why does (g^2 – 1)/(g – 1) = g + 1 ?

Originally posted by @eladar
That was what I was going to point out. They are not equal.

They are different at one point, therefore are not equal.

Exclude that one point then they are equal.


The same kind of screwed up thinking makes people think the square root of x squared equals x.

Expand on that, (X^.5)^2 does not equal X? What does it equal then?

Originally posted by @sonhouse
Expand on that, (X^.5)^2 does not equal X? What does it equal then?

Actually the square root of x then squared is x. The square root has a domain restriction, no negatives.

Squaring first does not have a domain restriction. Squaring first then square rooting results in the absolute value of x.

Originally posted by @eladar
Actually the square root of x then squared is x. The square root has a domain restriction, no negatives.

Squaring first does not have a domain restriction. Squaring first then square rooting results in the absolute value of x.

(-5)^2=25 25^.5=5 so absolute value of -5.

(-5)^.5 does not exist. Same problem for all negatives. So square rooting first only allows positive numbers and 0, those square roots would match the original numbers.

Originally posted by @eladar
(-5)^2=25 25^.5=5 so absolute value of -5.

(-5)^.5 does not exist. Same problem for all negatives. So square rooting first only allows positive numbers and 0, those square roots would match the original numbers.

"Same problem for all negatives." What???
sqrt(-1) = i does indeed exist!
Haven't you been around for the last hundreds of years?

Originally posted by @fabianfnas
"Same problem for all negatives." What???
sqrt(-1) = i does indeed exist!
Haven't you been around for the last hundreds of years?

The imaginary unit is often taken to be equal to the square root of -1, loosely speaking, but they are not strictly equal. i is often defined as i² = -1. See the introductory text to:

https://en.wikipedia.org/wiki/Imaginary_unit

In real space, the square root function is not defined for negative numbers. In complex space, the square root function has a so-called "branch cut" on the negative real axis. The square root for a complex number (not equal to 0) always has two solutions, similar to how the solution to the real equation x² = a² (a real and positive) has the solutions x = +/- a. The two solutions for the square root of -1 are i and -i.

Originally posted by @kazetnagorra
The imaginary unit is often taken to be equal to the square root of -1, loosely speaking, but they are not strictly equal. i is often defined as i² = -1. See the introductory text to:

https://en.wikipedia.org/wiki/Imaginary_unit

In real space, the square root function is not defined for negative numbers. In complex space, the square root function ...[text shortened]... positive) has the solutions x = +/- a. The two solutions for the square root of -1 are i and -i.

You say that sqrt(-1) = i is false?
Then we really have to rewrite a lot of math.

Originally posted by @kazetnagorra
As Eladar correctly points out, there is no paradox.

Off course that ALWAYS must necessarily be true!
If you always have totally flawless reasoning then to you there NEVER is a paradox!
All paradoxes have a solution and the 'solution' to ANY paradox is to show how it doesn't exist!
All paradoxes are illusionary and exist only in the mind and come from some kind of confusion in the reasoning process.
Unless you define 'paradox' as an illusion of a contradiction (which is my preferred way of defining it), no paradoxes exist!

Originally posted by @humy
Off course that ALWAYS must necessarily be true!
If you always have totally flawless reasoning then to you there NEVER is a paradox!
All paradoxes have a solution and the 'solution' to ANY paradox is to show how it doesn't exist!
All paradoxes are illusionary and exist only in the mind and come from some kind of confusion in the reasoning process.
So in one narrow sense, no paradoxes exist!

Agree. Totally.
To believe in a paradox is just to show you don't know enough.

Originally posted by @fabianfnas
You say that sqrt(-1) = i is false?
Then we really have to rewrite a lot of math.

It's not "false," but rather a loose, colloquial way of understanding i. Mathematics, as it is taught in high school, often avoids formal mathematical definitions and derivations in favour of simpler but less rigorous notations.

No math needs to be rewritten since i is not defined as sqrt(-1).

Originally posted by @fabianfnas
"Same problem for all negatives." What???
sqrt(-1) = i does indeed exist!
Haven't you been around for the last hundreds of years?

I is imaginary, which does not exist under the real numbers.

Try graphing the square root function to see what I am trying to explain.

Or imagine the grapg of y=x^2. To find its inverse you need to turn that function 90 degrees clockwise then reflect it anout the x axis. Since a parabola is symmetric, the relecting does nothing. It results in a parabola on its side being split by the positive x axis. The vertex of yhe parabola is still at the origin. This graph is not a function, it does not pass the vertical line test. Or stated another way, if you go to x=25, you will find solutiins at both y=5 and y=-5. To make this relation into a function, somebody said lets ignore all the negative solutions and have people simply realize that the opposite solution also exists.

But in any case, the inverse of x squared when graphed never extends into the second or third quadrants because negative values of x don't exist, why? Because that would mean if you squared a number you'd result in a negative number.

If you can accept thae fact that if you multiply a number by itself the answer can't be negative, then you should understand that you can only take square roots of non negative numbers.

Originally posted by @kazetnagorra
It's not "false," but rather a loose, colloquial way of understanding i. Mathematics, as it is taught in high school, often avoids formal mathematical definitions and derivations in favour of simpler but less rigorous notations.

No math needs to be rewritten since i is not defined as sqrt(-1).

I don't agree with this. Complex algebra is set up in such a way that roots of -1 exist. If you are trying to say that one can only take the square root of negative definite numbers if the real line is regarded as a subset of the complex numbers, so we are doing this operation on -1 + 0i, then consider that one can always specify the square root as a mapping between the set of real numbers and the complex ones. A mapping is not required to be surjective. So I don't think that there is anything non-rigorous about the statement that the square root of minus one is i.

Originally posted by @deepthought
I don't agree with this. Complex algebra is set up in such a way that roots of -1 exist. If you are trying to say that one can only take the square root of negative definite numbers if the real line is regarded as a subset of the complex numbers, so we are doing this operation on -1 + 0i, then consider that one can always specify the square root as a m ...[text shortened]... that there is anything non-rigorous about the statement that the square root of minus one is i.

But you agree that numbers involving i are imaginary and not real.

In the context of real world application such as lengths of sides of triangles, i does not apply.

Originally posted by @deepthought
I don't agree with this. Complex algebra is set up in such a way that roots of -1 exist. If you are trying to say that one can only take the square root of negative definite numbers if the real line is regarded as a subset of the complex numbers, so we are doing this operation on -1 + 0i, then consider that one can always specify the square root as a m ...[text shortened]... that there is anything non-rigorous about the statement that the square root of minus one is i.

I should have said...

If we are talking about real numbers


Real numbers are what most people think of as numbers.

Imaginary numbers were a reaction to the need to represent negative square roots which were not the result from simply squaring a real number.

Originally posted by @eladar
But you agree that numbers involving i are imaginary and not real.

In the context of real world application such as lengths of sides of triangles, i does not apply.

Observable physical quantities are required to be real, yes. Imaginary numbers are useful for describing phases, so there are plenty of applications in electronics and physics, but one always finishes the calculation by taking the real part or finding the magnitude of the complex quantity.

Originally posted by @deepthought
I don't agree with this. Complex algebra is set up in such a way that roots of -1 exist. If you are trying to say that one can only take the square root of negative definite numbers if the real line is regarded as a subset of the complex numbers, so we are doing this operation on -1 + 0i, then consider that one can always specify the square root as a m ...[text shortened]... that there is anything non-rigorous about the statement that the square root of minus one is i.

Thinking about it, the square root is the inverse image of the square function, so in the above I'm assuming some consistent way of choosing a principal value, such as being in the union of the upper half plane, treated as an open subset, and the nonnegative real line, to make sure my mapping isn't one to many.