Originally Posted by Xei
Things are getting messy so I am going to recapitulate again.
We are talking about the identity sqrt(a)sqrt(b) = sqrt(ab) for a, b nonnegative.
This statement is not an axiom of arithmetic; it is proven from the axioms of arithmetic, and the proof only works for when a and b are nonnegative reals. When a and b are not nonnegative reals, the proof does not work and the identity is in fact incorrect.
Therefore, I reiterate my reiteration: there was no creation of a special case. The truth for a and b nonnegative and the falsity otherwise are both deduced from the axioms of arithmetic.
There is nothing wrong with this and it is not unusual.
To help show this, here is another identity:
a^2 = a*a for a nonnegative.
a means the absolute value of a, which means its size irrespective of whether it is positive or negative. You can write it as sqrt(a^2) if you want.
This is exactly analogous. The identity is not an axiom of arithmetic: it is proven from the axioms of arithmetic, and the proof only works when a is nonnegative. If a is negative, the proof fails, and the correct identity is in fact
a^2 = a*a
What your argument says is that we have invented a special case because the identity doesn't work for negative numbers; and therefore, negative numbers are somehow fake. As you don't believe that negative numbers have the same problem as complex numbers, hopefully you now see why your argument doesn't make sense.
I am not disagreeing with your stance that axioms existed and conclusions followed from them. What happened is that axioms existed, and then conclusions based on them were added to by the introduction of imaginary numbers. The conclusion that there is a negative number exception to the radicand product rule is based on the original axioms PLUS the assumption that imaginary numbers exist. I don't think imaginary numbers exist, so I don't agree with the exception rule. However, if I did believe in imaginary numbers, I would agree that the exception is legitimate. There would have to be that exception if imaginary numbers were real.
Originally Posted by Xei
'How do you multiply a mug by something?'; it depends on which abstraction you are using. Again this goes back to my exposition of the second pitfall.
One abstraction you could be using is abstracting from the number of mugs. This abstraction is encapsulated by the natural numbers. You can multiply the mug by 10; the result is 10 mugs.
A different abstraction could be the orientation of the mug. This abstraction is encapsulated by the complex numbers. The centre of the mug is at 0 and the handle is pointing in the direction of 1. If you multiply the mug by i, the handle now points towards i, which is 90 degrees counter clockwise. If you multiply by it by i again, the handle points towards 1, which I'm sure you can see represents a 180 degree turn. Thus we have an entity which squares to give 1.
That's just one example. There are plenty of other examples of complex numbers being in onetoone correspondence with some kind of intuitive physical basis. If you know basic stuff about matrices, you will know that 2*2 matrices represent linear transformations of the plane (rotations, reflections, enlargements, skews), with addition representing the addition of the transformations and multiplication representing one transformation followed by the other. Surely you grant that these are fine. Well, complex numbers are just a special type of linear transformation.
There is also the issue of formal construction which you didn't respond to. Here is the crux of it again: if you grant that real numbers, along with addition, multiplication, and ordered pairs all 'exist', and that anything made out of them 'exists', then you are in fact compelled to concede that complex numbers 'exist', because you can construct the complex numbers using only these objects.
Okay, now I see what you are saying. I think you are saying something along the lines of what Olysseus said. I boldfaced the three comments that highlight where I disagree with you. I agree that you can multiply by imaginary and complex numbers and get imaginary and complex figures. You can multiply coordinates and matrix figures by i and numbers that involve i, and from there you can get other numbers involving i. However, I see that as using fiction to get more fiction. I can multiply 3 by unicorn and get 3 unicorns, but that doesn't mean the product represents something actual. If you throw in a fictitious factor, you can get a fictitious product. You cannot prove the existence of something fictitious when the premise involves something similarly fictitious.
The solutions to equations can be imaginary or complex, but that is only because the numbers are used as hypotheticals. 3i and 3i might be hypothetical x values that would make an equation work, but that does not mean they are actualities. They are fictitious principles that, if actual, would be actual values of x. I don't think x really can be 3i or 3i. There is only the hypothetical scenario that x would be 3i or 3i if there were such thing as i. I can see where you would say that such a situation proves that 3i and 3i are actual in the sense that they work as solutions (which they can) and it proves that they are actual in that way. What I am saying is that (to use an example and illustrate my overall point) such numbers can be used only in a hypothetical sense. The premise that such numbers are actual is what I think is false.
This is a complicated and sticky issue, but I will try to clear up my position with an analogy. Suppose a detective is given information about a crime. He can come to the conclusion that there are three characters who fit as suspects. Now suppose that a hypothetical person who does not exist is one of them. That fictitious person would be a viable suspect if he were real. In that way, he is a "solution" to the crime scenario but not an actual suspect. If the detective assumes the existence of a fictitious character named Orzog, and the defining characteristics of Orzog make him a person who fits the scenario, the detective could say, "There are three suspects Bob Smith, Al Johnson, and Orzog. Orzog fits the scenario just like Bob and Fred, but that does not make him an actuality. He is just a hypothetical that fits the situation. Equations can work the same way. 3i and 3i may be solutions to an equation, but that does not make them actual. They exist only as fictitious concepts that fit scenarios as hypotheticals.
Originally Posted by Olysseus
Don't know if this will help or not...but hope so.
Would it be correct to say that another way to see why the "paradox" 1 = i*i = sqrt(1)*sqrt(1) = sqrt(1*1) = sqrt(1) = 1 is that each number has two real square roots?
Thus the step that is wrong is the step that says sqrt (1) = 1, because it really needs to be written as +/(1). Its really as simple as saying that if x^2 = (16), for example, it is not correct to say the answer is 4, rather 4 is an answer, while the answer is +/ 4.
This way you can resolve the problem without appearing to negate an established rule just to make things fit. (And please note I am using the word "appearing" intentionally here, I am not saying there is anything wrong with other's explanations.)
That was my initial hunch when I came across the general issue, but I don't think that's the resolution. 1 does have two square roots, but it does not mean that if you have 1 = 1 you can get the sqare root of each side and put +/ a the beginning of one side. It only makes sense to do that when a variable is involved. If x^2 = 25, x can be 5 or 5. However, if you have the equation 25 = 25, you cannot get the square root of each side and come to the conclusion that 5 possibly equals 5. You can only come to the conclusions that 5 = 5 and =5 = 5. +/ only makes sense when an unknown is involved.
Originally Posted by Olysseus
Another thing that has helped me explain the "reality" of imaginary numbers to students is the following:
Imagine a number line, what happens if you multiply each number on the line by two? by three?
A: The number line will contract.(What was once 1 will now be 2, what was once 2 will be 4 and so on if you multiply each point by two.)
What happens if you multiply each point on the number line by 1/2?
A: The number line will dilate (what was once 2 will now be 1, what was once 4 will now be 2.)
What happens when you multiply each pint on the number line by (1)?
A: The number line will rotate 180 degrees! (What was once 1 will now be 1)
If multiplying by a negative implies rotating 180 degrees around the number line, what happens if you multiply by the square root of (1)?
A: You will rotate by 90 degrees.
I know this is not a formal proof of anything. But it gives me an intuitive idea that if we think of numbers as transformations, i is as "real" as any other number. The concept of i simply requires us to consider numbers in a new way. Rather than being thought of as static objects, numbers have to be thought of as something else. I would readily concede that i does not correspond to anything "out there" in the world of objects. But it does correspond to a "real" transformative action on the number plane.
I mentioned you where I responded to Xei. What I said there is my response to your above point. You can mulitiply anything by i and potentially get a product with i, but that does not mean i is real in the first place. The symbol is real, and the concept obviously exists because we are talking about it, but it is a fictitious concept.
Originally Posted by khh
I think the reason for this argument is that you have a fundamentally different view on what exactly math is. Seem UM is arguing from the standpoint that math is a natural science, describing the real world, while Xei is arguing from the standpoint that math is a tool that can be used by science. So when Xei says something is "real", he means that it can be used to correctly model occurring phenomenon (which it certainly can: There's a reason all engineers have to learn how to work with complex numbers).
So perhaps which is the correct interpretation of math is what should be argued about? Either that, or I've completely misunderstood what you're trying to achieve.
That is a good point. It goes along with what I said to Xei. I agree that imaginary and complex numbers can be used as good hypotheticals that can help people understand reality, but that is all they are. Hypotheticals can be used in law and other areas, but they are still just hypotheticals.


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