So I've trying to get my head around the concept for a while now.
As far as I can tell it is simply ignoring the fact you're dividing a negative number by a negative number for the ease of calculation.
Is that seriously it, or at an I missing something? It seems like it should just be a footnote about expressing double negatives
No that's not it, it gets more complicated when you start doing operations with i. purplemath.com
Define a binary operation * on two ordered pairs (a,b)*(c,d)=(ac-bd,ad+cb). No need to invoke sqrt(-1) at all.
Our definition of the natural numbers is axiomatic rather than constructive. We have not told you what the natural numbers are (so we do not address such questions as what the numbers are made of, are they physical objects, what do they measure, etc.) -
we have only listed some things you can do with them and some of the properties that they have. This is how mathematics works- it treats its objects abstractly, caring only about what properties the objects have, not what the objects are or what they mean. If one wants
to do mathematics, it does not matter whether a natural number means a certain arrangement of beads on an abacus, or a certain organization
of bits in a computer’s memory, or some more abstract concept with no physical substance; as long as you can increment them, see if two of them are equal, and later on do other arithmetic operations such as add and
multiply, they qualify as numbers for mathematical purposes (provided
they obey the requisite axioms, of course). It is possible to construct the natural numbers from other mathematical objects - from sets, for
instance - but there are multiple ways to construct a working model of the natural numbers, and it is pointless, at least from a mathematician’s standpoint, as to argue about which model is the “true” one - as long as
it obeys all the axioms and does all the right things, that’s good enough to do maths.
>cont.
Historically, the realization that numbers could be treated axiomatically is very recent, not much more than a hundred years old. Before then, numbers were generally understood to be inextricably connected to some external concept, such as counting the cardinality of a set, measuring the length of a line segment, or the mass of a physical object, etc. This worked reasonably well, until one was forced to move from one number system to another; for instance, understanding numbers in terms of counting beads, for instance, is great for
conceptualizing the numbers 3 and 5, but doesn’t work so well for −3 or 1/3 or √2 or 3+4i; thus each great advance in the theory of numbers - negative numbers, irrational numbers, complex numbers, even
the number zero - led to a lot of unnecessary philosophical anguish. The great discovery of the late nineteenth century was that numbers
can be understood abstractly via axioms, without necessarily needing a concrete model; of course a mathematician can use any of these models when it is convenient, to aid his or her intuition and understanding, but they can also be just as easily discarded when they begin to get in the way.
Terence Tao. Analysis I
I thought about simplifying this but what he wrote was too good to do that.
Okay now we are getting into it.
I have to admit the first paraphrasing that came to my mind was simply saying 'yes we all have aspergers'.
So you probably made the right call there.
You think this idea of mathematics is some sort of pursuit of 'natural truth' has ultimately held the field back?
Because honestly some of these 'controversies in mathematics' feels like watching schoolyard kids arguing the definition of infinity.
OP, it is unfortunate that complex numbers are taught in such an isolated an un-intuitive way, the way it is taught is that its just some random thing people thought up and called a number.
Instead, complex numbers are the ordinary numbers (real numbers) extended with an algebraic term which we call 'i'. There is some technicality in what is meant by "extended", but imagine for one moment that the only numbers that existed were rational numbers.
Imagine then that we wanted to extend the "rational numbers" further, and so we introduced a term called "[math]\alpha[/math]", so that we can investigate the kind of numbers we get when we only have [math]\alpha[/math] along with the rational numbers. That is, we are looking at the set:
[math]\{ a_0 + a_1 \alpha + a_2 \alpha^2 + \dots + a_n \alpha^n | \forall n \in \mathbb{N}^+, a_i \in \mathbb{Q} \} [/math]
Now lets say that this [math]\alpha[/math] is special in that whenever we multiply two [math]\alpha[/math] together we get [math]2[/math], that is: [math]\alpha^2 = 2[/math] (resist the temptation to 'solve' for [math]\alpha[/math] we are pretending only the rational numbers exist). But then this means our new set of numbers looks like:
[math]\{ a_0 + a_1 \alpha | a_0, a_1 \in \mathbb{Q} \} [/math], this is because whenever we get higher powers of [math]\alpha [/math] we can reduce them to get smaller powers.
Now back to reality, we "know" that what we mean by [math]\alpha[/math] was really [math]\sqrt{2}[/math], but the idea is that I can use this to extend the rational numbers, to get a new FIELD of numbers that look like: [math]3 + 5\sqrt{2}, 2 + \frac{1}{2}\sqrt{2} [/math] and etc.
What complex numbers is, is EXACTLY the same, instead of letting [math]\alpha^2 = 2[/math] we let [math]\alpha^2 = -1 [/math], and we investigate the system we get when constructing these new algebraic objects
>maths is just our daydreaming that has no connection to reality
when will this meme end
>maths is just our daydreaming that has no connection to reality
> when will this meme end
When it stops the being true.
Mathematics is just a 'language' we use when referring to numbers. It has the same room for ambiguity as any other.
A English teacher might tell you the Oxford comma is technically grammatically incorrect, but he recognises and understands it's intended purpose and meaning, even if he doesn't agree with.
A mathematician on the other hand would try to write a proof explaining why commas don't exist
This. The complex plane is literally just R^2 with the added structure of well-defined multiplication. Like, define the unit vectors 1=(1,0) and i=(0,1), and then the rule for multiplication
>(a,b)*(c,d)=(ac-bd,ad+cb)
can be expressed simply by defining i^2=-1. So multiplying by i is the same as applying a counterclockwise rotation by pi/2 (90 degrees,) and multiplying by any point on the unit circle in this plane (this can be written in the form e^it) is the same as counterclockwise rotation by its angle from (1,0), or by t.
So when solving for roots of some equation, this is really what's going on. Say you have y=x^2+1=0. Clearly this has no real solution, so now you're looking for its roots in the whole complex plane (which contains the real line.) Under these rules of multiplication, you have i^2+1=(-1)+1=0, and i, that is, the point (0,1) is a solution, when the function is defined over the whole plane.
See: mathworld.wolfram.com
The multiplication rule is conveniently set up so that you can get answers to these equations in the form a+bi by taking square roots of negative numbers, which isn't otherwise allowed in the real numbers. Try to visualize the plane, that helps make it more intuitive. If you're more advanced you can think of your function as a "rule" sending every point in the plane (where it's defined) to another point in the complex plane. This is basically what real functions do, send points from the real line to other places on the real line.
[math]\sqrt{-1}=\pm i[/math] :^)
Would it be better if we just use whole numbers and replace what is behind the comma with a fraction of the whole number?
> line ab = 3
A square with line ab = 3cm
> abcd = ab^(4/1)
youtube.com
Start with this and continue...It is a decent explanation....
Imagine 'i' to be a muddy river. It's not really useful on its own but sometimes you have to cross it to find more fertile land (solutions). For a long time people couldn't see past the river but eventually we learned to wade through it, see what happens and then you find dry land and normality returns.
I stopped caring about math when I was introduced to the concept of imaginary numbers. What a crock of shit. If your equation can only be solved by inventing numbers that can't exist, like some kind of math deity , then you are fucking wrong and the math is flawed. Same for algebra solutions that basically say "the correct answer is whatever the correct answer is". Thats what the math said transcribed to words but god forbid if i wrote in down in english instead of the ancient math runes the teacher word mark me wrong.
Math is logical and numbers never lie my ass. Math is just as flawed as any other human construct.
The basic thing that you have to understand, OP, is that it's a very strange thing at first blush, /but deciding that it's okay to do does not lead to any contradictions/, especially where the 'smaller' number systems that came before are involved.
That's why it's accepted and that's why it's valid. Its albebra and geometry turn out to be internally consistent with, insofar as they are extensions of, the previous number systems. If we were to instead do something like divide by zero, then we would produce contradictions very quickly, which would indicate to us that there's something horribly wrong with our reasoning process (which allows division by zero). Nothing of the sort crops up with complex numbers.
Nope. Nope. Nope.
Complex numbers aren't the same type as Real numbers.
It is a stupid mistake to think that the proposition:
forsome x : Real, x = -1
is the same as as the proposition
forsome x : Complex, x * x = -1
Besides there are many possible alternate solutions to the puzzle. Suppose we identify -1 with the matrix
-1 0 0
0 -1 0
0 0 -1
then we can equally well ask the proposition
forsome x : Matrix (3, 3), x * x = -1
It turns out that there are many more square roots of -1 then just the imaginary ones.
If you want to go further you can use Tensor types.
the problem with this argument is it relies on the denotation of "imaginary".
If I called 'i' the second quaternion unit, then I'm sure you wouldn't disregard it as non-existent and just consider it advanced (technically all numbers do not exist, they are abstract concepts, 'i' is no exception)
Never mind what it "means" op.
Just realize it's a useful concept when dealing with certain phenomena like electricity and signal processing.
It just is a concept that makes calculation easier.
Just think of math like a framework to model things in. You simply didn't need the number j to describe things. That's the only reason the idea is strange.
You don't question "but WTF is a binary and" or "WTF is a set really" do you?
That being said those ideas can represent physical things
>the number j
I beat all your friends are jmaginary
see this
Math does not describe anything realistic in the world until we apply it.
Turns out that imaginary numbers are really useful for describing rotations, like those of electrons in QM.
>Turns out that imaginary numbers are really useful for describing rotations, like those of electrons in QM.
Just use a matrix faggot.
> Just use a matrix faggot.
I do
[eqn]
i=
\begin{pmatrix}
0&1\\
-1&0
\end{pmatrix}
[/eqn]
>Let's make all complex numbers linear transformations on R^2 instead of elements within it subject to additional algebraic structure
fags
Your picture is false.
[math]1=\sqrt{1} = \sqrt{(-1)(-1)} = \sqrt{-1}\sqrt{-1} = i \cdot i = -1[/math]
Fantastic explanation.
You guys are fucking illiterate. Read what he said carefully. He said number systems are defined axiomatically. Relating it to the real world can help someone understand how its defined but its unnecessarily to mathematics as a whole.
>And hes right.
Naturals, Integers, rationals, reals and complex numbers are defined AXIOMATICALLY not by physical objects.
i*i=1 faggot
>
Re read your post, mate.
You can't split up sqrt((-1)(-1)) into sqrt(-1)sqrt(-1).
You might like the work of Gödel. He was famous for his incompleteness theorems which state that any system that was complicated enough can't be consistent and complete.
The proofs are neat, I'm sure there's a simple popsci version somewhere on youtube.
Hello first-time poster, welcome to Veeky Forums.
Perhaps if exponents were seen as vectors of numbers, rather than multiples, there is a rational explanation for square root of -x. Maybe squares and square roots are ultimately seen as useless and a stupid concept.
negative numbers are a spook
People tend to forget that we live on a round planet. Every square will eventualy bend due to gravity. Making it a square on a round surface
Many properties of real numbers are unintuitive but you can't really debate them on that ground. Before the wildassburgers defense force comes to imply my post let me tell you this. Maybe you will not accept that anything infinite exists, but that really isn't math. Lets look at a more down to earth example. Do you think that there are the same "number" of rationals than naturals? Well, depens obviously if you accept that two sets have the same cardinality if there is a one to one correspondence between them. Again you are getting a non-intuitive result from accepting an intuitive definition. So why should you look down on someone who says x^2+1=0 has a solution defined as "i"? Why would an unintuitive definition cause you trouble if an intuitive definition sill leads you to unintuitive shit? If you want some visual shit consider complex numbers as the 2D plane with some extra rules, again, you will stills return to shit that isn't clear. When you get maturity on math many things will get clearer and more insightfull, obviously the hard part is to theb explain someone with little background why imaginary numbers make "sense". That's the downfall of modern mathematical curriculum; we learn to visualize sicks till the point where it isn't enough and teachers just show us some retarded analogy and asks us to regurgitate some stupid crap.
X^2+1=0
An Object multiplied by the same object adds 1 equal 0.
0 equals an object multiplied by the same object and one (of what?), if the result is 0
Thus, the value of adding 1 creates 2 objects that would otherwise be 0
When you do this with say a negative number it would be like this. 0 equals Two objects that are the same when you withdraw 1(of what) from both. The result is that the value of the 1 being there states that both object are 0 without withdrawing 1 (of anything) from them
you have to imagine the numbers are complex or imaginary.
the huge false in here is, that you take the square root as it is for real numbers. although it can be the complex square root, too.
and this is how it looks:
i = -1 , because complex or principal square root means: i = i * x
so if i=0 it`s true.
so i = i * x = -1
try i = -1 and x = 1
i = -1 * 1 = -1
-1 = -1 q.e.d.
it`s ca. 25% of complex numbers which have negativ square roots i think. can you help me with that theorie?
25% excluded 0 ;D
damn. it did`nt to the square, because I copied them .____________. so ggimme sec. to do it once again...................
you have to imagine the numbers are complex or imaginary.
the huge false in here is, that you take the square root as it is for real numbers. although it can be the complex square root, too.
and this is how it looks:
sqrt i = -1 , because complex or principal square root means: sqrt i = i * sqrt x
so if i=0 : it`s true.
so sqrt i = i *sqrt x = -1
try i = -1 and x = 1
i = -1 *sqrt 1 = -1
-1 = -1 q.e.d.
it`s ca. 25% of complex numbers which have negativ square roots i think. can you help me with that theorie?
25% excluded 0 ;D
I have no idea wtf you are trying to do.
>i
how do i write this number with marks on a whiteboard?
1^(1/4)=1
1×0.25=0.25
Exactly, because i != sqrt(-1).
i^2 = -1.
Draw a real number line. It is the mark that is 1 unit vertical to 0.
This
Imaginary numbers from real non complex zeros in a real non complex non polynomial rational inequality
really makes me think
Here's the simplest explanation I can come up with:
The sole reason we need it is to have a way to logically flip between 4 different states of the same number using the same operation. For an example, we naturally can flip through two states by using a negative number (-1^n -- either +1 or -1), but to create another dimension in which we can operate, we need not two but four more states. i is the only case of a value being able to cycle through four states by repeating the same operation (i, -1, -i, 1). The interesting thing is that there is no value discovered yet that can flip through 8 states to allow us to work in a dimension higher, and people rather use quaternions as a substitute. Discovering some similar trick to i that would allows us to cycle through 8 values would probably slingshot humanity some 100 years of technological progress into the future
I don't get it either. It just seems like an elaborate way of doing co-ordinates. Like 4 + 3i is just (4, 3) in Cartesian, isn't it?
Then why not just say that instead of poncing about with numbers that don't exist?
>falling for this ancient pasta
A 2D coordinate plane on its own doesn't tell you how to add and multiply two points. If I gave you the points
(4, 3)
(9, -4)
and told you to add them, it's reasonable to suppose the answer is
(13, -1)
But we could just as well add all four elements together to get 4+3+9-4 = 12. That's probably not useful for anything, but there's nothing strictly saying we can't. The difficulty is more obvious with multiplication, where we have lots of choices that we might actually use. For instance, we could multiply element by element to get another point in 2D:
(4*9, 3*-4) = (36, -12)
Or we could do a inner product resulting in a scalar:
4*9 + 3*-4 = 36 - 12 = 24
Or an outer product resulting in a matrix (hard to format here):
4*9, 4*-4 = 36, -16
3*9, 3*-4 27, -12
Or use complex multiplication:
(4*9 - 3*-4, 4*-4 + 3*9) = (48, 11)
The set of points doesn't tell you how they relate to each other, you have to impose those operations on it before you can talk about arithmetic.
Yeah Veeky Forums butchered my formatting for the matrix, it's meant to be a 2x2 square matrix with 36, -16 in the top row and 27, -12 in the bottom row.
Oh and if you suppose the 2D plane is embedded in 3D space you could define a cross product where the resulting vector is perpendicular to the plane and proportional to the sine of the angle.
So why do we use complex multiplication a lot?
- If you're a mathematician it's because it makes the complex numbers an algebraically closed field. This basically means you can write a polynomial a + bx + cx^2 + dx^3 + ... = 0 and the roots will always lie in the complex plane (not true of the real line)
- If you're a physicist or engineer it's because multiplication by exp(i*theta) means a rotation around the origin by an angle theta. This makes it super convenient to represent cyclical processes like signals.
>You can't split up sqrt((-1)(-1)) into sqrt(-1)sqrt(-1).
Wait, why not?
The mathematical language that we use is a language. We have decided on a well defined symbology that represents complex ideas. But the ideas themselves are completely unambiguous. Traditional language doesn't have this concreteness of meaning.
>If you're a physicist or engineer it's because multiplication by exp(i*theta) means a rotation around the origin by an angle theta
But how can i be used like that if it can't be expressed at all?
Not sure what you mean exactly. We don't have to "express it" as anything, it is what it is.
Say if you have Euler's identity, if the result is 1 then can't you work backwards to find a numeric representation of i?
Or in your example: what is the operand given to exp ()? So if I want to rotate by 1, then it's exp (i * 1), so the angle is simply i? Given that, whatever the result of exp (i * 1) is, can't you do the inverse of the exp operation and get the actual value for i?
[eqn]\sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b} \quad a, b \in \mathbb{R}[/eqn]
It's a subtle point, it has to do with how sqrt(x) is defined. The equation
x^2 = 9
and
x = sqrt(9)
are NOT saying the same thing. The first equation has two solutions +3 and -3, the second has only one: +3. The reason is that sqrt(x) is a function, and functions (by definition) can only be one-to-one or many-to-one, never one-to-many or many-to-many. Therefore, we DEFINE sqrt(x) to be the POSITIVE solution to y^2 = x. In other words
sqrt(x) = x^(0.5)
From this, it is possible to prove that sqrt(a)*sqrt(b) = sqrt(ab):
sqrt(a)*sqrt(b) = a^(0.5)*b^(0.5) = (ab)^(0.5) = sqrt(ab)
For complex numbers you can't do this. Let's look at i:
i^2 = -1 fine, but also
(-i)^2 = (-1)^2 * i^2 = 1*-1 = -1
So what is sqrt(-1)? Is it i or -i? We can't use a simple rule like "take the positive one", because such a thing is ill-defined on the complex plane. The rule we use to make sqrt(z) single-valued on the complex domain is to express it as polar coordinates
z = r*exp(i*theta)
where r is the distance from the origin and theta is the angle from the x axis.
And we define sqrt(z) as sqrt(r)*exp(i*theta/2) ("take the square root of the distance from the origin, and halve the angle from the x-axis"). If we do this, then:
sqrt(a)*sqrt(b) = sqrt(r_a)sqrt(r_b)*exp(i*theta_a/2)*exp(i*theta_b/2) = sqrt(r_a*r_b)*exp(i*(theta_a+theta_b)/2)
Which might superficially look like sqrt(a*b), but you must remember that theta is periodic in 2pi: if you go 2pi radians around the origin you end up back where you started. This is fundamentally what "breaks" the equivalence between sqrt(-1)sqrt(-1) and sqrt(-1*-1). In the first, we're doing the square roots first and then multiplying:
sqrt(-1)*sqrt(-1) = exp(i*pi/2)*exp(i*pi/2) = exp(2pi*i/2) = exp(i*pi) = -1
In the second we multiply first and take the square root afterwards:
sqrt(-1*-1) = sqrt(exp(i*pi)*exp(i*pi)) = sqrt(exp(i*2pi)) = exp(0/2) = 1
doing this from memory so there might be mistakes; look up "branch cuts" for more
There isn't any such thing as a "numeric representation of i".
i is i is i, it can't be represented as a real number because it ISN'T a real number.
>So if I want to rotate by 1, then it's exp (i * 1), so the angle is simply i?
No, the angle is 1 radian. If you do the complex log of the complex exponential, you get*
ln(exp(ix)) = ix
*not entirely true, since you run into similar problems when trying to define a complex logarithm as you do for square root.
The "many-to-many" clause is, at best, confusing. But yea.
What you're asking is like trying to write a fraction as an integer; it's impossible by definition. The fraction 1/3 is DEFINED TO BE "the number that, when multiplied by 3, yields 1".
Similarly, the complex number i is DEFINED TO BE "the number that, when squared, gives -1"
So write it as a decimal with as many digits as you deem sufficient.
This just doesn't make sense. If it's 1 radian then the inverse must be the square root of -1. Just like if I feed the exp function 2.1, then doing the reverse on the result will yield 2.1.
>So write it as a decimal with as many digits as you deem sufficient.
I said integer. A decimal isn't an integer.
>If it's 1 radian then the inverse must be the square root of -1.
No, look at the equation again. Rotation by theta is:
exp(i*theta)
Taking the logarithm of that gives you i*theta +/- 2npi. It doesn't give you i.
>Taking the logarithm of that gives you i*theta +/- 2npi. It doesn't give you i.
So that must be i then.
Wait, I think I see what your confusion is.
theta = 1 radian
exp(i*theta) = exp(i*1) = exp(i)
So taking the logarithm gives i (+/- 2npi, but we can ignore that).
This doesn't make it a real number though. It's still on the complex plane, because we did a complex logarithm of a complex exponential.
Basically you have
i = ln(exp(i))
which a) is tautological and b) has i's on both sides of the equation. It doesn't give you a representation of i in terms of something non-imaginary, because such a thing isn't possible.
I don't get it. If you can't even begin to express it then I don't see how you can do anything with it. It's as arbitrary to me as saying exp (theta * banana).
is there a word for fear of imaginary numbers?
I always thought this explained it very well.
youtube.com
You just can't express it directly as in i = [something real], otherwise the definition would be rather useless. You can however express it with [math] i^2 = - 1[/math]
Are you in high school or something?
Anyways watch this series on imaginary numbers: youtu.be
>If you can't even begin to express it
This is how you express i:
i^2 = -1
That's it. That's the definition. What more do you want?
A number. Like the square root of any positive number.
i is a number.
I want digits. Like you can say pi or 3.1415926535. Give me numerals.
That's like saying "I want pi expressed as a fraction of integers".
i = (0,1)
Forget it then. I don't need to know, but it would have been nice to.
What you're asking is just impossible. If we were able to express i as a real number then that would mean x^2 = -1 has a real solution. It doesn't, so we can't.
We can't express 1/2 as an integer. We can't express sqrt(2) as a rational number. We can't express i as a real number.
I don't know how to state it in simpler terms than that.