i suck at math-stil funny though

[Bethan]

I don't know the exact definition in English, but what we call "square root" in French is the "positive square root" (we don't even ever specify "positive"), and I was allways taught "x²=a -> |x| = sqrt(a) ", or in case of your "physical" example, that the module of the velocity is sqrt(u²+2as), but it give you no information about it's direction. Indeed having the module of a vector equal to -2m/s would be totally meaningless.

[Badpaladin]

OK, I had one prepared . I will insert smileys to keep this argument friendly .

This example doesn't help your case - the kinematics formula you're using isn't used on vectors, it's used to find the magnitude. Direction is applied afterward, which is were the positive/negative comes in. The square root (function) is not what gives us "the ball travels 2 m/s in the positive or negative direction". What you did was provide a single input, and it produced a single output (hence it behaves exactly as a function should). You forced a "second" output by realizing the ball could be moving in either direction at 2 m/s.

In all honestly, I'm interested to find out if the Square Root Function is really a farce. If so, that's news to me. Note that I am not confusing this with the square root, which is just a number. It was never the number that was in contention.

[Twoddle]

This example doesn't help your case - the kinematics formula you're using isn't used on vectors, it's used to find the magnitude. Direction is applied afterward, which is were the positive/negative comes in. The square root (function) is not what gives us "the ball travels 2 m/s in the positive or negative direction". What you did was provide a single input, and it produced a single output (hence it behaves exactly as a function should). You forced a "second" output by realizing the ball could be moving in either direction at 2 m/s.

Makes me a bit sad to realize some parts of the world see it like this. Maths is pure, maths is complete, in fact complex numbers were invented to fill a hole. The complete answer is given in the formula, there are 2 solutions. There is no "forcing" of a second output, it's built into the formula.

You forced a "second" output by realizing the ball could be moving in either direction at 2 m/s.

The realization usually comes from when you see the negative value of the square root and you work out how it applies. In this case you think, oh yeah that's for when the ball is traveling back down.

[Deleted]

I like this mathematical solution to Girls

We state the fact that girls take both Time and Money

Girls = Time * Money

Then we declare that Time equals Money

Time = Money

And we all know that Money is the root of all evil

Money = √(Evil)

So that equates to this

Girls = Money * Money = Money²

Then

Girls = (√(Evil))²

Which leaves us with the irrefutable proof

Girls = Evil

[Badpaladin]

Makes me a bit sad to realize some parts of the world see it like this. Maths is pure, maths is complete, in fact complex numbers were invented to fill a hole. The complete answer is given in the formula, there are 2 solutions. There is no "forcing" of a second output, it's built into the formula.

No, it's not built into the formula. You are using a Square Root Function, sometimes denoted as a Principle Square Root and saying it produces two results. This is absolutely, 100% incorrect. It's not sad, it's not more than that and it doesn't involve complex numbers. It is not any simpler than this. This function is the function described in the first place. There is no mention of the number square root, only the function. The function is called the Square Root Function because it's used so much more often than the Square Root Relation. This was outlined beforehand.

The realization usually comes from when you see the negative value of the square root and you work out how it applies. In this case you think, oh yeah that's for when the ball is traveling back down.

No. There is no realization from the formula. Kinematics equations such as what you presented as used to determine the magnitude, and information known about the object beforehand is then applied to give it a direction. This is how it is used in physics, engineering and applied mathematics. In "pure" mathematics, the Square Root Function also behaves exactly like this. The square root relation does not. You seem to think that there is no such thing as the Square Root Function. There is, and it's not the Square Root Relation.

Note:

I want to make it absolutely and unambiguously clear that it is not the definition of the square root that I am talking about, it is the definition of the square root function. This is because it was the square root function that was mentioned, not the square root. I am completely aware that the relation of a square root that maps elements of X to elements of Y is not a function, and I want to be sure you're clear that it's not what's under discussion.

[schwarzkopf]

No, what I am saying is that if x² = 4, then x = +sqrt(4) or -sqrt(4), but the sqrt(4) itself always equals 2. It is by definition - the Square Root function is defined that way..

Maybe in programming, but definitely not in maths.... sqrt(4) = +2/-2, sqrt(-4) = +2j/-2j

OT: ? = 2b or (not 2b)

[Twoddle]

The function is called the Square Root Function because it's used so much more often than the Square Root Relation. This was outlined beforehand.

There is no such thing as the square root function because square root is not a function, this was also outlined above.

Kinematics equations such as what you presented as used to determine the magnitude, and information known about the object beforehand is then applied to give it a direction. This is how it is used in physics, engineering and applied mathematics. In "pure" mathematics, the Square Root Function also behaves exactly like this. The square root relation does not. You seem to think that there is no such thing as the Square Root Function. There is, and it's not the Square Root Relation.

This is also wrong. The formula I quoted is used to calculate the velocity v (notice I put "not speed" in brackets to make it even clearer) which has both magnitude and direction and can therefore be negative in the frame of reference. If I was calculating the speed you would be right since speed has magnitude and no direction and can only be positive.

I want to make it absolutely and unambiguously clear that it is not the definition of the square root that I am talking about, it is the definition of the square root function. This is because it was the square root function that was mentioned, not the square root. I am completely aware that the relation of a square root that maps elements of X to elements of Y is not a function, and I want to be sure you're clear that it's not what's under discussion.

Again, there is no such thing as the square root function and it does not make sense to define it.

[Badpaladin]

There are mounds of articles and definitions to state that there is a Square Root Function, "officially" denoted as the Principle Square Root Function. Denying its existence is just bizarre to me. But hey, I'm not a pure math guy. Perhaps they do weird stuff like that (which is why I stay firmly within the bounds of what's useful computationally).

[Auloria]

I believe this is the problem:

There is a definition of "function" that f(x) must be a single-valued, one-to-one mapping of numbers. Therefore, the square root is NOT a function in that sense (more accurately it is a multi-valued function). However the radical (which you would call a square root) can be used to implicitly refer to the range of positive real numbers, which IS a function. I just checked all four calculus books I have sitting on my shelf, and they all describe this very problem in the introductory chapter. In fact, I often see the explicit ±√x in definitions that involve a radical.

Therefore, the "square root" is not a function, but the "square root" is. It just depends on what the guy saying "square root" happens to be thinking of. :-)


HOWEVER, there is a big difference between

f(x) = √(x), where f(x) is a "function"

and taking the square root of something. If you encounter a squared variable in the course of doing algebra and you want to take the root, but fail to consider both the positive and negative roots, that's wrong, even in those calculus texts. You either need an explicit radical in the definition of the problem (which seems to be the calculus textbook standard for the principal root) or an explicit definition of the allowed range.


(Note that every scientist/mathematician I've met does not use a radical or "square root function" or any other such thing to mean only the positive root. It is never a good idea to make assumptions.)

[xoolex]

Well I feel the need to chime in here, as a math professor, and say there is a lot of confusion in this thread.

First off, the idea of A square root of a number is defined. This can refer to the positive or negative number whose square is the given number. Then for convenience/notational simplicity we define THE square root of a nonnegative real number to be the unique nonnegative number whose square is the given number. We also agree that when you use the square root symbol you mean the nonnegative value and if you want the non positive value you put a minus sign in front of the square root symbol. This is entirely arbitrary but it is what we agree to use so that everyone is on the same page. This allows us to create a function on the nonnegative real numbers called the square root function.

Then we extend this definition to sqare roots of negative numbers by introducing the imaginary number i but again by default we take the positive part when using the square root symbol, sqrt(-4) = 2i not -2i by agreement. After this we can then find sqare roots of complex numbers using the complex logarithm. However the logarithm is multiple valued and so you must choose a branch of it to make sense as a function. As a result the complex square root function has multiple branches as well. But even so we can agree on which branch to take, which we call the principal branch so that we can have an actual function since most of mathematics is built around functions.

[Deleted]

Well I feel the need to chime in here, as a math professor, and say there is a lot of confusion in this thread.

First off, the idea of A square root of a number is defined. This can refer to the positive or negative number whose square is the given number. Then for convenience/notational simplicity we define THE square root of a nonnegative real number to be the unique nonnegative number whose square is the given number. We also agree that when you use the square root symbol you mean the nonnegative value and if you want the non positive value you put a minus sign in front of the square root symbol. This is entirely arbitrary but it is what we agree to use so that everyone is on the same page. This allows us to create a function on the nonnegative real numbers called the square root function.

Then we extend this definition to sqare roots of negative numbers by introducing the imaginary number i but again by default we take the positive part when using the square root symbol, sqrt(-4) = 2i not -2i by agreement. After this we can then find sqare roots of complex numbers using the complex logarithm. However the logarithm is multiple valued and so you must choose a branch of it to make sense as a function. As a result the complex square root function has multiple branches as well. But even so we can agree on which branch to take, which we call the principal branch so that we can have an actual function since most of mathematics is built around functions.

Excellent reply! Kudos to you.

This is best demostrated (imo) by the root of unity. I don't think many people would immediately argue that 'naturally' 1^(1/n) has n solutions and not just 1. Especially when n is odd.

(^ means power and subsequently ^(1/n) means n th root.)

[ecthrund]

OK, I had one prepared . I will insert smileys to keep this argument friendly .

Simplest example I could find, there are many others, the formula for the final velocity (not speed) in terms of initial velocity, acceleration and final distance from the ground of a ball thrown into the air from the ground.

v² = u² + 2as
or v = sqrt(u² + 2as)

That is final velocity = sqrt(initial velocity squared + acceleration * final distance)
We will let a = -10m/s², negative because it's downwards.

So let's assume we throw a ball into the air at velocity 8m/s and we want the know the velocity when the ball is 3m off the ground.

The answer has 2 solutions! One for when the ball is traveling up and one for when the ball is traveling down.

sqrt(8*8-2*10*3) = ±2m/s

The thing is it does not make sense to define the "Square root function" because it's not a function. You might find this talked about on very elementary level but it's wrong to do so.

Actually

v² = u² + 2as
or v = +sqrt(u² + 2as) or -sqrt(u² + 2as)

You can check out any standard calculus book. As mentioned a couple of posts before me, the reason why the positive value is chosen for the square root function is to avoid confusion and to have a standardization.

For example, in your particular illustration, if we were discussing speed instead of velocity? As speed cannot be negative (maybe you would like to argue with this too), the only way to give out the correct formula would be to use the absolute value notation. Instead of doing that everywhere a square root is involved, we simply define a square root function.

This confusion regarding the sign of the square root has spoiled the grades of limitless Maths students. When I started out, I was also one of them. But luckily, my teacher spent a month just on functions. Now I can see why.

[Puck]

All the numbers, they hurt my brain.

[Twoddle]

Don't touch my formula ecthrund it was right the first time. Everything I have said in this thread is correct, you are still arguing for the sake of it. Apples and oranges.

[ecthrund]

Don't touch my formula ecthrund it was right the first time. Everything I have said in this thread is correct, you are still arguing for the sake of it. Apples and oranges.

You can define things yourself and do what you want, but that doesn't make it right. Anyway, I am done arguing with you - people who seem more knowledgeable than me have already tried.

[Twoddle]

You can define things yourself and do what you want, but that doesn't make it right.

I would say that if the dictionary definition of square root agrees with me then I am right. The only places I see mention of "square root function" are on wikipedia and math sites for kids.

[ecthrund]

You can pick up any basic calculus book and find it out. or just look at how quadratic equations are solved. Or you can choose to continue to be obdurate and confuse between the concept of Square Root and the accepted conventions relating to the use of the symbol denoting square root.

Also, for something as basic and universally known as square roots, Wikipedia is unlikely to have the wrong answer.

[Szemere]

1] a=b
2] a+a=a+b
3] 2a=a+b
4] 2a-2b=a+b-2b
5] 2a-2b=a-b
6] 2(a-b)=(a-b)
7] 2=1

Two things:
1) between lines 6 and 7 I skipped a simple but important notation. That cancellation here is equivalent to division by (a-b)
the line should be 6.5] 2(a-b)/a(-b)=(a-b)/(a-b)
2) the first line states that a=b therefore (a-b)=0 and threfore the line 6.5] is division by zero which leads to an inconsistent result.

Starting at 4, both ends of the equasion was 0. At that point you were just multiplying 0 by random things/adding other random things to the equasion, and through stealthy division by 0, you could basically make any point 7] seem credible, as long as the division by 0 was stealthy and the random things added to the equasion were stealthy too, preferably by adding them in before you even made both sides of the equasion 0, to be precise at 3].
Nice trick to confuse the hell out of people that just skim through the first few steps because they are relatively nowhere near as complex as 4-7, setting that up as distraction.

[Deleted]

1/3=.3333...

1=.9999...

Voodoo magic!

Except 1/3 isn't possible to write in decimal, 0.33333~ is just the closest number possible.

[Twoddle]

You can pick up any basic calculus book and find it out. or just look at how quadratic equations are solved. Or you can choose to continue to be obdurate and confuse between the concept of Square Root and the accepted conventions relating to the use of the symbol denoting square root.

That's what I meant by apples and oranges, I am arguing over the fundamentals because I am a purist, in isolation sqrt(x) has 2 values and that's why your paradox fails, you are arguing otherwise. xoolex gave the diplomatic answer, something we call sqrt(x) which has positive values only by agreement only.