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Hoofbeat
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#1
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I was reading fermat's last theorem and in it he suggestions thinking of all real no's as one axis (say x-axis) and the imaginaries as another axis (say the y-axis).

Now, I don't have knowledge of imaginary and complexes no's beyond the fact that sqrt-1 is i (I'm going to be teaching myself P4 as soon as I've ordered the book!), but it got me thinking. As we normally think of things in 3D and possessing 3axes: x, y and z, is their a whole other set of no's that we haven't discovered yet?

I'm probably completely off track with my thoughts as, like I said, I don't know details behind imaginary no's and complex no's, but I just got this crazy notion in my head that there should be another whole set of no's out there we haven't discovered yet!
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rts
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(Original post by Hoofbeat)
I was reading fermat's last theorem and in it he suggestions thinking of all real no's as one axis (say x-axis) and the imaginaries as another axis (say the y-axis).

Now, I don't have knowledge of imaginary and complexes no's beyond the fact that sqrt-1 is i (I'm going to be teaching myself P4 as soon as I've ordered the book!), but it got me thinking. As we normally think of things in 3D and possessing 3axes: x, y and z, is their a whole other set of no's that we haven't discovered yet?

I'm probably completely off track with my thoughts as, like I said, I don't know details behind imaginary no's and complex no's, but I just got this crazy notion in my head that there should be another whole set of no's out there we haven't discovered yet!
Graphs with an x-axis as the real component of a number and the y-axis as the imaginary component are called Argand Diagrams and have been around for hundreds of years. I don't really understand how you made the jump to 3 dimensions and another kind of imaginary number. The Argand diagram was created mathematically to solve problems, it isn't connected to the physical y-dimension.
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fishpaste
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Maybe if you find some condition which holds true for all real numbers, e.g. square a real number and you'll always get a positive real number. Then 'invent' a number which doesn't obey this condition, and you have a new set. But I don't think it would automatically contain the real set, like the complex numbers do.
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Squishy
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Look up quaternions if you're interested. They're numbers of the form a + bi + cj + dk where a, b, c and d are real, ij = k, i^2 = j^2 = -1, and ij = -ji.

A bit like even more complex complex numbers.
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beauford
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(Original post by Hoofbeat)
I'm probably completely off track with my thoughts as, like I said, I don't know details behind imaginary no's and complex no's, but I just got this crazy notion in my head that there should be another whole set of no's out there we haven't discovered yet!
Well it is possible to show that any function involving powers of just real numbers does always give real numbers. Whereas complex numbers are 'closed' in the sense that given any function of a complex number, the answer is always another complex number (i think)
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shiny
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(Original post by Squishy)
Look up quaternions if you're interested. They're numbers of the form a + bi + cj + dk where a, b, c and d are real, ij = k, i^2 = j^2 = -1, and ij = -ji. A bit like even more complex complex numbers.
:eek: Vector Spaces :eek:
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nollaig
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Well it is possible to show that any function involving powers of just real numbers does always give real numbers. Whereas complex numbers are 'closed' in the sense that given any function of a complex number, the answer is always another complex number (i think)
.

You need to be careful when examining (binary) operations for closure. Real numbers are not closed under every operation: (-1)^0.5 is a binary operation involving real numbers but the result [often termed the product] is i, which is not a real number. Also with complex numbers: they are not closed under addition or multiplication. eg (a + ib)(a - ib) and (a + ib) + (a - ib) are both real numbers. I always am amazed that i^i is a real number !
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jpowell
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We are used to thinking in reals. Mathematicians denote the set of all real numbers to be R. But it is quite possible to have R^2, and R^3, which you will no doubt be used to as 2 and 3 dimensional vector space.

C, the set of all complex numbers can be thought of as a 2 dimensional vector space, but we can also have C^2 i.e. a vector composed of two complex numbers, and which would need 4 dimensions to be shown (which is of course very hard in a 3 dimensional space).

Naturally all these vector spaces can be expanded and you can study the geometry of R^n and C^n, where n is any number.
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JamesF
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(Original post by nollaig)
.

You need to be careful when examining (binary) operations for closure. Real numbers are not closed under every operation: (-1)^0.5 is a binary operation involving real numbers but the result [often termed the product] is i, which is not a real number. Also with complex numbers: they are not closed under addition or multiplication. eg (a + ib)(a - ib) and (a + ib) + (a - ib) are both real numbers. I always am amazed that i^i is a real number !
Im afraid not, The complex numbers are of the form a+bi, where a and b are real, and CAN be 0. Therefore the field of complex numbers contain all the real numbers and they are closed under addition and multiplication.

In addition to being closed under the definition of a field, as beauford said, the field of complex numbers is 'algebraically closed' which is a consequence of the fundamental theorem of algebra.
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Hoofbeat
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Thanks for all your guys help. I think I just had a silly idea into my head and because I've only studied up to P3 I'm a little "blinkered" or narrow-minded! I'm just ordering P4-6 books off amazon now so hopefully in a good few weeks I'll at least know the basics of Complex No's!
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elpaw
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(Original post by Squishy)
Look up quaternions if you're interested. They're numbers of the form a + bi + cj + dk where a, b, c and d are real, ij = k, i^2 = j^2 = -1, and ij = -ji.

A bit like even more complex complex numbers.
then there's octerions (or something like that) with 8 components - i read it in new scientist a while back, how they did some stuff to do with quantum mechanics
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jpowell
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Octonians.

A lot of the uses of quartonians have dissapeared with modern vector calculus though, so there isn't much mainstream use of them within physics.
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Squishy
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(Original post by AntiMagicMan)
Octonians.

A lot of the uses of quartonians have dissapeared with modern vector calculus though, so there isn't much mainstream use of them within physics.
Octonions...

Yeah, they have 8 components (sedenions have 16)...I gave up after complex numbers. The reason I prefer applied maths to pure maths is because I don't have to learn about things so abstract I'll never be able to put them to any use in this lifetime...also, I hate all the confusing symbols ...but applied maths has many of those as well.
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