Square Root of a Negative Number?

[JMBS]

I'm pretty sure it's not possible but if anybody has any kind of theory on finding the square root of a negative number go ahead share. Sorry if I sound a bit imature or "noobish". I'm new here and probably younger then a majority of the users here.

Example: -100

 

[nicksauce]

That's what the set of complex numbers are for. One can define the squareroot of -1 to be i, then it follows that the squareroot of -100 is 10i (or -10i).

 

[JMBS]

Thanks, you obviously know more then my 7th grade math teacher.

 

[nicksauce]

Well in your teacher's defense, it is typical for teachers to tell 'lies' like "you can't take the squareroot of a negative number" because they don't want the class to get even more confused. It does possibly hurt the brighter students, but should help the others students understand better, as they would probably be even more confused if your teacher started talking about complex numbers.

 

[DaveC426913]

Or more specifically, imaginary numbers. A complex number is merely the combination of a real number and an imaginary number.

 

[nicksauce]

True, although I hesitate the use the nomenclature "imaginary" when introducing the concept because it often makes people think that they "don't exist" while real numbers "do exist".

For example, in the thread titled "Complex Infinity" someone wrote:
"When using complex math it can have applications because you are using non real numbers... which is exactly what they are... numbers that don't really exist, although they are defined."

My mother once said something like "Why take a class in imaginary numbers? That would be like taking a class on ghosts and bigfoot". I then told her "Ok I'll take a class on complex numbers" and she was OK with it.

So I usually try to avoid the word "imaginary".

 

[LukeD]

You teacher is correct in a sense when she said that the square root of a negative number doesn't exist.

You can think about it like this:
When you were just using the numbers 1, 2, 3, 4, and so on back in 2nd and 3rd grade

You'd have something like 1 + ___ = 3, and you'd fill in 2. Or you'd have something like 3 - ___ = 2, and you'd fill in 1.

If someone instead put 3 + ___ = 3, what would you put? None of the numbers 1, 2, 3, and so on satisfy that, so the number that would doesn't exist.

However, later you invented new numbers. You came up with 0 and negative numbers, and suddenly that question had an answer. The answer didn't exist in the old numbers, but it does in the new ones.

Likewise, when you were using just the integers, if someone cut a cupcake into 2 pieces and gave you one, then someone asked you how many cupcakes you were given, you couldn't give them an integer. No number existed that could answer that question. Later though, you defined the rational numbers and you could give a fraction as an answer.

Then you started doing multiplication and division. And at some point, someone asked you what numbers x have the property that x*x = 9, and in the rational numbers, this would be -3 and 3. But what if instead your teacher asked you for the numbers x such that x*x=2? Then you could sit there and try to work it out on paper, and you might be able to find some number that almost worked, but you'd never be able to get an answer, because in the rational numbers, the square root of 2 does not exist, so you invent the real numbers, and they give you numbers that answer some questions like that.

But then someone asks for the numbers x such that x^2 = -9.
Well, you can prove that for any real numbers, if x is not 0, then x^2 > 0, so there can't be any real number x such that x^2 = -9.

Ah, but whenever we had that problem before, we just invented new numbers, so maybe we can just do that again and our question will have an answer. In fact, we can: that is what the complex numbers do.

 

[kts123]

To add to what nicksauce said, it's also often dangerous to introduce students to advanced mathematical concepts before they are ready. For example, we can "divide by zero" in (rare) special cases. Once again, your teachers probably never mentioned that because there's no point in you learning it, not to mention it might lead you into believing something you shouldn't.

 

[uman]

It's all just a game with definitions. The square root whose domain is all real numbers is not the same function as the square root whose domain is the nonnegative real numbers. Similarly, whatever kts123 is talking about surely involves a different definition of "divide" and "zero".

 

[kts123]

There's a thread discussing division by zero, if anyone here is interested:

https://www.physicsforums.com/showthread.php?t=236748

 

[DaveC426913]

There's a thread discussing division by zero...

Just. One?

 

[MadScientist 1000]

My 9th grade math teacher failed me on a square roots test because instead of saying "No solutions" for the square root of a negative number, I wrote "Imaginary". It's horrible how teachers can't just tell the class "Imaginary means no rational solutions".

 

[d_leet]

My 9th grade math teacher failed me on a square roots test because instead of saying "No solutions" for the square root of a negative number, I wrote "Imaginary". It's horrible how teachers can't just tell the class "Imaginary means no rational solutions".

Real, not rational.

 

[JMBS]

A rational number is a number that can be expressed as a ratio of two integers. Non-integer rational numbers "fractions" are usually written as the vulgar fraction.




a/b A:Numerator B:Denominator

Right?

 

[HallsofIvy]

A rational number is a number that can be expressed as a ratio of two integers. Non-integer rational numbers "fractions" are usually written as the vulgar fraction.




a/b A:Numerator B:Denominator

Right?

No, a: Numerator and B: denominator.

But what does this have to do with the discussion?

 

[matt grime]

I got so frustrated at the number of times this kind of question was asked, and the refusal to have a FAQ in maths here, that I wrote this:

http://www.maths.bris.ac.uk/~maxmg/maths/introductory/complex.html

 

[D H]

Hello, Matt.

First and foremost, welcome back! We have missed you.

Secondly, we now have a place for FAQ-like things: The https://www.physicsforums.com/library.php" ).

I am remiss in adding to the PF library, being totally overwhelmed at work and in real-life (kids graduating from college, kids getting married, kids finally moving out of the house). When I have some "free time" I will start some entries ...

 

[LukeD]

Actually, back when I didn't think complex numbers "existed", it wasn't quite for that reason. It was actually all of the word problems that was given in high school.

In word problems, you're given real world applications of the real numbers, and every time you use numbers, it's to measure something "real". Usually the things had dimensions: length, time, etc. So the numbers had to be ordered. There weren't any infinities or "infinitesimals", just large numbers and really tiny numbers (unless they were "infinitesimals" used to illustrate Calculus problems, but those didn't "exist" because they 1) weren't measurable and 2) didn't actually exist as numbers in a rigorous framework of the form we were using). So my only experience with numbers was with things that could be measured, and these things already had names as far as being able to apply the real numbers to them. So when I'd try to extend it to the complex numbers, i'd think, "ok, i have this problem here and this thing has length 2, but what about if i keep everything else the same and try to think of the thing as having length 1 + i", which would be complete nonsense. There are of course real world applications to the complex numbers, but that's not the right way to think about them because you'd always just think "well there's no way for anything to have a length 1+i, and I've never seen any situation where I can actually a real number with a complex one and have it be meaningful, so the complex numbers don't exist in any 'real' sense."

Of course the problem was that I'd already got in my mind an idea of how problems should be setup and complex numbers just didn't fit in there. Once I was shown how other mathematical structures can be applied to solve problems and how many of them could be thought of as "numbers", but that they didn't have the same properties as what I used to think of as "numbers", then I realized that the whole concept was artificial and no numbers existed in any sense.

 

[JMBS]

No, a: Numerator and B: denominator.

But what does this have to do with the discussion?

What do you mean by no?

 

[uman]

Well MadScientist1000, if your teacher had introduced the square root function as a function from the non-negative real numbers to the non-negative real numbers, "imaginary" is clearly a wrong answer, and you should have failed.

I'm not trying to be harsh, just saying that things in math are basically whatever you define them to be.

 

[LukeD]

Yes, but when someone says "square root", they are being ambiguous because you do not know immediately what the domain of the function is. If someone asks you "what is the square root of a negative number" then it depends on whether or not the the function is defined for the negative reals.

But if the function isn't defined for the negative reals, then the question is meaningless.

You're contemplating the value of f(y) when f: X -> R and y is not in X.

Usually when I'm asked a question, I assume that the question is well formed, so I probably would have answered imaginary as well (since that would be the answer if the square root function was defined for the negative reals)