sqrt(1)!=-1?

(-1)(-1) does equal 1, however when you square both sides of an equation it may introduce extraneous solutions. By your same logic, -1 == 1 because -1^2 == 1^2.

Edit: So basically, always check your solutions in the original equation.

Wow, you sound just like my professor but you explained a bit better...

So...just another question..if you do the same to both sides, it still remains true

x==x
x+2==x+2

or

x == 6
x + 2 == 6 + 2

why is it different with squares (^2) than with that simple addition (+)?

I would like to say, I am sorry if I am sounding dense, but maybe another example could help?

Edit: crap, it just hit me like a ton of bricks...thanks for the response.

Quote:

Original post by Imgelling I would like to say, I am sorry if I am sounding dense, but maybe another example could help?

Here's a proof that 3 = 4:

3 = 4
3*0 = 4*0
0 = 0

So obviously 3 = 4!

Quote:

Original post by juturnas (-1)(-1) does equal 1, however when you square both sides of an equation it may introduce extraneous solutions. By your same logic, -1 == 1 because -1^2 == 1^2. Edit: So basically, always check your solutions in the original equation.

Okay, so, I am still confused the more I think about your answer. You start with a false equation.

By your logic based on mine (and a partridge in a pear tree) :
3 = 5 // false
3^2 = 5^2 // false, no crap

but what if I gave you as a math test question:
sqrt(1) = -1
you would find that to be true right?

I am not trying to be dense, I am really not, and this might be an obscure rule that I don't remember.

But really, if I presented you with:
sqrt(1) = -1
you would equate that to be true right? Or am I missing the whole point. Once again, I am sorry for being annoying.

Quote:

Original post by Sneftel Quote: Original post by Imgelling I would like to say, I am sorry if I am sounding dense, but maybe another example could help? Here's a proof that 3 = 4: 3 = 4 3*0 = 4*0 0 = 0 So obviously 3 = 4!

Once again (you replied as I was), you started with a false statement.

edit: if I am missing something obvious, let me know please!

Quote:

Original post by Imgelling Once again (you replied as I was), you started with a false statement.

It's the same thing, however. By starting with sqrt(1) = -1, you're also starting with an incorrect assumption, and then using the "power" operator to "prove" your incorrect assumption is correct.

So, you can say:

sqrt(1) = -1
(sqrt(1))² = (-1)²
1 = 1

Or you can say:

3 = 4
3*0 = 4*0
0 = 0

They both start with an incorrect assumption and then proceed to "prove" that it's "true".

Quote:

but what if I gave you as a math test question: sqrt(1) = -1 you would find that to be true right?

No. sqrt is a function. A function, by definition, has only one possible output for a given input. sqrt(1) cannot be -1, because it's already busy being 1.

Of course, -1 IS a square root of 1. Nevertheless, the sqrt function returns the positive square root of the number.

Quote:

but what if I gave you as a math test question: sqrt(1) = -1 you would find that to be true right?

No. sqrt is a function. A function, by definition, has only one possible output for a given input. sqrt(1) cannot be -1, because it's already busy being 1.

Of course, -1 IS a square root of 1. Nevertheless, the sqrt function returns the positive square root of the number.

Quote:

Original post by Sneftel Quote: but what if I gave you as a math test question: sqrt(1) = -1 you would find that to be true right? No. sqrt is a function. A function, by definition, has only one possible output for a given input. sqrt(1) cannot be -1, because it's already busy being 1. Of course, -1 IS a square root of 1. Nevertheless, the sqrt function returns the positive square root of the number.

Sneftel:
I'm sorry, I see your point with the function, but I didn't know how to draw (write) the sqrt() text. I didn't mean it as a function with programming. Sorry for the misunderstanding.

Codeka:
1. How are you writing super/sub script?
2. What is the incorrect assumption in the fact that the square root of 1 = {1,-1}? I assume the fact of -1 * -1 = 1 and 1 * 1 = 1, both true right? Therefore (can't make that symbol either), the square root of 1 = {1,-1}.

Basically, should I just realize, as a rule, that if the starting equation has a variable on both sides and when I solve and it becomes a square root of a number that equals a negative number, I should answer that number is not a solution?

Once again, sorry, I am not trying to be difficult, just trying to understand.

Edit: I think my main problem is the fact the numeral 1 comes up as a possible answer and just throwing my world in a spin.

my 2 cents,

x^2 = 1
x^2 - 1 = 0
(x - 1)(x + 1) = 0
x = 1 or x = -1

Quote:

Original post by Imgelling 1. How are you writing super/sub script?

Regular HTML works on these boards.

Quote:

Original post by Imgelling 2. What is the incorrect assumption in the fact that the square root of 1 = {1,-1}? I assume the fact of -1 * -1 = 1 and 1 * 1 = 1, both true right? Therefore (can't make that symbol either), the square root of 1 = {1,-1}.

We are also talking about the mathmatical function square root (√). Mathematical functions can only be defined to return one value for a given input, and sqrt(x) is defined as returning the primary square root of x. Wikipedia explains it better than I could:

Quote:

Every non-negative real number x has a unique non-negative square root, called the principal square root, which is denoted with a radical symbol as sqrt{x}, or, using exponent notation, as x1/2. For example, the principal square root of 9 is 3, denoted sqrt{9} = 3, because 32 = 3 × 3 = 9 and 3 is non-negative. The principal square root of a positive number, however, is only one of its two square roots.

(Edited so it looks OK without wikipedia's funky TeX support)

The difference is that equations are not the same as functions. While equations involving square roots have two solutions, the mathematical function sqrt(x) has only one output.

Quote:

Original post by Imgelling Sneftel: I'm sorry, I see your point with the function, but I didn't know how to draw (write) the sqrt() text. I didn't mean it as a function with programming. Sorry for the misunderstanding.

Neither did Sneftel: He means that sqrt is a function in the mathematical sense (i.e., each input maps to a single output).

Quote:

2. What is the incorrect assumption in the fact that the square root of 1 = {1,-1}? I assume the fact of -1 * -1 = 1 and 1 * 1 = 1, both true right? Therefore (can't make that symbol either), the square root of 1 = {1,-1}.

The convention is that sqrt(1) is 1. See this page for more info.

Quote:

Basically, should I just realize, as a rule, that if the starting equation has a variable on both sides and when I solve and it becomes a square root of a number that equals a negative number, I should answer that number is not a solution?

No, you should get used to verifying that what you think are solutions are actually solutions, by plugging in the values in the original equation.

Quote:

Once again, sorry, I am not trying to be difficult, just trying to understand.

No worries: This is a common source of confusion.

Quote:

The principal square root of a positive number, however, is only one of its two square roots.

So, I am stuck at my beginning question.

Okay, looking at what alvaro linked to.

Okay, I think I got it. Enough to pass my exam Wednesday, I think. More than likely I will be back to pick your brains some more and I appreciate EVERYONE'S time and understanding.

Thanks again.

rate++ for everyone! woot!

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

just to throw you off further

Quote:

Original post by Sneftel Quote: Original post by Imgelling I would like to say, I am sorry if I am sounding dense, but maybe another example could help? Here's a proof that 3 = 4: 3 = 4 3*0 = 4*0 0 = 0 So obviously 3 = 4!

Here's a link of 1 = 2 that goes step-by-step of the above common mistake if you still don't get it.
Basically all you have to remember is that if you square,multiply or divide both sides of an equation by a variable you have to check the solutions since you don't know if that variable might be zero as above and screw up your answer LOL.

To clearly see why squaring is the same as multiplying see this very simple example:
Let me give a very simple example. Suppose you have the
equation

x = 3

for which the solution is obviously 3. Now multiply both sides of
this equation by x which gives

x^2 = 3x

One of the roots of this equation is still 3, but there is also
another root, namely zero, which has been introduced. {0,3} are roots
(or the solution set as you'll sometimes see) of the second equation, but only 3 is a root of the first. Often, in
the process of solving radical equations, we square both sides the equivalent of multiplying by an expression which includes the
variable. We find roots to this new equation, but sometimes not all
roots of the new equation are also roots of the original equation.
This should make all clear now :)

p.s. You need a new teacher!

Folks, please take this as a friendly reminder that the forum has a policy about homework/schoolwork problems. Technically they are off topic (this is a game development forum). They can be harmful to the person asking (there have been cheaters here). It is my decision, as forum moderator, whether to allow such posts to remain open to to close them. There are other, dedicated forums that are meant for homework/schoolwork. See Forum FAQ for more details.

I don't think this thread did any harm, and in fact I think it really was a beneficial discussion for Imgelling. I'm going to leave the thread open until tomorrow, since it seems to be really helping Imgelling to study for the test tomorrow.

I will say, my view of the whole sqrt(1) = -1 thing is that hellknows2008 gave the best answer so far. He proved algebraically that the sqrt(1) can be either 1 or -1. (And of course I'm referring to the mathematical operator rather than to the computer library sqrt function.) I also like the Wikipedia article that someone linked to.