Imaginary Numbers

[noz92]

I know that imaginary numbers are numbers that are based on

[math]i[/math]

[math]\sqrt{-1}[/math],

but why are they imaginary. Why can't there be [math]\sqrt{-n}[/math]? Why can't you have a product a negative number, yet you can have a sum?

[Rasori]

I have limited knowledge on imaginary numbers, but the reason that there can't be a square root of a negative number is simple enough: to get a negative number by multiplication, you need one negative and one positive number. This means that there isn't one number that can be multiplied by itself to equal a negative number.

 

1*1 is 1, but -1*-1 isn't -1, it's 1.

 

-1*1 would equal -1, but then that isn't a square because it is two different numbers.

 

Hopefully this helps a little, until someone like Sayo comes in to be really specific.

[bloodhound]

the term imaginary is highly misleading.

[swansont]

the term imaginary is highly misleading.

 

How so? Especially once you've accepted the term "real numbers."

[bloodhound]

the terms, "real" "imaginary" and "complex" are not now intended to suggest anything about the reality or the absense.

[ecoli]

I've always believed we use the term imaginary as a way to force impossible numbers (such as the sqaure root of -1) to fit into our number system.

[noz92]

What are complex numbers? I think I read somewhere that they where the same thing as imaginary numbers. Either the book was wrong (not likely) or I misread it?

 

And why do you have to have both a positive and negative integer when it comes to multiplying the negative integers. Why does [math]-5^2=25[/math], and

[math]i[/math]

[math]\sqrt{-1}[/math]?

[bloodhound]

i = sqrt (-1) is just defined to extend the real system into a complex system. u just stick in [math]\sqrt{-1}[/math] in [math]\mathbb{R}[/math]. this extended system sort of has less structure that [math]\mathbb{R}[/math] as the ordering property doesnt extend to [math]\mathbb{C}[/math](i.e the relation |z|<|w| doesnt make sense.). But this system makes it indespensable fot understanding part of mathematics.

 

[math]\mathbb{C}[/math] is a field (i.e its a set with two binary functions + and . such that (F,+) is abelian group, so is (F^*,.) and multiplication is distribution over addition.) C also contains R as a subfield.

 

Basically C just consists of numvers

[math]z=x+iy[/math] where [math]x,y\in \mathbb{R}[/math]

they are added and multiplied according to the usual rules of algebra, along with [math]i^2=-1[/math]

 

maybe someone like matt grime or MandrakeRoot could explain more about the properties of complex numbers.

[Gilded]

"What are complex numbers?"

 

Isn't the complex number group just real numbers + imaginary numbers together?

[noz92]

Sorry Bloodhound, but I didn't exactly understand what you were saying. Could you explain that in simpler terms.

[JaKiri]

Imaginary numbers are called imaginary numbers because they don't fit on the numberline infinity to negative infinity. They are defined as being multiples of the square root of -1, or, more generally, the square roots of negative numbers.

 

For example:

 

What is the square root of 4? Plus/minus (+/-) 2, obviously, because +2*+2 = 4 = -2*-2.

 

What about -4? It must be some constant multiplied by 2, that much is clear, because 2^2 = 4. It can't be +/- 1, because that's been used. It needs to be something that, when squared, is equal to -1. Since that's impossible on our numberline, we define this number to be i. Therefore, the squareroot of -4 is +/- 2i; multiplying out...

 

+2i*+2i = + 4 (-1) = -4

-2i*-2i = + 4 (-1) = -4.

 

See?

 

Complex numbers are numbers on an Argand plane (think of a graph, with the x axis being the 'real' number line and y being the 'imaginary' number line), generally defined as numbers which have both real and imaginary elements, written in the form 'a + bi'.

[bloodhound]

dont worry about complex numbers too much now. if ur going to do higher maths. physics or even chemistry you should encouter them and study them in detail.

 

for now just think of the comlex numbers as the numbers of form

[math]z=x+iy[/math] where [math]x,y\in \mathbb{R}[/math]

x is called the real part of z and y is the imaginary part.

and are written

[math]$Re$(z)=x[/math]and

[math]$Im$(z)=y[/math]

 

The complex field is basically [math]\mathbb{C}:=\{x+iy \colon x,y\in \mathbb{R}\}[/math]

 

A field is basically a set which is an abelian group with respect to two binary operations, addition and multiplication. and multiplication being distributive over addition. I am not going to go into detail, if u want to know more about groups and fields, then u can look up on the net. I recommend that once you know the group axioms etc. you go and show that [math]\mathbb{R}&\mathbb{C}[/math] are fields.

 

Also R is a subfield of C , i.e R is contained in C , and R is a field in its own respect.

 

[edit] like jakiri said its helpful to visualise the complex number in the x,y plane. this is because there is a one to one correspondence between points in [math]\mathbb{C}[/math] and [math]\mathbb{R}^2[/math]. then the addition will follow the normal vector addition. etc.

[noz92]

dont worry about complex numbers too much now. if ur going to do higher maths. physics or even chemistry you should encouter them and study them in detail.

You know, physics is my area of study, that's what I'm best at, so, I still insist on learning them.

[bloodhound]

You know, physics is my area of study, that's what I'm best at, so, I still insist on learning them.

I havent done much complex numbers myself, but hopefully i will be able to tell you something about them once i finish my complex analysis module next term. For now best bet would be to wait for matt or mandrake or more experience mathematicians to reply.

[noz92]

Okay.

[Dave]

It's unlikely you'll encouter fields/groups on the general undergraduate physics course. Group theory is quite an important part of higher quantum mechanics (or so I've heard).

[CPL.Luke]

as far as I have studied in algebra (not very far yet)

 

imaginary numbers come in the form of bi

 

for a number like the squrt of -4 you can break it up into the squrt of 4 multiplied by the squrt of -1

 

you then define the squrt of -1 as i

 

so then you have

squrt(4) *i

 

which reduces to

2i

or bi

 

complex numbers come in the form a+bi (this is used to handle the quadratic formula when you encounter a negative number underneath the radical sign)

 

in order to really understand complex numbers you need to understand the quadratic formula.

 

as i don't know how to input the html to create those fancy looking equations you'll have to wait till someone who can comes along

[JaKiri]

in order to really understand complex numbers you need to understand the quadratic formula.

 

...Not really.

 

You can understand what they are perfectly well without knowing that it is possible for them to be the square root of something.

 

[-b +- SQRT(b^2-4ac)] /2a

 

Screw latex!

[noz92]

as i don't know how to input the html to create those fancy looking equations you'll have to wait till someone who can comes along

I just learned how to do those' date=' there really fun. Try visiting the Latex tutorial.

[Dave]

It should be said that the motivation for studying imaginary numbers originated from solving quadratic equations; when you have a negative discriminant, you get complex roots for the quadratic.

[VazScep]

How so? Especially once you've accepted the term "real numbers."

The way we view and do mathematics has changed over the centuries, and so mathematical names which are retained for the sake of tradition are often misleading. IIRC, the term "imaginary" was coined by Descartes, when complex numbers had not long appeared on the scene as a "cheat" for solving cubic equations, and when their legitimacy was very much in doubt.

[VazScep]

It should be said that the motivation for studying imaginary numbers originated from solving quadratic equations; when you have a negative discriminant, you get complex roots for the quadratic.

While this was observed, the interesting thing is that Cardano's method of solving cubic equations with real solutions sometimes required an intermediary use of complex numbers.