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What are imaginary numbers?

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Re: What are imaginary numbers?

#71

Post by alok-g »

ColinWright wrote: post #60
alok-g wrote: post #53

Earlier quoted context omitted.

Here is what I am confused about with regards to diagonalisation: 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 since the way these are iteratively generated includes all possibilities. This is also an enumerable set by definition. Let's now reverse the bits and put them after a decimal. These are just real num…

> 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. >…

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, we cover both zero and one together with all combinations for the first (n) bits. As n -> infinite, all possibilities get covered.

The question is if 111111111... is also there in this set. But isn't it there too?

Re: What are imaginary numbers?

#72

Post by alok-g »

pndmnm wrote: post #62
alok-g wrote: post #53

Earlier quoted context omitted.

Here is what I am confused about with regards to diagonalisation: 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 since the way these are iteratively generated includes all possibilities. This is also an enumerable set by definition. Let's now reverse the bits and put them after a decimal. These are just real num…

> This set now includes all possible bit strings of infinite length since the way these are iteratively generated includes all possibilities. That's not correct -- in fact, it contains no infinite bit strings at all (to prove this to yourself, ask at what position the first infinite string appears).

Please see my response to Colin. Sorry for me not having this stated right the first time.

Re: What are imaginary numbers?

#74

Post by psykotic »

Jun8 wrote: post #52
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)…

"A complex "number" (don't think of it as of a number! ..." Excellent point! Another example of misleading mathematical terminology is "random variable", which are not random or variable but instead are well-defined mappings.

> Another example of misleading mathematical terminology is "random variable", which are not random or variable but instead are well-defined mappings.

Indeed. Kolmogorov's random variables are a special case of observables in mathematical physics.

In classical physics, the state space for a free particle is S = R^3 x R^3. As an observable we might take the particle's velocity in some particular direction, which defines a mapping f : S -> R. We want such mappings to respect the relevant structure of the state space. A classical state space usually has the structure of a smooth manifold, so the mappings should be smooth as well.

In classical stochastic physics, the states now have the structure of a probability space. We want the mappings to be measurable so we can take the preimage of a measurable set of observable values to calculate its probability. This is exactly the Kolmogorov definition of a random variable.

Quantum theory doesn't quite fit into the above scheme, but there are several ways these three cases can be unified. For example, from the viewpoint of C* algebras of observables, classical systems have commutative algebras and quantum systems have noncommutative algebras.

By the way, there is a connection to monads that I find enlightening. With monads, there is both an internal semantics and an external semantics. From the internal point of view, random variables are indeed random and variable. From the external point of view, they are neither. (This can be given a Bayesian spin by thinking of the internal and external observers as someone with respectively imperfect and perfect knowledge.) This is analogous to someone's perspective on the state monad from the internal and external points of view. From the viewpoint of someone who lives in the state monad, the same expression can return different values depending on context. But from an outsider's explicit state-passing point of view, everything is referentially transparent.

Re: What are imaginary numbers?

#75

Post by alok-g »

ColinWright wrote: post #60
alok-g wrote: post #53

Earlier quoted context omitted.

Here is what I am confused about with regards to diagonalisation: 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 since the way these are iteratively generated includes all possibilities. This is also an enumerable set by definition. Let's now reverse the bits and put them after a decimal. These are just real num…

> 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?

Re: What are imaginary numbers?

#76

Post by btilly »

diminish wrote: post #63
yequalsx wrote: post #59

Earlier quoted context omitted.

Note that 2575.97 also has the property that raised to the ith power gives i. There are infinitely many such numbers.

or 0.008983291.. Are there countably infinite solutions or otherwise? See infinity discussion few days ago at HN http://news.ycombinator.com/item?id=4526049 PS: Just curious, i am not a mathematician.

It is countably infinite. Let's try to find them all.

A relatively simple way to understand this is that i^i = e^(i * log(i)) for every possible log of i. So all we need to do is understand what values log(i) could have (there are actually many), and then we can work it out. But log(z) just undoes e^z, so we need to understand e^z.

Now let's work backwards. If z = x + y i with x and y real, then x tells us the absolute value of e^z and y tells us the angle. The absolute value of i is 1, so any possible solution to log(i) has real part 0. The angle that we want to wind up with is 90 degrees, or pi/2. Therefore y can be ..., -3.5 pi, -1.5 pi, .5 pi, 2.5 pi, 4.5 pi, ... .

Therefore log(i) has to be one of 1.5 pi i, -.5 pi i, -2.5 pi i, -4.5 pi i, ... .

Now i^i is e^(i log(i)) so it can be any of ..., e^(3.5 pi), e^(1.5 pi), e^(-.5 pi), e^(-2.5 pi), e^(-4.5 pi), ... .

Unless I've made a trivial calculation error, that is the whole list.

Re: What are imaginary numbers?

#77

Post by alok-g »

esrauch wrote: post #68
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…

Regarding 4, don't believe anyone who claims it's an intuitive result; it is simply a properly of real numbers that you cannot have nonzero infinitesimals and that any two distinct numbers have a number between them that is not equal to either (infinitely many in fact). You can construct alternate number lines that do allow nonzero infinitesimals and then .9999... actually is not equal to 1 under that number line; th…

Do you know of something that I can read further on this? I wonder about the impact this may have on mathematics.

I did not know of this property of real numbers of nonzero infinitesimals. (Possibly there is where my confusion is starting from.)

Re: What are imaginary numbers?

#78

Post by jerf »

JadeNB wrote: post #8
T-hawk wrote: post #4

Here is an even better discussion on the same topic, and the HN thread from last year: http://betterexplained.com/articles/a-visual-intuitive-guide... https://news.ycombinator.com/item?id=2712575 One great conclusion from this approach is how intuitive it becomes to understand the square root of i . I always thought you'd need another dimension to describe that, and another dimension for the square root of that unit,…

> We've arrived back at the axiom that all numbers have two square roots of opposite signs. While I like the geometric content of the rest of your post, I have to quibble on two points: * that a number has two (usually distinct) square roots is a theorem, not an axiom. That is, it's not 'built in' (unlike, say, commutativity of addition), but rather can be proven on the basis of 'built-in' properties. * while one may…

In higher maths, commutativity of addition can be proved from the Peano axioms without taking it as an axiom. It's a common homework problem.

Re: What are imaginary numbers?

#79

Post by ColinWright »

alok-g wrote: post #71
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. >…

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 you want to talk about an infinite "decimal" string, you need to talk about the things that come after the decimal point, in order. As such, they come in order, and you can't have infinitely many zeros and then a finite string on the end.

  > The string is still enumerable since this is just binary
  > encoding mapping to the set {0, 1, 2, 3, ...}
You need to be more careful about how you actually define the strings. Strings have a start, then they go on one place by one place.

  > The question still is if it covers all possible bit strings of
  > infinite length.
Well, you haven't actually properly defined strings, but even so, no. Everything you have starts with a zero.

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

  > For (n+1)th place, we cover both zero and one
  > together with all combinations for the first (n) bits.
  > As n -> infinite, all possibilities get covered.
No, because as soon as you have a one in your expansion the string is finite, so not all possibilities are covered.

  > The question is if 111111111... is also there in this set.
  > But isn't it there too?
You defined the set - tell me where it is. Even leaving alone the fact that these aren't proper strings, it doesn't appear to be there.

Re: What are imaginary numbers?

#80

Post by alok-g »

shou4577 wrote: post #56
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…

I can try: 1. Matrix multiplication is indeed defined oddly. To understand why, it can help to think of matrices as "linear transformations". For example, the matrix | 0 -1 | | 1 0 | corresponds to counterclockwise rotation of vectors by 90 degrees (go ahead, try multiplying the vector by this matrix - the answer should be ). Let's call this matrix A. Also, the matrix |-1 0 | | 0 1 | corresponds to reflection over th…

>> 0.333... = 1/3

Indeed, I was aware of this. And as you said, it does not really answer since it just leads to the same question for 0.3333...

I understand the piece on calculus too. My maths background is mostly related to engineering mathematics, and I have generally excelled at it. I later discovered that I am not clear on some "pure" mathematics things. Hypothetically, if counting were taught to me in adulthood, I may have asked a lot of relevant questions that I did not ask in my pre-school and later lost them without knowing them. I am rediscovering some of these questions now.

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