The word "the" is not in the original question.

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Sqrt(4) = x
4= x^2

All I did is square both sides. You've admitted x^2=4 has 2 solutions. So what changes?

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In short: sqrt(x^2)=abs(x).

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The word "the" is not in the original question.

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Really? Clearly it's common usage.

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The word "the" is not in the original question.

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Really? Clearly it's common usage.

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In math, when you have to use clearly to cover your shoddy work, you've lost.

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The word "the" is not in the original question.

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Really? Clearly it's common usage.

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In math, when you have to use clearly to cover your shoddy work, you've lost.

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You doubt that a question on TexAgs (general board) is common usage? You are smoking crack.

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From your link:

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In common usage, unless otherwise specified, "the" square root is generally taken to mean the principal square root.

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(which is the positive one)

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That would be common usage, not mathematical usage and certainly not in any math course from algebra II on.

Finding a principal value is a convention we use, it doesn't mean one value is more correct than another.

For instance, as I stated above, the arctan function produced an infinite amount of answers (not just 1) but it's range is restricted to make it a function when needed. That doesn't mean the other answers are incorrect.

I'm sure you googled the square root definition by now and you've probably noticed not a single page contradicts what I've stated here.

Common usage would also say sqrt(-4) doesn't exist. See what happens when you lose precision by bringing semantics into the argument?

Also, I'm out for the night, pick up in the morning.

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From your link:

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In common usage, unless otherwise specified, "the" square root is generally taken to mean the principal square root.

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(which is the positive one)

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That would be common usage, not mathematical usage and certainly not in any math course from algebra II on.

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This is a load of crap. I worked for the math department at A&M teaching math, have been an engineer for decades, and been programming for longer than that. Under none of those circumstances does anybody use anything like sqrt(4)= +-2. Mathematica and Maple, both of which are used by mathematicians and scientists world wide does not treat and both return multiple roots when proper. Every computer language I know of, C, C++, Java, Python, Ruby, javascript, and countless others ALL treat sqrt(x) as having a single answer. To pretend that 99.99999% of people use the common usage of sqrt as the principal square root is delusional. Clearly a question asked on the general board of TexAgs is common usage.

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I'm sure you googled the square root definition by now and you've probably noticed not a single page contradicts what I've stated here.

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I have googled square root definition and have noticed that you are dead wrong. Even your own link from wolfram shows you to be wrong.

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Common usage would also say sqrt(-4) doesn't exist. See what happens when you lose precision by bringing semantics into the argument?

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Sure it does. Imaginary numbers are used all the time in my field, and NOBODY I have ever met uses sqrt(x)=+-y.

The fight is on. There is no quit in atmag's failures. They're quite entertaining.

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The fight is on. There is no quit in atmag's failures. They're quite entertaining.

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You must be a fan of that crappy ass movie Avatar.

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The fight is on. There is no quit in atmag's failures. They're quite entertaining.

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You must be a fan of that crappy ass movie Avatar.

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Nope. But your inane rambling on that thread was amazing.

BTW, here's a quote from wiki:

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Every positive number a has two square roots: sqrt(a), which is positive, and -sqrt(a), which is negative. Together, these two roots are denoted +/-sqrt(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.

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So sqrt(a) is positive, and -sqrt(a) is negative. Furthermore, when denoting both roots, one uses +/-sqrt(a).

So in English vernacular one would say 4 has two roots, -2 and 2. But writing sqrt(4) denotes ONLY the positive root.


Edit: Wiki used the square root symbol, but TexAgs can't handle that, so I changed it to sqrt(...)

The plane will not take off.

Pro wrestling isn't fake, it's only predetermined

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The radical definitionally refers to the principle root.

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Every non-negative real number a has a unique non-negative square root, called the principal square root, which is denoted by radical(a), where radical is called the radical sign or radix.

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You can't just change radical(a) to sqrt(a).

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The function sqrt(..) returns the principle square root in practically every programming and scripting language on Earth including Mathematica and Maple. That is what it is DEFINED as. It's not shorthand for the English phrase: "both square roots of".

That's how programming was made to play nice with math.

It makes it a pain in the ass to get correct answers since you have to always run sqrt() twice.

Think about programming a solver for the quadratic equation...

The original question seemed to be merely evaluating a complex expression and not solving an equation. That does matter.

Sqrt (-4) sqrt (-9) as an expression has only one answer, which is -6.

As far as the algebraic equivalency argument of squaring, think about it from a graphing standpoint:

Sqrt (x) = 2 has one and only one solution. x = 4.
Sqrt (x) = -2 has no real solutions.

x^2 = 4 has two solutions, +/-(2). The graph of the parabola will clearly support this.

Powering both sides of an equation introduces the possibility of extraneous solutions. Happens regularly with trig equations when squaring is done to generate Pythagorean identities...you will often get angles that do not satisfy the original equation, so those solutions are discarded.

Something that I think got glossed over is aTmAg's correct statement of sqrt (x^2) = abs (x). This is what will introduce both the positive and the negative solution.

So, all of that to say "where's the equation being solved in the OP?" There isn't one. (...and, no, the = after the expression doesn't make it an equation. There has to be a quantity on the other side to which it is equal.) Since there's only an expression, you merely evaluate it as it appears.

While sqrt(x^2) = abs(x) is true...

sqrt(x^2) = x is also true

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sqrt (x) = -2 has no real solutions.

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This is incorrect. Square both sides to solve for x. Hint: it's 4.

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While sqrt(x^2) = abs(x) is true...

sqrt(x^2) = x is also true

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sqrt (x) = -2 has no real solutions.

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This is incorrect. Square both sides to solve for x. Hint: it's 4.

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No, 4 is not a solution to sqrt (x) = -2.

Sqrt (4) = 2
- sqrt (4) = -2

(4,-2) is not on the graph of sqrt(x), which it would have to be if 4 were a solution to sqrt (x) = -2.

135 degrees is not a solution to sin x = cos x, but if you were to square both sides and solve, 135 would show up as a solution. But, it is extraneous. Same thing with sqrt (x) = -2 and x = 4. It is an extraneous solution.

3 pages of deliberation and no one posted tits?!

Btw I'm in.

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That's how programming was made to play nice with math.

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No it is not. You are dead wrong. The function sqrt(x) is defined to be:



Which itself is defined as the principle square root, in both software and general mathematics. Which means they both result in only one single value, and is non-negative for any non-negative x. Always. The link on wiki says it. Bobo's wolfram link says it. Nowhere does anything say that either of those two refer to "both square roots of x". So, in short, sqrt(x) is matches the math, it does not merely "play nice with math".

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It makes it a pain in the ass to get correct answers since you have to always run sqrt() twice.

Think about programming a solver for the quadratic equation...

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I've written "programming solvers" for the quadratic equation many times. It would be a HUGE pain in the ass if sqrt(x) returned 2 values. As would the mathematical radix operator represented 2 values. They would, in fact, both be worthless. That is why both in mathematics and in programming they only represent the single principle square root.

What's the square root of "this thread sucks"?

I'll hang up and listen...

I don't think he understands the difference between an operation and a function.

It's 8th grade algebra so I understand how he could be having problems.

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Wait, MULTIPLE people are trying to argue that only the positive square root is a possible answer? I'm a libarts tard and even I know the solution is always positive and negative because they both give the same square

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Most numbers have 2 roots, BUT only represents one of them: the principle root. "sqrt(x)" is the same thing as . So the expression "sqrt(-4)*sqrt(-9)" is the same thing as "THE principle root of -4 multiplied by THE principle root of -9", which results in ONE number. Not two.

His own quote from wiki says, as the first sentence, that sqrt(x) always has two values. One positive and one negative.

I also don't think he's ever plotted y=sqrt(x) by hand... Do they even teach children to plot by hand anymore?

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I don't think he understands the difference between an operation and a function.

It's 8th grade algebra so I understand how he could be having problems.

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You laugh, but you owned yourself on your own link. The radix operator ( and sqrt(x) ) only represents the principle root. Not both. So maybe you should go back to 8th grade algebra again, since you clearly have no idea what you are talking about.

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His own quote from wiki says, as the first sentence, that sqrt(x) always has two values. One positive and one negative.

I also don't think he's ever plotted y=sqrt(x) by hand... Do they even teach children to plot by hand anymore?

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It says no such thing. Nowhere does "sqrt" appear on the entire page. The first sentence says:

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In mathematics, a square root of a number a is a number y such that y^2 = a, in other words, a number y whose square (the result of multiplying the number by itself, or y y) is a.[url=https://en.wikipedia.org/wiki/Square_root#cite_note-1][1][/url] For example, 4 and 4 are square roots of 16 because 4^2 = (4)^2 = 16.

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There is a difference between "sqrt(x)" and "the square roots of 16". The statement "sqrt(x)" is the same thing as which is ONLY the principle root. It is not both roots. Which is why it later says:

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Every positive number x has two square roots: , which is positive, and -, which is negative

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(I had to plug in images because texAgs can't handle special characters)

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I don't think he understands the difference between an operation and a function.

It's 8th grade algebra so I understand how he could be having problems.

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You laugh, but you owned yourself on your own link. The radix operator ( and sqrt(x) ) only represents the principle root. Not both. So maybe you should go back to 8th grade algebra again, since you clearly have no idea what you are talking about.

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No, there is a difference between the sqrt(x) as an operation and the sqrt(x) as a function which is what the article is talking about. Again, you haven't grasped that idea in this entire thread.

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I don't think he understands the difference between an operation and a function.

It's 8th grade algebra so I understand how he could be having problems.

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You laugh, but you owned yourself on your own link. The radix operator ( and sqrt(x) ) only represents the principle root. Not both. So maybe you should go back to 8th grade algebra again, since you clearly have no idea what you are talking about.

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No, there is a difference between the sqrt(x) as an operation and the sqrt(x) as a function which is what the article is talking about. Again, you haven't grasped that idea in this entire thread.

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No there is not. You cannot be more wrong on this. The operation sqrt(x) IS A FUNCTION. It is the same exact thing as which is also a function. Both ALWAYS return ONLY the principle root of x. That is the reason Mathematica and Maple only return a single value for sqrt(x). They aren't software limited to single answers. Both an return multiple answers when it is proper to do so. The reason they never return multiple answers for sqrt(x) is because it is never proper to do so.

Keep doubling down on your foolishness though.