Imaginary Number Paradox

[Quantum Mechanic]

I've glanced over the .33333......=1/3 type threads and I thought I'd post a math riddle slightly more interesting.

Here it goes:

-1=-1

sqrt(-1)=sqrt(-1) [sqrt=square root]

sqrt(-1/1)=sqrt(1/-1) since -1/1 = 1/-1

sqrt(-1)/sqrt(1)=sqrt(1)/sqrt(-1)

And so we cross-multiply and get:

sqrt(-1)sqrt(-1)=sqrt(1)sqrt(1)

-1=1

Edited March 21, 2005 by Quantum Mechanic

[EC]

In the third step your essentially saying i = 1/i which isn't true. I think that's your mistake.

[Quantum Mechanic]

In the third step your essentially saying i = 1/i which isn't true. I think that's your mistake.

Not my mistake; that's the crux of this problem.

But for those not familiar with i, it appears completely valid.

Edited March 21, 2005 by Quantum Mechanic

[slave]

sqrt(-1/1)=sqrt(1/-1) since -1/1 = 1/-1

sqrt(-1)/sqrt(1)=sqrt(1)/sqrt(-1)

You said -1/1 = 1/-1. You never said j = 1/j or j=j**-1

[realitycheck44]

3. sqrt(-1/1)=sqrt(1/-1) since -1/1 = 1/-1

4. sqrt(-1)/sqrt(1)=sqrt(1)/sqrt(-1)

This is the invalid step.  Separating sqrt(ab) into sqrt(a) times sqrt( is only valid if both (a) and ( are positive real numbers.

If you have time: Why?

Thanks in advance.

Zak

PS: Did your problem have to do with proving i* = 1? (*=squared)

[EC]

realitycheck44-- The answer to your question is the one I gave earlier, i.e., that i does not equal 1/i so the quantity under the radical cannot be seperated with complex numbers. By the way, i* = -1 not 1 -- [sqrt(-1)][sqrt(-1)] = -1

[TomL]

Yea, the associative property of the square root function only applies to real numbers. Sorry.

[punk]

I've glanced over the .33333......=1/3 type threads and I thought I'd post a math riddle slightly more interesting.

Here it goes:

-1=-1

sqrt(-1)=sqrt(-1)                  [sqrt=square root]

sqrt(-1/1)=sqrt(1/-1)          since -1/1 = 1/-1

sqrt(-1)/sqrt(1)=sqrt(1)/sqrt(-1)

And so we cross-multiply and get:

sqrt(-1)sqrt(-1)=sqrt(1)sqrt(1)

-1=1

Incidentally,

sqrt(-1) = +/- i

Or:

if y^2 = x, then y = +/- sqrt(x)

So you really should have written:

-1 = -1

sqrt(-1) = +/- sqrt(-1)

...

no problem here.

So, you should really write it as:

-1 = -1

i*i = i*i , (-i)*(-i) = (-i)*(-i) , (-i)*(-i) = i*i

i = i , -i = -i, -(-i) = i

So no problem here.

Remember, numbers typically have two roots, so you have to be careful taking the square root of both sides of an equation.

[Hal]

3 = 3

sqr_root(9) = sqr_root(9)

3 = -3

the imaginary number part just adds a bit of complexity: thats all you're essentially doing. It's purely convention that sqr_root(a) is assumed to be the positive root when a is positive, but that convention doesnt extend to complex numbers.

Edited March 22, 2005 by Hal

[Hal]

post deleted: I wasnt thinking properly.

Edited March 22, 2005 by Hal

[plaintext]

I never studied imaginary numbers. I have a question about negative numbers. Are they "real"? A negative number is a positive number that I am looking at upside down. I mean, like height and depth. A depth of 1 unit is a height of -1 units.

Edited March 22, 2005 by plaintext

[punk]

3 = 3

sqr_root(9) = sqr_root(9)

3 = -3

the imaginary number part just adds a bit of complexity: thats all you're essentially doing. It's purely convention that sqr_root(a) is assumed to be the positive root when a is positive, but that convention doesnt extend to complex numbers.

No, no, no

Complex numbers are not treated any differently than real numbers, or positive real numbers.

The issue here has nothing to do with not understanding complex numbers. The issue is not understanding the square root.

The square root is NOT a function (a function being a mapping from a number to ONLY one other number). The square root is a mapping from a number to TWO other numbers.

It is no a "convention" but rather an abuse of notation to write something like:

sqrt(4) = 2

Rigorously we should write something like:

sqrt(4) = {2,-2}

Where {x,y} is the set containing just the number x and y.

Books have a very bad habit of putting in abuses of notation like:

sqrt(-1) = i

The correct definition of i is:

i^2 = -1 (read "i squared equals -1")

And as above:

sqrt(-1) = {i,-i}

The original "paradox" above stems from misusing the square root, and cleverly selecting only one of the roots. We might as well abbreviate it (and make it more general) as:

sqrt(x^2) = x

sqrt(x^2) = -x

x = sqrt(x^2) = -x

x = -x

Again, the problem has nothing to do with "i". But since people are intimidated by the notion of "i" they are more likely to miss the real problem.

[HaloNoble6]

Rigorously we should write something like:

sqrt(4) = {2,-2}

Where {x,y} is the set containing just the number x and y.

Actually, your expression above sets a scalar quantity equal to a set of scalar quantities. You did it a few times, but it should be something like this:

Which means the scalar expression sqrt(4) "can take on" any of the two values in the set {2,-2}, and not that it's equal to this set.

[HaloNoble6]

Right, though my mathematical statement itself is not correct (I'm not setting an element equal to a set), my explanation is. I should've just written .

[punk]

Actually, your expression above sets a scalar quantity equal to a set of scalar quantities.  You did it a few times, but it should be something like this:

Which means the scalar expression sqrt(4) "can take on" any of the two values in the set {2,-2}, and not that it's equal to this set.

No, you are still thinking of the square root as a conventional function (i.e. a number valued function).

The square root is a "multivalued function" (in abuse of terminology), but it is really a set valued function. The square root of a number is a set of numbers. The square root isn't one or the other, but both.

To say:

sqrt(4) = +/- 2

Is really the same thing as saying:

sqrt(4) = {2,-2}

Think of the square root as a mapping from the set of complex numbers to the set of pairs of complex numbers.

[Nate T.]

Extracting square roots was originally invoked as a way to solve equations which look like "x^2 = a," for x. The solution set of this equation is the set of all x which satisfy this equation. Hence, if you want "the square root of a" to mean a number and not a solution set, you have to specify (through convention) which solution to the equation is meant. Mathematicians choose the positive root and express other roots in terms of that one.

Given that the squaring function is a two-to-one function (except at zero), it has no inverse, and so there is no "square root" function unless one restricts the domain of the squaring function so that it is one-to-one. People usually do this by restricting the domain of the squaring function to nonnegative values (which amounts to choosing the positive solution above).

So no more of this "numbers are equal to sets" business.

[Optimizer]

I never studied imaginary numbers. I have a question about negative numbers. Are they "real"? A negative number is a positive number that I am looking at upside down. I mean, like height and depth. A depth of 1 unit is a height of -1 units.

It sounds like you don't mean to be philosophical, so the answer would be: "Yes, negative numbers are real numbers". "...a positive number that I am looking at upside down..." seems a bit clumsy in describing it, though. I would suggest a similar, but alternative, visualization - that real numbers have direction. Every (American) football fan, for example, recognizes that "positive yards" means "going forward" and "negative yards" means "going backwards".

Going further with this, let's recognize that a complex number is essentially two-dimensional - it has both a real and imaginary component. In a more abstract treatment, they are actually treated as "ordered pairs", if I'm not mistaken.

In practice, I think the radical symbol is always used to denote the positive square root. We can use the "positive (or negative) square root" if we want to define a function, so let's not get hung up about that sort of thing. Obviously, the negative square root is denoted by putting a negative out front, and the "+/-" is used when it is ambiguous. Aside from the basic i=sqrt(-1), I think it's unusual to apply the radical symbol to a complex number - probably either because of the problem we see with this paradox, or because of the two-dimensional nature involved (or perhaps these two are the same thing). The exponential form is probably used instead. After all, what does "positive square root" mean when you're talking about complex numbers?

This is a good paradox - it really makes you think about the basics you thought you already knew.

[punk]

Extracting square roots was originally invoked as a way to solve equations which look like "x^2 = a," for x.  The solution set of this equation is the set of all x which satisfy this equation.  Hence, if you want "the square root of a" to mean a number and not a solution set, you have to specify (through convention) which solution to the equation is meant.  Mathematicians choose the positive root and express other roots in terms of that one.

Given that the squaring function is a two-to-one function (except at zero), it has no inverse, and so there is no "square root" function unless one restricts the domain of the squaring function so that it is one-to-one.  People usually do this by restricting the domain of the squaring function to nonnegative values (which amounts to choosing the positive solution above).

So no more of this "numbers are equal to sets" business. 

No the squaring function does not have an inverse which is a *function mapping into the set of numbers*. But it does have a perfectly definable inverse.

We can either say the inverse is a mapping which maps numbers into several numbers.

Or if we include sets of numbers in the space in which we are working, then we can define an inverse which is a *function mapping into the set of pairs of numbers*.

I happen to perfer the latter formulation trying to keep functions and defining the set of things comprising the domain and range of the functions more broadly. If you want to keep the domain and range in terms of numbers in the more conventional sense, then you can just as well include mappings which are not functions.

It is 6 of one a half dozen of the other.

It is really a personal question of how you want to define the arithmetic system you want to work in.

But again, the square function has a perfectly definable inverse in either case.

Conventions exist for convenience and nothing else. They have no bearing at all on the rigorous system, or how the system works. They should not be included in a rigorous argument unless everyone agrees to allow them.

[Nate T.]

We can either say the inverse is a mapping which maps numbers into several numbers.

Depending on where you look in mathematical literature, a "mapping" is either a function or a continuous function. In either case, by the definition of a function, you cannot have a function which maps one number into "several numbers." Such a "function" mapping one number into many is called a relation.

Or if we include sets of numbers in the space in which we are working, then we can define an inverse which is a *function mapping into the set of pairs of numbers*.

You could do this, but in practice no one ever does. If you'd like to do it, I suppose that's okay, but you yourself would have to extend the arithmetical rules operating on these pairs of real numbers, since I don't know of anyone else who has done it.

One problem with this is that if you wanted to extract roots of roots, you would have pairs of pairs of real numbers, and so on. Since you have to account for every possible operation in this way (i.e., you want the algebraic operation to be closed), you would have to extend the real numbers to a set which is notationally nightmarish.

[Hal]

The issue here has nothing to do with not understanding complex numbers.  The issue is not understanding the square root.

The square root is NOT a function (a function being a mapping from a number to ONLY one other number).  The square root is a mapping from a number to TWO other numbers.

It is not a "convention"

Yes it is.

root(4) is often used as a conventional shorthand for 4^(1/2) which is defined as exp(1/2 log(4)), which is well-defined and equal to 2, not -2. Yes a number always has 2 square roots, but its an often used convention to treat root(x) as referring to the positive one.

Edited March 23, 2005 by Hal

[Hal]

But again, the square function has a perfectly definable inverse in either case.

Only if you restrict the domain.

Edited March 23, 2005 by Hal

[Hal]

I never studied imaginary numbers. I have a question about negative numbers. Are they "real"? A negative number is a positive number that I am looking at upside down. I mean, like height and depth. A depth of 1 unit is a height of -1 units.

Mathematically speaking, complex numbers are defined as pairs of real numbers - you dont actually need to talk about "square roots of -1" or whatever, although it is conceptually simpler to view them that way. If you have 2 + 3i, then this would be written (2,3), 3 + 5i would be written (3,5), and so on. Real numbers are then defined to be complex numbers with no imaginary part, for instance 5 would be written 5 + 0i, or (5,0). The imaginary number i would hence be (0,1). The operations of complex arithmetic are then defined on these pairs.

I suspect that isnt what you mean though. Whether imaginary numbers are real simply depends on how you are defining 'real'. You cant have 5i apples for instance. What is your criteria for judging something to be 'real'?

Edited March 23, 2005 by Hal

[Bryan]

Mathematically speaking, complex numbers are defined as pairs of real numbers - you dont actually need to talk about "square roots of -1" or whatever, although it is conceptually simpler to view them that way. If you have 2 + 3i, then this would be written (2,3), 3 + 5i would be written (3,5), and so on. Real numbers are then defined to be complex numbers with no imaginary part, for instance 5 would be written 5 + 0i, or (5,0). The imaginary number i would hence be (0,1). The operations of complex arithmetic are then defined on these pairs.

After being exposed to imaginary/complex numbers several times over my educational career, they never conceptually "clicked" until I started taking electrical engineering courses.

The easiest way for me to conceptualize complex numbers is to view them as two-dimensional vectors (a number with a magnitude and a direction). You can think of the pairs of numbers as "legs" of a right triangle, the real part being the horizontal leg, and the imaginary part being the vertical leg. The magnitude of the vector is simply the hypotenuse of the triangle. The sign of each number in the pair will define the vector’s direction.

[Hal]

After being exposed to imaginary/complex numbers several times over my educational career, they never conceptually "clicked" until I started taking electrical engineering courses. 

The easiest way for me to conceptualize complex numbers is to view them as two-dimensional vectors (a number with a magnitude and a direction).   You can think of the pairs of numbers as "legs" of a right triangle, the real part being the horizontal leg, and the imaginary part being the vertical leg.  The magnitude of the vector is simply the hypotenuse of the triangle.  The sign of each number in the pair will define the vector’s direction.

I agree; I dont think most people really understand complex numbers until they are introduced to the geometry of the complex plane (I certainly never). I think that historically this was how things occurred too; people generally became less suspicious towards complex numbers once the Argand diagram was introduced and the beauty of the theory started to shine through. Edited March 23, 2005 by Hal

[punk]

Depending on where you look in mathematical literature, a "mapping" is either a function or a continuous function.  In either case, by the definition of a function, you cannot have a function which maps one number into "several numbers."  Such a "function" mapping one number into many is called a relation.

You could do this, but in practice no one ever does.  If you'd like to do it, I suppose that's okay, but you yourself would have to extend the arithmetical rules operating on these pairs of real numbers, since I don't know of anyone else who has done it.

One problem with this is that if you wanted to extract roots of roots, you would have pairs of pairs of real numbers, and so on.  Since you have to account for every possible operation in this way (i.e., you want the algebraic operation to be closed), you would have to extend the real numbers to a set which is notationally nightmarish.

If you go to Category Theory, a "mapping" is used in the sense I am using it above. It is nothing more than a set of arrows from a domain set to a codomain (range) set.

If I am mistaken in this definition of "mapping" then there is another term I should be using. The point is we are taking something more primitive than a function which can simply be represented by arrows from a domain set to a codomain set. Some of these "mappings" will be functions, most wont.

This is all perfectly rigorously definable.