Complex numbers are cool, but automatic differentiation using dual numbers is even cooler.
What are imaginary numbers?
101–110 of 143 posts
Re: What are imaginary numbers?
#102I 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…
> For example, what is multiplication by i+1? Without further understanding, I don't think such a question can be answered.
i+1 on the plane is a vector at 45 degrees and a scaling of sqrt(2). So multiplying by it scales your multiplier and rotates it 45 degrees. This is a direct extension of the intuitive understanding given.
Re: What are imaginary numbers?
#103A 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)…
I saw, and experienced, this approach in school and see no value in it for improving your understanding complex numbers. The point of the approach is to teach students about abstraction and formalism, and complex numbers happen to be a convenient place to do it. But the formalism is a barrier for building up a more convenient mental model about what is really going on. Before disputing this, in any calculation that y…
> But the formalism is a barrier for building up a more convenient mental model about what is really going on.
Actually, I'd say that complex numbers are a wonderful opportunity for illustrating that math is nothing but a mental model, not necessarily with anything "really going on", AND at the same time incredibly useful.
IMO the best way to think about complex numbers is to start with the observation that sqrt(-1) cannot exist in the realm of real numbers, quite easily provable. Then you make the bold-assed assumption that it exists anyway and simply call it i. Then play around with it a little, and find out that adding the axiom "there is something called i, where i * i == -1" does not lead to any contradictions; instead you can work with this i just fine, and numbers of the form a+bi are closed under all your everyday operations, just by applying your everyday arithmetic rules. Neat!
Then you notice that those numbers can be interpreted as 2D vectors and some operations on them as geometrical transformations thereof. Super Neat!
Then someone notices that those numbers and the operations on them can actually be used to model certain aspects of the physical world. Holy Crap!
Math is FREAKING AWESOME!
Re: What are imaginary numbers?
#104Why 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 ar…
Beautiful exposition. Even in the late 1700s, many mathematicians rejected the use of mere negative numbers, viewing them as anomalies which indicated that one had phrased a problem wrong to begin with. On the other hand, Euler understood everything very well and even calmly explained how to take logarithms of complex numbers, which bewildered most of his contemporaries.
Someone once said that a lot of confusion could have been avoided if, instead of the terms positive, negative, and imaginary, they had instead used the terms forward, backward, and lateral.
Re: What are imaginary numbers?
#105A 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)…
OK, my main point restated:
A number is whatever we want it to be, whatever it is useful to treat as a number.
That's it. That's the whole gag. Saying something isn't a number is pointless: It is if it can be, and it isn't if it's more interesting to treat it as something else.
This leads to something more fundamental:
Math is all about modeling. Math is a language for making models that are logically consistent.
Confusing a model with the thing being modeled is wrong. Saying complex numbers aren't numbers because they're rotations is going at it backwards: Complex numbers can be used to model rotations. That doesn't tie them to that one model. They can be used in other ways, too.
Re: What are imaginary numbers?
#106A 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)…
I saw, and experienced, this approach in school and see no value in it for improving your understanding complex numbers. The point of the approach is to teach students about abstraction and formalism, and complex numbers happen to be a convenient place to do it. But the formalism is a barrier for building up a more convenient mental model about what is really going on. Before disputing this, in any calculation that y…
"Young man, in mathematics you don't understand things. You just get used to them."
-- John von Neumann
For what it's worth, I have sometimes manipulated complex numbers as (a, b), usually when using them as 2-D vectors. But I make no claims to being a mathematician.
Re: What are imaginary numbers?
#107Earlier quoted context omitted.
> don't think of it as of a number! By the same token, you shouldn't think of a real "number" as a number, but as an infinite sequence of rational numbers. Yes, all of the reals are infinite sequences of rationals. Also, a rational isn't a number but an ordered pair of integers. (Technically, the set of the rationals is the set of all ordered pairs of integers (excluding the ones with zero in the second position) wit…
One minor correction. The integers are more naturally defined as an equivalence class of pairs of natural numbers, with (a, b) equivalent to (c, d) if and only if a+d = b+c. (Your approach does not eliminate the equivalence class issue, because you have to specify that +0 and -0 are the same.) In this formulation, you define (a, b) + (c, d) to be (a+c, b+d). And you define (a, b) * (c, d) to be (a * c + b * d, a * d…
For those who find the above so-called Grothendieck construction somewhat puzzling, here's a little motivation and background. It is the group completion of a monoid. The simplest case is the group completion of the free monoid on one generator. The monoid has as elements all the finite strings (including the empty string) built from the symbol x. Its operation is concatenation. Any such string is uniquely defined by the number of x symbols that occur, so this monoid is isomorphic to the natural numbers.
Its group completion must add an inverse for all elements. We will denote the formal inverse of x by the symbol x' with the relations xx' = x'x = 1 where 1 denotes the empty string. This cancellative concatenation is commutative because xx' and x'x both equal 1. Thus a string can be identified (though not uniquely) by a pair of natural numbers (m,n) that count the number of x and x' occurrences.
To see the connection with the sign representation of integers, add an orientation to the relations xx' = x'x = 1 to get the length-shortening reduction rules xx' -> 1, x'x -> 1. It's then easy to see that every string has a unique normal form that is either empty or consists entirely of x symbols or entirely of x' symbols, corresponding to the cases 0, +n and -n, respectively. For if a string is not of this form it must have at least one x and one x' element. But then there must be at least one adjacent pair of x and x' elements. Hence the string admits a reduction and cannot be a normal form.
Re: What are imaginary numbers?
#108Aside from people's very worthwhile answers describing the complex number system, I think it is worth mentioning that the use of the term "imaginary" is an unfortunate historical remnant. In experience, a lot of the average student confusion comes from their trying to get their head around the naive meaning of imaginary. Now that modern mathematics understands that all number systems are more or less games with axiom…
Those are the mundane ones. Almost all real numbers (except for set of set of measure 0) cannot even be given names, and are thus beyond imagintion.
Re: What are imaginary numbers?
#109Earlier quoted context omitted.
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.)
Look up nonstandard analysis. It is a formalism that can define calculus, but doesn't have any practical difference on physical computations on real world entities.
Re: What are imaginary numbers?
#110Earlier quoted context omitted.
One minor correction. The integers are more naturally defined as an equivalence class of pairs of natural numbers, with (a, b) equivalent to (c, d) if and only if a+d = b+c. (Your approach does not eliminate the equivalence class issue, because you have to specify that +0 and -0 are the same.) In this formulation, you define (a, b) + (c, d) to be (a+c, b+d). And you define (a, b) * (c, d) to be (a * c + b * d, a * d…
For those who find the above so-called Grothendieck construction somewhat puzzling, here's a little motivation and background. It is the group completion of a monoid. The simplest case is the group completion of the free monoid on one generator. The monoid has as elements all the finite strings (including the empty string) built from the symbol x. Its operation is concatenation. Any such string is uniquely defined by…
It should be noted for non-mathematical readers that a group has a single operation called multiplication, and the multiplicative identity is called 1.
The natural numbers form a group with "multiplication" being addition, and "1" being 0. In which case the Grothendieck construction on the natural numbers gives the integers with addition.
If calling the basic operation multiplication instead of addition is confusing, remember that one of the inspirations for group theory are permutations of a set of things, which can be represented by matrices using matrix multiplication to perform the permutations.