Isn't 4 rotations 4i? Why is it i^4
Rotation is multiplication. Translation is addition.
What are imaginary numbers?
111–120 of 143 posts
Re: What are imaginary numbers?
#112A 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…
I disagree with this for several reasons.
First, I find that having more different ways to look at a problem, the better I am able to deal with it. I can look at an equation algebraically, or as a graph, e.g. I can use rectangular coordinates or polar coordinates. I can look at complex numbers as abstract entities or as points in a plane.
Second, if you look at the history of complex numbers, mathematicians were just not sure what to make of them, and had no way to have confidence that what they were doing was even consistent. Being able to interpret them as point in a plane with intuitive geometric operations gave them a huge boost.
Third, thinking of them this way led to the search for generalizations. Gauss and Hamilton tried to find a way to do arithmetic in three dimensions, or prove that it couldn't be done. Hamilton eventually found the four-dimensional quaternions. And the (ac-bd,bc+ad) definition was generalized to the Cayley-Dickson construction.
Re: What are imaginary numbers?
#113Earlier 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…
>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
Yes, that clears it up totally...
Re: What are imaginary numbers?
#114Earlier quoted context omitted.
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…
> 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 Yes, that clears it up totally...
Hopefully you kept reading since that part was just the nomenclature for those interested in looking up this stuff.
Re: What are imaginary numbers?
#115Earlier quoted context omitted.
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…
I disagree with this for several reasons. First, I find that having more different ways to look at a problem, the better I am able to deal with it. I can look at an equation algebraically, or as a graph, e.g. I can use rectangular coordinates or polar coordinates. I can look at complex numbers as abstract entities or as points in a plane. Second, if you look at the history of complex numbers, mathematicians were just…
And not long afterwards, Clifford generalized real numbers, complex numbers, and quaternions into what are now known as "Clifford algebras". Handy stuff. Certain algebras allow you to express geometric shapes such as points and lines using very simple equations. Other algebras show how quaternions (for example) arise naturally as the even subalgebra of Cl0,3(R). The "spacetime algebra" appears as CL1,3(R), which makes it easy to express special relativity.
Re: What are imaginary numbers?
#116A 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've always viewed complex numbers in the opposite way.
Start with the set of real numbers. Then join to that set the symbol i, with the property that i*i=-1, (probably other stuff in formal math).
Then, out of convenience, treat expressions of the form a+bi as a single number.
Re: What are imaginary numbers?
#117Earlier quoted context omitted.
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.
Additive inverses are unique in any ring. The subtle things I skipped over because of the technicalities involved is how do I know the distributive property holds for negative integers. Indeed, what is a negative integer? How does one get them from the natural numbers? Suffice it to say that this can all be defined in a consistent, precise way and everything works out.
Actually, we only think that it can be described in a consistent way. We have proven it consistent using set theory , but Godwell's theorems tell us that we cannot prove a system to be consistent without using something outside of said system (unless the system is inconsistent). This means that there is some level in our chain of proofs that cannot be proven consistent.
Re: What are imaginary numbers?
#118A 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. Con…
Math isn't about moddeling, moddeling is applied math.
Math is about making up rules and figuring out what stuff happens with said rules.
Re: What are imaginary numbers?
#119A 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 was going to post something like this but you did a much better job of it. In some sense, TLDR: "Imaginary number" is a misnomer that confuses people. 1+2i is also just a convenient but potentially confusing notation to those who don't know what it is a short-hand for. Complex numbers are basically 2D vectors with a funny multiply operation ((a,b) (c,d) = (ac-bd,bc+ad)) and it just so happens that defining it this…
Complex numbers are not vectors. If you have (a + b) in an expression, you can do algebra with it. If you then say b=ci, (a + b) would be a complex number, but the work you just did as if it were 2 real numbers is still valid.
Re: What are imaginary numbers?
#120The title statement is meaningless. "Imaginary numbers" are not "multiply" by "90°". What does it even mean to "multiply" by an angle? Correct statement: "*i is equivalent to a 90° rotation in some contexts." Which is kind of obvious to anybody who has done some maths or physics at some point?
No it's not. i is a number such that i^2 = -1 by definition.
And in some contexts -1 is rotating by 180 degrees, which happens to be 290, or i*i.