Assuming we’re talking real numbers, what is (sqrt-2)^2? I can see it being undefined assuming you have to do the sqrt first; or I can see it being -2 assuming there is nothing that says you have to do the sqrt first.
If you do the square first, it’s +2 or -2, not just -2.
It’s 2i.
Huh? OP guessed it right the first time.
(√-2)[sup]2[/sup] is undefined if we are restricted to real numbers.
Remember your bare-bones basic algebra? Do the work inside parentheses first. That includes evaluating functions before you do any work with the value you get thereby. √-2 must be evaluated first, and as that is undefined, any further evaluation comes to a crashing halt right there.
If you are imaginative enough to imagine imaginary numbers, you could proceed.
√-2 gives you i√2, and (that)[sup]2[/sup] = (i√2)[sup]2[/sup] = -2.
If you do the computations in the reverse order, you get:
√( (-2)[sup]2[/sup] ) = √(4) = +2.
ETA: BTW, the usual definition of the square root function is the principal square root.
Thus, √4 for example is just +2, not ±2.
You tell me what you mean by “talking real numbers”, “sqrt”, etc., and then the answer to what you mean by “(sqrt -2)^2 [assuming we’re talking real numbers]” will be in what you tell me. No one else can tell you what you mean; you get to be the arbiter of that.
[I don’t think there’s much value talking about sqrt without saying that (sqrt x)^2 = x, though. There are very few contexts where I’d be motivated to write (sqrt -2)^2 and not take it to express -2 [not that there aren’t any, but they are few]. It’s true that sqrt -2 will not describe a “real number”, as such, but so what? You’re the one who introduced it. And there’s nothing magical about the quantities which happen to be called “real numbers”, such that there’s something wrong with considering other quantities.]
Well, but there kind of is. When imaginary numbers are introduced to algebra students, they are warned that you have to take the square root first when negative numbers are involved. If i is defined to be sqrt(-1), so that i² = -1, it would be inconsistent to allow you to do
I² = (sqrt(-1))² = sqrt((-1)²) = sqrt(1) = 1.
Interestingly, the earliest “practical” use for imaginary/complex numbers may have been in finding real-number solutions to cubic equations, via a method that turned out to involve square roots of negative numbers in the intermediate steps only to have them drop out at the end. Here’s one web site I found that gives some details.
So, as Indistinguishable said, you have to specify what you mean by things like “talking real numbers.” If you’re not allowed to ever consider non-real numbers, then what meaning, if any, does sqrt(-2) have? But if you’re just saying that the final value has to be a real number, then we have something we can work with.
Thank you!
What I meant by “talking real numbers” was assuming all the steps in the calculation used numbers that were drawn from the set of real numbers, can the square root of negative 2, raised to the second power be solved. Turns out it can’t. Ignorance fought.
Let’s back the question up a bit, to something the OP is probably more comfortable with. If we’re dealing with just integers, what’s (1÷2)*2? This is exactly the same sort of question the OP is asking: It’s looking at a restricted domain of numbers, together with an operation that would take us out of that domain, and another that would take us back into it.
Yes, but if I understand correctly, there’s nothing in the rule book that says an elephant can’t pitch or that you can’t change 1/22 into 12/2, yes? But it seems in my original question you have to do the square root of -2 first. I intend that to be a question even though it doesn’t look like one.
I’m not a mathematician, but it’s not obvious to me that you have to evaluate SQRT(-2) before anything else.
If you’re purely doing arithmetic it’s usually easier to do it that way. But you can follow the algebraic rules for manipulating expressions and get the same answer.
(x[sup]1/2[/sup])[sup]2[/sup] = xsup*2[/sup] = x works in algebra and I don’t see why it wouldn’t work in arithmetic when x = -2
Consider an even simpler case. Suppose we work with a number system consisting only of non-negative integers. (That is, the positive integers along with 0.) No fractions and no negative numbers.
What is 5 - 7? Undefined.
What is 5 - 7 + 4? Yes, that’s undefined too.
But isn’t that just the same as 5 + 4 - 7?
Not necessarily. By the definition of our standard notation conventions, 5 - 7 + 4
means exactly (5 - 7) + 4 and nothing but. Sure, you could use some combination
of commutative law and associative law and other rules to re-write it as (5 + 4) - 7
BUT that’s changing the problem. The relevant rules simply allow you to prove
that these two distinct problems are equivalent, BUT thos relevant rules work
in the full real number system (and in this case, even in the more limited system
of all integers), not in the system of non-negative integers.
If Chronos writes “what’s (1÷2)*2?” we must assume he means exactly what he
wrote, (1÷2)*2, and not some other permutation that might happen to be equivalent
in some other number system. You can’t use some rules to re-arrange it unless you
are working in a number system where those rules work.
ETA: Same with:
This only works because certain rules says it does, and when x = -2, those rules only apply when you are working in the complex number system.
There is an interesting bit of history behind this. Why are REAL numbers called REAL numbers? Are there also unreal numbers? Okay, if imaginary numbers really exist, why are they called imaginary?
According to some history that I read, this all came up before mathematicians had “invented” imaginary and complex numbers. Thudlow Boink pointed out, above, that they came up as intermediate results in some kinds of problems, where they all canceled themselves out by the end of the problem. Did that make it kosher to use square roots of negative numbers? Square roots of negative numbers also started showing up when physicists began analyzing electric circuits, which entail a lot of trigonometry.
As I read the history, they all decided to just pretend that expressions like √-1 were valid even at a time when everyone “knew” they weren’t. They just treated them like some weird algebraic quantity. And Surprise! Surprise! They found that they could solve problems and get answers that seemed to work out right in the real world.
So they called these bizarre numbers “imaginary” and just pretended that they work. But how do you do the arithmetic (or algebra) with them? How do you even know what their rules are? Here, mathematicians (or more probably, physicists) made a daring leap: They just assumed that these new fictitious numbers would follow the same laws as the actual bona fide actual numbers do. They assumed that the commutative, associative, and distributive laws work the same.
Experience showed that this seemed to work, and problems could be solved to give answers that seemed right. So mathematicians decided to accept square roots of negative numbers as being legitimate, and furthermore extended that to complex numbers.
But the terminology was entrenched by then. We still call them REAL and IMAGINARY numbers to this day.
I’d say you can, and in many contexts will have good use to, consider an interpretation of your expressions in which “(sqrt(-2))^2” and “sqrt((-2)^2)” mean the same thing [here, I am using parentheses to indicate “ordering”, in the typical fashion]. In both cases, in some sense, we are raising -2 to the power 1/2 * 2.
Some will object that (-2)^2 = 4, and 4^(1/2) = 2, while, distinct from that, we have (-2)^1 = -2, so that ((-2)^2)^(1/2) cannot be taken to equal (-2)^(2 * 1/2).
Well, I shall quibble with this objection. Everything always depends on what one means, and where a natural rule raises its head in mathematics, it is natural to consider interpretations of terms in which such rules continue as far as possible.
One might say, for example, that for each nonzero ordinary “real” number (or, indeed, complex number), there are multiple corresponding “leal” numbers, where a leal number contains additionally the data of a suitably coherent system of designated choices of nth roots for arbitrary n (that is, a leal number is a real number with a choice of complex logarithm).
It is clear how to define multiplication of leal numbers by leal numbers to yield leal numbers, and exponentiation of leal numbers to real powers to yield leal numbers, and we will find in this context that the rule (a^b)^c = a^(b * c) is universal.
And we will find that, for any particular leal number -2 [the fundamental such choice being the complex number corresponding to “get twice as large and rotate 180 degrees”], that ((-2)^2)^(1/2) = -2 = ((-2)^(1/2))^2 [with (-2)^2 = “get 4 times as large and rotate 360 degrees”, which is different in a fine-grained sense from “get 4 times as large and rotate 0 degrees”, and thus has a different square root].
We can perfectly well work within a context where there is no need to choose whether to take the square root first or square first. Where we will find that sqrt(-2)^2 = sqrt((-2)^2) = -2.
And this context will be very close to simply working within the ordinary real numbers. But it will be as above (or something like that), and if that is to count to you as not “talking real numbers”, and it is for some unfathomable reason deeply important that you be “talking real numbers” through and through, well, then, them’s the breaks.
As always when I ask my rudimentary math questions, I learn so much more than I expected to! Thank you, everyone.