, on the other hand, uses the definition which describes the "algebraic root", which is a different thing, I'd say the proper definition of the operation of taking the square root, or the half-power (I prefer to use ^(1/2) for that, to avoid this confusion).

Edited November 7, 2014 by SasQ

[CloudFyre 848]

I wasn't talking about taking means. When I said "if you mean the answers", I meant the answers you have in mind. I'll try to put it another way:

XD I totally thought you meant "if you mean (or average) the answers".

 

tl;dr version of this thread for those just joining us: mathematical definitions are important. Define everything before working on a problem.

Edited November 7, 2014 by CloudFyre

[Admiral Regulus 2,769]

I don't know where you got this idea from, but it is wrong.  The square root of n, is by definition, the number r such that r^2 = n.  So if n = 2, then r = -2 is a valid solution because (-2)^2 = 4.  Don't rely on the conventions written into machines, they typically assume that you already understand the mathematical nuances of the square root function.

 

But the square root function is a function. It returns only one value, and it is a positive value.

 

The square root of n, is by definition, the number r such that r^2 = n.

 

 

No, this is wrong.

 

If r² = n, then r = sqrt(n) or -sqrt(n). If sqrt(n) could be positive or negative, then saying that would be irrelevant. The thing is, you're assuming the functions are inverses of each other when they're really not.

 

The square root function, f(x) = sqrt(x), only exists for f(x) >= 0 and x >= 0.

 

See Wikipedia:

 

Every positive number a has two square roots: √a, which is positive, and −√a, which is negative. Together, these two roots are denoted ± √a (see ± shorthand). Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive a, the principal square root can also be written in exponent notation, as a^1/2.^[2]

 

Edited November 7, 2014 by Regulus

[CloudFyre 848]

The square root of n, is by definition, the number r such that r^2 = n.  So if n = 2, then r = -2 is a valid solution because (-2)^2 = 4.

 

 Agreed. This is the textbook definition. WolframAlpha actually says that in it's definition (and also makes the point that there are two different definitions that circulate, a generic "square root" which most people think of as the positive value, and the algebraic square root, in which there are 2 answers).

[Snuggie Lord 170]

I am too confused to write a proper reply.

[honk friend 1,978]

Sometimes they are five, sometimes they are three. You must try harder, it isn't easy to become sane.

Hmm, wait, what?

[Admiral Regulus 2,769]

Sometimes they are five, sometimes they are three. You must try harder, it isn't easy to become sane.

Hmm, wait, what?

 

That depends on the term of n in the nth power.  A square number has a positive and negative root, and a cube number has 3 roots, each separated by 120 degrees in the real-imaginary plane. A fifth-power has five roots, each separated by 72 degrees in the real-imaginary plane.

 

So, what you said actually does make sense.

Edited November 7, 2014 by Regulus

[Twilight Dirac 721]

No, this is wrong.   If r2 = n, then r = sqrt(n) or -sqrt(n). If sqrt(n) could be positive or negative, then saying that would be irrelevant. The thing is, you're assuming the functions are inverses of each other when they're really not.   The square root function, f(x) = sqrt(x), only exists for f(x) >= 0 and x >= 0.   See Wikipedia:   Quote Every positive number a has two square roots: √a, which is positive, and −√a, which is negative. Together, these two roots are denoted ± √a (see ± shorthand). Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive a, the principal square root can also be written in exponent notation, as a1/2.[2]  

 

Ok, this is a notational issue.  If you want to use this notation, you still need to keep track of the fact that their are two distinct roots and that OP flipped them on you when he moved the minus one in the radical into the denominator in steps 4 through 7.  

Edited November 7, 2014 by Twilight Dirac