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What are imaginary numbers?

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Re: What are imaginary numbers?

#81

Post by ColinWright »

alok-g wrote: post #75
ColinWright wrote: post #60

Earlier quoted context omitted.

> Start with binary non-negative integers: > 000 001 010 011 100 101 ... (goes to infinity) > This set now includes all possible bit strings of > infinite length No, it only contains the strings of finite length. There are infinitely many of them, but each one stops after a while. In particular, then n^th one only has log2(n) places before it then becomes all 0s. > This is also an enumerable set by definition. Yes. >…

Edit: This comment should be read after my comment below this. It shows up first on HN. If infinity was in the original set, what would diagonalisation produce? [Genuinely asking, I am unclear on this.] I am flipping all the bits along a diagonal and they are all zeros before flipping. If 0.99999... = 1, then using my argument of flipping the bits around the decimal, wouldn't infinity be in the set?

  > If infinity was in the original set,
What does this mean? Your complete imprecision is making it impossible to answer the questions, despite wanting to help, because they don't make sense.

What set? Define it clearly. Don't talk about infinite strings of zeros followed by stuff, because in the context of decimal or binary expansions that doesn't make sense. Strings have a start, then they go on one place at a time.,

  > I am flipping all the bits along a diagonal and they
  > are all zeros before flipping.
If they are all zero before flipping then 0.11111... isn't there. You've stated that the n^th number has 0 in the n^th place. That means 0.11111... is not the n^th number for any n.

If 0.99999... = 1, then using my argument of flipping the bits around the decimal, wouldn't infinity be in the set?

Re: What are imaginary numbers?

#82

Post by ColinWright »

alok-g wrote: post #67
ColinWright wrote: post #55

Earlier quoted context omitted.

I don't have time now to provide complete answers of the type you want, but I can give you a little insight. > Matrices, especially matrix multiplication. Consider a transformation of space, specifically a shear, or a rotation, or an expansion (or contraction). All of these leave the origin unmoved, and a line will always become a line. They are what we call linear transformations. So given a point (x,y,z) in 3D spac…

Thanks for a detailed explanation. The one on matrices helps too even though others also posted the same idea. >> We have a universe of discourse that has a bunch of atomic things. A "set" is then a collection of things This is where I get stuck. In set theory, they define numbers (and addition) starting from a set and element-of operator. I cannot imagine a bunch of atomic things without imagining numbers/counting f…

  >> We have a universe of discourse that has a bunch
  >> of atomic things. A "set" is then a collection of things

  > In set theory, they define numbers (and addition)
  > starting from a set and element-of operator. I cannot
  > imagine a bunch of atomic things without imagining
  > numbers/counting first.
The point is that you can start with nothing, and define the set that has nothing in it. That's {}. Now we have one thing. We can define a set containing that, so we have { {} }. Now we have two things, and we can definea set containing both of them, and so on.

This is what happens when people want to construct a model of numbers and arithmetic using set theory as the basis. It proves that we can useset theory as the foundation. It doesn't mean it's a sensible thing to do in real life- it's a lot like programming in machine code.

  > So it seems to me that unless set and element-of are left
  > undefined (as others here have suggested), natural numbers
  > are more fundamental than sets.
What are numbers? What is your model for numbers? What is "723"?

  >> 0,9999... = 1

  > What I am getting is that the "limit" is implicit in the
  > statement above even though often unstated. The
  > ellipsis is what signifies the limit there, being otherwise
  > not mathematically defined.
So what is your question? You seem to be saying that when you write down "0.99999..." that is intended to represent the limit of the sequence 0.9, 0.99, 0.999, ... Define what you mean by limit. Once you make a careful definition of "limit" wou find that the limit of the above sequence is 1.

  >> Call that collection A

  > I get the precise definition of sack B. I am missing it for sack A.
Sack A has a cube with the number 1 written on it. And it has a cube with the number 2 written on it. And it has a cube with the number 3 written on it. And it has a cube with the number 4 written on it. And it has a cube with the number 5 written on it. And so on.

  > I understand what happens on a given cube,
I assume you mean a cube in sack B.

  > ... and that cubes in sack A are numbered 1 though infinity.
You can't say that with precision, because it is infinity you are struggling with. What you can say is that for every number n there is a cube with n on it, and that every cube has exactly one number written on it.

This level of detail matters.

  > Is there anything to say about the numbers written on
  > two different cubes in sack A?
Yes - they're different.

  > Basically what you are saying is that sac A cannot have
  > all possible combinations like sack B is defined to have.
No. I'm saying I have two collections of objects. I have very carefully defined what these objects are. And I'm saying that you cannot pair them off, one-to-one, without having things from sack B left over.

  > I am not sure why that is.
I haven't proved it yet, so I haven't explained why this is so. I have merely claimed it is true to see where your understanding fails so far.

Re: What are imaginary numbers?

#83

Post by lmkg »

stiff wrote: post #49

A complex "number" (don't think of it as of a number! think of it like you would think of a vector, group, ring or any other abstract structure) is just an ordered pair of real numbers that behaves in a certain predefined way when being added to another complex number or multiplied by it. For an introduction, to avoid unnecessary confusion, it is best to write such "numbers" as ordered pairs using the notation: (a,b)…

Understanding a mathematical concept doesn't mean being able to perform computations using it. It means having an intuitive understanding for what it can represent and how to use it, and how to interpret concepts that use it.

Defining complex numbers as a bunch of arbitrary arithmetic operations on tuples lends nearly zero understanding, no matter how good you get at performing that arithmetic. Understanding complex numbers as rotations is much more useful in being able to understand, say, electromagnetism. In fact, I'll go so far as to say it makes the physics easier even if it makes the math more cumbersome.

Teaching mathematics as simple rules for manipulation of symbols is basically like telling only the punchline to a joke. Not even saying the punchline early, but just saying it on its own and providing no context. The fact that math can be (not is) just a bunch of mechanical rules is amazing when those rules can be mapped to complex and abstract phenomenon, and used to provide deeper understanding of them. The mechanical rules, in themselves, are boring. Their boringness is actually what makes them a good punchline, when contrasted with the intricacy of the systems they describe, but without that set-up, they're just boring.

Re: What are imaginary numbers?

#84

Post by agumonkey »

shou4577 wrote: post #37

I have mixed feelings about this explanation. On the one hand, the mathematician inside of me is raging "this is neither specific, nor rigorous!" In my opinion, this definition is as close to useless as a mathematical definition can be, since the person who has this (and only this) will be hard pressed to answer any other questions involving complex numbers. For example, what is multiplication by i+1? Without further…

What I love about it is the by-necessity introduction. Limits of previous concepts drive the next idea, I believe that it's a very efficient and general approach to communicate.

Re: What are imaginary numbers?

#85

Post by alok-g »

ColinWright wrote: post #79
alok-g wrote: post #71

Earlier quoted context omitted.

For the first set, I meant to write: [Prepend each string with infinite zeroes] ...000 ...001 ...010 ...011 ...100 ...101 ...000 Now all bit strings here have infinite length. The set is still enumerable since this is just binary encoding mapping to the set {0, 1, 2, 3, ...} The question still is if it covers all possible bit strings of infinite length. For units place, we covered both zero and one. For (n+1)th place…

> For the first set, I meant to write: > [Prepend each string with infinite zeroes] > ...000 > ...001 > ...010 ... If there are infinitely many zeros on the front, you can't actually append anything. That doesn't end up being well-defined. (Well, actually, there are transfinite ordinals, but that would confuse the issue. It's not what you mean, and it doesn't help) > Now all bit strings here have infinite length. If…

>>> [Prepend each string with infinite zeroes] > ...000 > ...001 > ...010 >> If there are infinitely many zeros on the front, you can't actually append anything.

I am lost. In this first set, I do not have a decimal point anywhere. Why cannot I have an infinitely many zeros to the left of 1. It will still be just one when looked at as a number.

>> Strings have a start, then they go on one place by one place

As far as representing it as a string, I may still start from the right and work towards the left.

I understand your point for the decimal case, infinitely many zeroes on the right of decimal cannot be followed a finite string. My argument does not require this however. (I change ...00000011010 from the first set to .0101100000000... in the second set.)

>> > For units place, we covered both zero and one. >> No, you don't seem to have.

The units place is the rightmost below. Both zero and one are covered.

...000 ...001

Stating the above for the decimal case, let n=1 be the place right after the decimal, n=2 to the right of it, and so on. Now for place n=1, the both zero and one are covered (first two cases below). For n=2 place, again both zero and one are covered for all possible combinations above for n=1 place (first four cases below).

0.000000000000... 0.100000000000... 0.010000000000... 0.110000000000... 0.001000000000... 0.101000000000...

Using the mathematical induction argument, all combinations are covered. This must include 0.1111111111... It sits exactly where (simple) infinity sits in the enumerable set {0, 1, 2, 3, ... }

Re: What are imaginary numbers?

#86

Post by ColinWright »

esrauch wrote: post #66
yequalsx wrote: post #41

Earlier quoted context omitted.

Shoot. I lost sight of the goal. Sorry. a·0 = 0 for all numbers a Thus it is true that –1·0 = 0 Replace 0 on left hand side with (–1 + 1) to get –1·(–1+1) = 0 Now use the distributive property –1·(–1) + (–1)·1 = 0 Keeping in mind that 1·a = a for all numbers gives us –1(–1) + (–1) = 0 Thus –1(–1) is a number such that when added to –1 gives 0. Hence –1(–1) must be the opposite of –1 which is 1.

You are assuming that additive inverse is unique. Also you are assuming that -1(-1) is some value at all; you could write an extremely similar proof that leaves you with 0/0 = 1 and it would be faulty to conclude that 0/0 actually is 1.

  > You are assuming that additive inverse is unique
Suppose that some number a has two additive inverses,x and y.

    a + x = 0 = x + a

    a + y = 0 = y + a
Consider

    x + a + y = (x + a) + y = 0 + y = y
    x + a + y = x + (a + y) = x + 0 = x
Thus x=y, and so the additive inverse is unique.

  > you are assuming that -1(-1) is some value at all;
We can define it to have a value and then derive the properties of the value. We can show it to be consistent by creating an explicit model. Such things are relatively easy to do, but require a level of detail inappropriate for this context.

  > you could write an extremely similar proof that
  > leaves you with 0/0 = 1 and it would be faulty to
  > conclude that 0/0 actually is 1.
Could you present such a proof for us?

Re: What are imaginary numbers?

#87

Post by alok-g »

ColinWright wrote: post #81
alok-g wrote: post #75

Earlier quoted context omitted.

Edit: This comment should be read after my comment below this. It shows up first on HN. If infinity was in the original set, what would diagonalisation produce? [Genuinely asking, I am unclear on this.] I am flipping all the bits along a diagonal and they are all zeros before flipping. If 0.99999... = 1, then using my argument of flipping the bits around the decimal, wouldn't infinity be in the set?

> If infinity was in the original set, What does this mean? Your complete imprecision is making it impossible to answer the questions, despite wanting to help, because they don't make sense. What set? Define it clearly. Don't talk about infinite strings of zeros followed by stuff, because in the context of decimal or binary expansions that doesn't make sense. Strings have a start, then they go on one place at a time.…

>> If they are all zero before flipping then 0.11111... isn't there.

This helps. Since the first set has infinitely many zeros on the left, flipping around the decimal means there would have to be infinitely many zeros somewhere on the right. 0.1111.... however has infinitely many ones before these supposed infinitely many zeroes, which cannot be for the same reason that a finite string with one cannot be after infinitely many zeroes at the right of a decimal.

This means that the first set does not have 111111...

So if I keep incrementing binary numbers, I'll never reach 11111... within the limits of enumerability.

I need to think and read more. The above seems to imply that the simple infinity is not in the enumerable set. But I may be confused again.

>> You've stated that the n^th number has 0 in the n^th place. That means 0.11111... is not the n^th number for any n.

I have been confused about this. Somehow 0.11111... is not in the set, in spite of the mathematical induction proof I supply. I need to think more. The proof must be incomplete (or imprecise as you say).

Re: What are imaginary numbers?

#88

Post by ColinWright »

jostmey wrote: post #18

Why does (-1)*(-1)=(+1) ? It is arbitrary, and there really is no good reason. We could construct number lines that work differently, so that imaginary numbers never appear. Such alternative number lines would still allow us to solve the exact same physics and engineering problems. Sure, the computations would work differently, but the way we would measure and use the initial conditions in our equations would be diff…

This sounds profound, but is wrong on so many levels. In a sense everything about mathematics is arbitrary, but there's a consistency and structure that makes such a statement unhelpful and misleading.

Consider.

If you're content with the counting numbers then we can construct the negative numbers. These have the specific property that when added to the positive number of the same size we get zero,

But most people are happy with the integers, so move on. We're reasonably happy with addition, but what is multiplication? If you think of it as repeated addition you're screwed when you want to multiply by 2 1/2. It's better to think of it as a scaling. Multiplying by 2 means that you scale things up to be twice as big. Thus 1 goes to 2, 2 goes to 4, and 5 goes to 10. Also, -1 goes to -2, -6 goes to -12, and so on.

So what do we mean when we scale by -1? We look at the sequence of scaling by 4, then by 3, then by 2, and so on, each time asking where the number 1 gets sent.

    Scale by 4 and 1 -> 4
    Scale by 3 and 1 -> 3
    Scale by 2 and 1 -> 2
    Scale by 1 and 1 -> 1
    Scale by 0 and 1 -> 0
Following this progression we see that it's natural in some sense to say that scaling by -1 means that 1goes to -1. And indeed, 2 goes to -2, and 73 goes to -73.

Scaling by -1 sends something to the same distance on the other side of zero.

So where does -1 get sent under a scaling of -1? It gets sent the same distance the other side of zero. -1 gets sent to 1.

Therefore it makes sense to say that -1 scaled by -1 is 1.

(-1) * (-1) = 1

Wecan use this to ask about the square root of -1. What geometric operation can we perform on the number line, such that doing it twice is the same as multiplying by -1?

An answer is to rotate anti-clockwise by 90 degrees. Another answer is to rotate clockwise by 90 degrees.

Pursue this, and you start to construct the Agrand diagram, and the complex numbers.

Re: What are imaginary numbers?

#89

Post by Natsu »

alok-g wrote: post #34

Can someone please also do this for: 1. Matrices, especially matrix multiplication. Unlike matrix addition, multiplication is defined in a very weird way. I think I understand where it is coming from -- defining it that way allows representing and solving linear equations. More insights, however, would help. 2. Dot and cross products. E.g., the magnitude of dot product in 3D is a.b.cos(theta), while for cross product…

Why is 0.999... one? It's an ambiguous decimal expansion.

1 - 0.999... = 0.000... Some part of you might think that there "must" be a 1 at the end of all those zeros. The problem is that there is no end at which to put a 1.

And you'd end up with tiny holes everywhere if they're not equal. 1 = 1/3 + 1/3 + 1/3 = 0.333... + 0.333... + 0.333... = 0.999... oops. Where did our missing one go this time? Surely it's clear from this that we have nothing but 9s in that expansion and that three 3s can never be larger than 9, even if you repeat them over and over forever? And there can't be any funny business going on at the end, because infinite lists do not have ends by definition.

At some point you go through the rules and just accept that this is how they play out and that using other rules just leads to weirdness (AKA "nonstandard analysis").

Re: What are imaginary numbers?

#90

Post by alok-g »

ColinWright wrote: post #82
alok-g wrote: post #67

Earlier quoted context omitted.

Thanks for a detailed explanation. The one on matrices helps too even though others also posted the same idea. >> We have a universe of discourse that has a bunch of atomic things. A "set" is then a collection of things This is where I get stuck. In set theory, they define numbers (and addition) starting from a set and element-of operator. I cannot imagine a bunch of atomic things without imagining numbers/counting f…

>> We have a universe of discourse that has a bunch >> of atomic things. A "set" is then a collection of things > In set theory, they define numbers (and addition) > starting from a set and element-of operator. I cannot > imagine a bunch of atomic things without imagining > numbers/counting first. The point is that you can start with nothing, and define the set that has nothing in it. That's {}. Now we have one thing…

>> You can't say that with precision, because it is infinity you are struggling with. What you can say is that for every number n there is a cube with n on it, and that every cube has exactly one number written on it.

This may be nailing it -- struggling with infinity.

When you say "every number n there is a cube with n on it", what does "every" mean. Does every number include infinity? Or should not not consider infinity to be a number? If the latter, this is probably where I went wrong.

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