Square root is positive or negative?

[vlad777]

I remembered something from some time ago.

Subject was linear algebra (relations,functions,groups,rings,polynomials,
complex numbers,...) .

Everybody in my class was apparently confused about
whether result of a square root is negative or positive.

When I came to the end of the lecture that I have missed
everybody was like: "oh now I get it!" .

The explanation was that operation is different if the number
was complex or real , or something like that.

If someone knows this, can you explain it to me?

Many thanks.

 

[ronsimpson]

Square root is positive or negative?

answer = yes it is positive or negative.

Square root of 25 is +5 and -5.
+5X+5=25 and -5X-5=25

 

[Ian Rogers]

If you graph [LATEX]X^{2}[/LATEX] Remember it crosses the X axis at 0.. A mirror of all the values on the positive side

 

[vlad777]

+5X+5=25 and -5X-5=25

I don't understand what your equations are representing. X1=4 X2=-6

If you graph [LATEX]X^{2}[/LATEX] Remember it crosses the X axis at 0.. A mirror of all the values on the positive side

If you exchange axes, you get square root.
Thanks, I know about that. This was something involving complex set or field or something.


Also operation is a function and as such cant give two values for one input value, but when complex
something is involved all math theory is satisfied.

 

[ericgibbs]

hi vlad,

This pdf covers complex numbers and roots.

It also gives examples.

E.

 

[vlad777]

Maybe it was like this:

If you consider your number real then root is positive. (operating in real set)
If you consider your number complex (although it has no imaginary part)
then result is both negative and positive.
(operating in complex set)

But I still don't understand why?

 

[jpanhalt]

With regard to Ron's post #4, if you read it as:

(+5) X (+5) = 25 and (-5) X (-5) = 25

maybe it will make more sense. The "X" denotes multiplication, not an unknown variable.

<snip>
Everybody in my class was apparently confused about
whether result of a square root is negative or positive.
<snip>
The explanation was that operation is different if the number
was complex or real , or something like that.

It is difficult to reconstruct what it is you don't understand from a lecture you missed. Have you tried asking the lecturer the question?

Are you asking what is the square root of a complex number? If so, maybe these links will help:

https://www.math.toronto.edu/mathnet/questionCorner/rootofi.html
https://en.wikipedia.org/wiki/Imaginary_unit

If that is not it, then maybe you can give an example to illustrate your question.

John

 

[vlad777]

(+5) X (+5) = 25 and (-5) X (-5) = 25
maybe it will make more sense. The "X" denotes multiplication, not an unknown variable.

I don't know how I didn't figure that out because I thought it might be something else.
But the keyboard has * sign and I always use that.

Have you tried asking the lecturer the question?

This was long time ago.

 

[vlad777]

The main point is that operation in maths is defined as a function, and a function is defined
as a relation that doesn't give two results for one input.

So for root function new definitions that build on these and complex set allow root to have two results.

I think this has something to do with this:

Equation a+b=c defines two operations + and - that are closed in integer set.
Equation a*b=c defines two operations * and / that are closed in rational set.

But equation a^b=c defines three operations: exponentiation, root and logarithm so it's not a field or ring...

(Another question: are ^, root, log closed in complex set?)

 

[3v0]

Maybe it was like this:

If you consider your number real then root is positive. (operating in real set)
If you consider your number complex (although it has no imaginary part)
then result is both negative and positive.
(operating in complex set)

But I still don't understand why?

There are always 2 square roots. But if we are working with lengths, ages, area etc the negative square root is discarded because it does not make sense.

For complex numbers used in vectors, electricity etc the negative square root is considered.

 

[Ratchit]

vlad7777,

In the thread linked below, complex or duplex numbers were discussed extensively. You were the last person to post in that thread. What did you not understand about numbers that have a orthogonal component?

https://www.electro-tech-online.com/threads/why-dont-imaginary-numbers-make-so-much-sense.121331/


Ratch

 

[vlad777]

You make a point, and reveal that I didn't read it all, but it has 7 pages.
(I'm sorry, but it's a common sin.)
Witch page do you think is most relevant?

 

[Ratchit]

vlad777,

The first three posts of page #1.

Ratch

 

[MrAl]

The main point is that operation in maths is defined as a function, and a function is defined
as a relation that doesn't give two results for one input.

So for root function new definitions that build on these and complex set allow root to have two results.

I think this has something to do with this:

Equation a+b=c defines two operations + and - that are closed in integer set.
Equation a*b=c defines two operations * and / that are closed in rational set.

But equation a^b=c defines three operations: exponentiation, root and logarithm so it's not a field or ring...

(Another question: are ^, root, log closed in complex set?)

Hello there,


This kind of question comes up a lot and i see this really a lot on the web all over the place.

The problem here stems from the way they teach math in school, or rather the way they have to teach it.

When we learn something new we usually learn the basics first, then start to expand on that with later years of study. That's because we cant learn every single thing about something right away sometimes because it is too far reaching to do that. First we learn the alphabet, then words, then pronunciation etc., but we dont learn every single word on the first day. It takes small steps to learn anything, so what happens is that we are usually taught one small step at a time, and unfortunately sometimes these smaller steps are a bit misleading.

And then there is the matter of how we speak. Sometimes we speak in a more causal way which has a way of encompassing things in groups that dont belong in groups or glancing over the little details. What is happening here is in the past you have glanced over the details due to the manner of teaching that was presented to you and now you are questioning that because you came across a contraction. This is typical so no reason for real alarm.

In this example you were told that a function can only have one y value for any one x value. Strictly speaking, this is 100 percent true. However, there are times when we stretch this definition to include some of the more common functions that we encounter that actually have two values for one x. Even though strictly speaking this is not a function, we might still think of it as a function because even though it has more than one value the number is still limited, and in the case here it is limited to two.

Another example is the circle. We often define the circle as:
R^2=x^2+y^2

but if this is solved for y we find we have to use two different functions to make a circle because the circle has two values for each x except at two distinct points. So we have:
y^2=R^2-x^2

and in order to solve this we have to take the square root of both sides and get:
y=sqrt(R^2-x^2)

but clearly this cant be the end of it because this means y would have to be positive always and that means we cant construct the part of the circle that is under the x axis, so we have to modify it to:
y=+/- sqrt(R^2-x^2)

and so we end up with two values of y, one positive and one negative, for each value of x.

This is really the learning procedure in action. We first learn some things, then later we learn how to modify them to fit a wider class of applications. Now that you are learning that a square root can possibly have two different values, you are starting to modify your old definitions to encompass a wider range of applications.

This isnt the last time this kind of thing is going to happen. You'll next want to look into how to find the square root of a negative number. Doing that you'll get into more theory and that will expand your understanding of what you can do with math. Before this you could only do positive square roots, now you'll be able to do much more as well as enter into the world of complex analysis where some more complicated problems can be simplified simply by knowing how to use complex numbers.

You'll run into the number which is called "j" or in some texts "i", which is the square root of minus 1.
Without going into this too deep right now, you can often think of 'j' as a number which can not actually be calculated but acts as a placeholder for the algebraic expression such as:
y=1+2*j

Note that if we look at this as a regular algebraic expression, we can not solve it directly because we dont know what j is really. If j was 3, we would have the answer: 7, and if j was 4 we would have the answer: 9, but j is not a number which is directly calculable so we have to leave that expression as is. When we do this we call the '1' the "real part" and the '2' the "imaginary part". I dont want to get too much into this right now as you have a lot to think about already.

 

[vlad777]

Number in this question doesn't have imaginary (orthogonal) part,
it is only that root operation can be executed in real set or complex set.

---
I read things posted in this tread and its basic knowledge on complex numbers,
although I didn't know that about root of i , which can explain my other interests.
I will look at it in more detail later.

I think that this lecture called "linear algebra" , in your country you call abstract algebra.

Hello MrAl.
I plotted my first circle on pc screen 13 years ago.
Little later I figured out its not strictly a function.
I was not a good student, but I have genuine interests that I do and will pursue (slowly ) .

 

[3v0]

Let me try again.

First all real numbers are complex numbers. This unifies things.

We simply ignore the negative root when it is does not fit what we are modeling with the math.

The complex number a + bi can be identified with the point (a, b). A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number.

so 6 = 6+0i

and in general

n = n + 0i

 

[vlad777]

Let me try again.

First all real numbers are complex numbers. This unifies things.

We simply ignore the negative root when it is does not fit what we are modeling with the math.



so 6 = 6+0i

and in general

n = n + 0i

I understand this completely.

So you don't think there's any truth to what I'm proposing,
and everything important is already said.

 

[3v0]

That is what I think and recall from my uni days with a little help brushing up via the web. But do not let that stop you.

I just want to note that is a bit off topic: A lot of people go nuts with numbers and do not relate it back to the problem they are solving, what they are modeling. They come up with nonsense answers that had the preserved the units and did some thinking they would see are clearly wrong.

 

[Hayato]

The main point is that operation in maths is defined as a function, and a function is defined
as a relation that doesn't give two results for one input.

The statement above is correct, and it is true for quadratic functions.

A function does not give different results for a input. But it doesn't mean that it cannot give the same results for different inputs.

Let f(x) be our function.
Being f(x) a function, it must be stated that:
For a particular 'x' value, f(x) can only and only assume a value f(x). That means that, for this same particular 'x' you cannot have f(x), f2(x), fn(x). Ok?

Nothing impeaches that for a set where X = {x1, x2,...,xn}, tou can have f(x) = {y1, y2,....,yn}, where any of those y are the same.

 

[Hayato]

Just o illustrate:
Not Function:
View attachment 66491
(X can result into 2 or more different values)

Function:
View attachment 66492
(Different X's can result into equal values)