B Real numbers and complex numbers

[pbuk]

@pbuk. As to whether "all of us" share this convention, I myself have been a professional mathematician all my life (over 40 years) and cannot recall ever hearing of this notation being restricted to the "principal square root of -2"

I can't remember whether it was as universally agreed on 40 years ago, but in 2021 I think you will find this is the common definition:

"Any nonnegative real number

has a unique nonnegative square root

; this is called the principal square root and is written

or

" etc. https://mathworld.wolfram.com/SquareRoot.html

"$\sqrt 2$: Positive number whose square is $2$". https://mathvault.ca/hub/higher-math/math-symbols/

"one defines $\sqrt 2$ ... by proving that there is exactly one positive real number that squares to 2. Then $\sqrt 2$ is defined to be this number." Princeton Companion to Mathematics

etc.

 

[fresh_42]

As usual, I like the version on nLab:
https://ncatlab.org/nlab/show/square+root

If $K$ is a monoid (which we write multiplicatively) and $x$ is an element of $K$, then the element $x^2$ is the square of $x$. Conversely, if $x^2=y$, then $x$ is a square root of $y$.

If $K$ is an integral domain, then (in classical mathematics) $x$ and $−x$ are the only square roots of $x^2$. If $y$ has a square root, then we often denote its square roots together as $\pm\sqrt{y}$, although there is no meaning of $\sqrt{y}$ itself.

If $K$ is a linearly ordered field, then every element $y$ has a unique nonnegative square root if it has a square root at all; this is the principal square root of $y$ and denoted $\sqrt{y}$.

 

[mathwonk]

I like the idea of a principal square root of a positive real (that's the one I was assuming in my post for sqrt(2), and I have heard about that) better than the idea of a "principal" complex square root of a negative number. In fact the more I think about it, it seems to me there is really no way to define a principal square root of a negative number without referring to a specific model of the complex numbers, one in which a preferred square root of -1 has been chosen and given a specific name. I.e. in the abstract complex numbers, defined merely as an algebraic closure of the reals, there is no preferred square root of -1, hence no distinguished element to call i.

When I defined the principal root in terms of "counter clockwise" motion on the unit circle, I was assuming we are taking as our model of the complex numbers the usual model defined by the set of ordered pairs of reals, and in which one designates the pair (0,1) as i. But there is no intrinsic reason to choose this as i, rather than (0,-1). So if our definition of the complex numbers is only an algebraically closed field, algebraic over the reals, there is no distinguished element i.

The situation in the reals is different, where one can distinguish a positive root, where positivity is a property of certain reals, as fresh42 lays out so clearly. Non negative reals are distinguished by the fact that they are squares (of reals). But there is no field theoretic property of the complexes that can be used to distinguish i from -i, for the simple reason that sending a+bi to a-bi is a field automorphism. Thus the idea of a "principal" root of a negative number is to me not a natural notion.

On the other hand, the idea of a "primitive" root of 1 is well defined, (a root whose powers yield all other roots), but not unique, and i and -i are both primitive (4th) roots (of 1). Thus I am more familiar with expressions like u.sqrt(2), or u^r.sqrt(2), for an nth root of 2, where u is a primitive nth root of 1, and sqrt(2) denotes the positive square root of 2.

Of course I readily admit I am not always that observant, and may simply have missed out on what most people say and write in this regard. But as Charlie Brown said, he was not that fond of math, but preferred subjects where the answers were more a matter of opinion. So perhaps even this discussion would suit him!

 

[pbuk]

I like the idea of a principal square root of a positive real (that's the one I was assuming in my post for sqrt(2), and I have heard about that) better than the idea of a "principal" complex square root of a negative number. In fact the more I think about it, it seems to me there is really no way to define a principal square root of a negative number without referring to a specific model of the complex numbers, one in which a preferred square root of -1 has been chosen and given a specific name. I.e. in the abstract complex numbers, defined merely as an algebraic closure of the reals, there is no preferred square root of -1, hence no distinguished element to call i.

All of that is true, however once you have chosen $i$ you can define the principal square root of a negative number $z = \sqrt {(-1)(r)} = i\sqrt {r}$.

When I defined the principal root in terms of "counter clockwise" motion on the unit circle, I was assuming we are taking as our model of the complex numbers the usual model defined by the set of ordered pairs of reals, and in which one designates the pair (0,1) as i. But there is no intrinsic reason to choose this as i, rather than (0,-1).

Absolutely: the definition of the principal square root of a negative number $z = \sqrt {(-1)(r)} = i\sqrt {r}$ still holds.

The situation in the reals is different, where one can distinguish a positive root, where positivity is a property of certain reals, as fresh42 lays out so clearly. Non negative reals are distinguished by the fact that they are squares (of reals). But there is no field theoretic property of the complexes that can be used to distinguish i from -i, for the simple reason that sending a+bi to a-bi is a field automorphism. Thus the idea of a "principal" root of a negative number is to me not a natural notion.

I think the idea of a natural notion for the complex numbers is somewhat ironic. There is no field theoretic property of $\mathbb C$ that permits complex logarithms either, but once you add them in you can do some neat stuff.

Of course I readily admit I am not always that observant, and may simply have missed out on what most people say and write in this regard. But as Charlie Brown said, he was not that fond of math, but preferred subjects where the answers were more a matter of opinion. So perhaps even this discussion would suit him!

yes it is true that opinions here (or at least the way they are expressed) are less consistent for complex square roots: Wolfram Alpha doesn't isn't even internally consistent!

"The concept of principal square root cannot be extended to real negative numbers since the two square roots of a negative number cannot be distinguished until one of the two is defined as the imaginary unit, at which point

and

can then be distinguished. Since either choice is possible, there is no ambiguity in defining

as "the" square root of

." https://mathworld.wolfram.com/PrincipalSquareRoot.html

"The principal square root of a number

is denoted

(as in the positive real case) and is returned by the Wolfram Language function Sqrt[z]." https://mathworld.wolfram.com/SquareRoot.html

 

[pbuk]

As usual, I like the version on nLab:
https://ncatlab.org/nlab/show/square+root

I don't think I do for this seems to imply that $x = \sqrt 4$ [edit: has a unique solution identifies a unique number] in $\mathbb R$ but not in $\mathbb Z$, however we have rehearsed our differences on foundations of mathematics before and I don't think it would be productive to do so again

 

[FactChecker]

I like the idea of a principal square root of a positive real (that's the one I was assuming in my post for sqrt(2), and I have heard about that) better than the idea of a "principal" complex square root of a negative number. In fact the more I think about it, it seems to me there is really no way to define a principal square root of a negative number without referring to a specific model of the complex numbers, one in which a preferred square root of -1 has been chosen and given a specific name. I.e. in the abstract complex numbers, defined merely as an algebraic closure of the reals, there is no preferred square root of -1, hence no distinguished element to call i.

When I defined the principal root in terms of "counter clockwise" motion on the unit circle, I was assuming we are taking as our model of the complex numbers the usual model defined by the set of ordered pairs of reals, and in which one designates the pair (0,1) as i. But there is no intrinsic reason to choose this as i, rather than (0,-1). So if our definition of the complex numbers is only an algebraically closed field, algebraic over the reals, there is no distinguished element i.

That sounds right, but there is practically no area of mathematics where there are not canonical representations and standard forms that are somewhat arbitrary. What version of this you prefer probably depends on where you are going with it. In complex analysis, applied math, and engineering, there are good reasons to introduce the standard definitions of i and the principle branch of the square root, logarithm, etc.

 

[swampwiz]

Summary:: Why don't the answers agree?

To find √(-2)√(-3).

Method 1.
√(-2)√(-3) = √( (-2)(-3) ) = √(6).

Method 2.
√(-2)√(-3) = √( (-1)(2) )√( (-1)(3) ) = √((-1)√(2)√(-1)√(3) = i√(2)i√(3) = (i)(i)√(2)√(3) = -1√( (2)(3) ) =-√6.

Why don't the two methods give the same answer?
Thanks for any help.

There are a few mistakes. And there is also the ambiguity of what is exactly meant by the radical. Technically, a radical could indicate any deMoivre root (which for the square root means one is the ± of the other), but a common usage is for any radical to represent the value that has a complex angle of the product of complex angle of the base and the reciprocal of the root degree - aka the principle root.

If the radical is to be regarded as the former, then there is no problem since both results are the same (i.e., ± of the principal square root of 6), and the steps just have a extra ± factors that are all redundant.

If the radical is to be regarded as the latter, then the proper analysis, using phasor notation (i.e., the phasor has the value of the complex exponent of e to the product of i & the angle (as radians), would be:

[ √(-2)√(-3) ] = [ 2 (∠180°) ]^1/2 [ 3 (∠180°) ]^1/2 ]

= [ 2^1/2 (∠90°) ] [ 3^1/2 (∠90°) ] = [ 6^1/2 (∠180°) ]

= - 6^1/2

Thus this step in Method 1 is incorrect; you cannot bring everything under the radical like you could under a general exponent.

√(-2)√(-3) = √( (-2)(-3) ) -> INCORRECT

[ (-2)^a (-3)^a ] = { (-2)(-3) }^a -> OK, if a is an integer

Method 2 is loaded with the same type of mistake, although obviously in the end, the result is correct.

 

[Svein]

Actually, taking a (square) root of something in the complex plane is not straightforward at all. It involves analytic continuation and the concept of Riemann surfaces. A short example involving the square root of two:
2=2e^{2n\pi} where n is any integer. Therefore \sqrt{2}=\sqrt{2}e^{n\pi} where n is any integer. Since e^{n\pi}=1 for any even n and e^{n\pi}=-1 for any odd n, the square root is not a function in the classical sense.

Generalizing slightly, \sqrt{z} is only single-valued for z=0. We can, however, define several single-valued functions for that square root by restricting arg(z) suitably. Wherever two of those functions are defined on a common interval for arg(z), they are identical and thus are analytic continuations of each other.

Granted, this is just a one-paragraph sketch of a subject requiring several pages to be anywhere near complete.

 

[wrobel]

Summary:: Why don't the answers agree?

To find √(-2)√(-3).

this statement seems to be incorrect by itself. $\sqrt{z}$ is a multivalued function
which branch is expected to choose?