Complex numbers as an abelian group

In summary, Complex numbers as an abelian group has the properties of commutativity, associativity, and closure.
  • #1
karnten07
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[SOLVED] Complex numbers as an abelian group

Homework Statement



Multiplication of complex numbers defines a binary operation on C^x:=C\{0} (complex numbers not including zero). Show that C^x together with this binary operation is an abelian group. (without further discussion you may use the usual laws of algebra for R,
such as associativity for addition and multiplication of real numbers)

p.s the wording may sound strange because my lecturer is european i think and english is his second language.

Homework Equations





The Attempt at a Solution



I know that for an abelian it must show commutativity, ie, x*y=y*x. For it to be a group there must be associativity of multiplication, that en element exists e, that is an identity element ie. e*x=x and x*e=x, and also that there is an element y, that is an inverse element ie. x*y=e and y*x=e

I have wrote this information out in my answer but i think i might need to show it in an example. What kind of example would someone suggest, maybe (1+i) x (1-i). Giving 1 +1=2?? Any ideas guys? Thanks
 
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  • #2
Examples don't show anything. You have the properties you want to prove. Now start showing them one by one. What's the identity, e? Can you show commutativity, (a+bi)*(c+di)=(c+di)*(a+bi)? I would next show there is an inverse, and then use that to prove closure by showing a*b=0 implies a*e=0. Associativity is more of a pain in the neck to write down than anything deep. Showing you have an inverse is the only proof with any real meat on the bones.
 
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  • #3
Dick said:
Examples don't show anything. You have the properties you want to prove. Now start showing them one by one. What's the identity, e? Can you show commutativity, (a+bi)*(c+di)=(c+di)*(a+bi)? I would next show there is an inverse, and then use that to prove closure by showing a*b=0 implies a*e=0. Associativity is more of a pain in the neck to write down than anything deep. Showing you have an inverse is the only proof with any real meat on the bones.

I have shown that commutativity exists, but how do i show there is an inverse, can you suggest how i compute the inverse of a complex number?
 
  • #4
Good start. If you think the identity is 1+0i, you're right. To get the inverse of (a+bi) which is 1/(a+bi) multiply 1/(a+bi) by (a-bi)/(a-bi)=1. You are new to complex numbers, aren't you?
 
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  • #5
Dick said:
Good start. If you think the identity is 1+0i, you're right. To get the inverse of (a+bi) which is 1/(a+bi) multiply 1/(a+bi) by (a-bi)/(a-bi)=1. You are new to complex numbers, aren't you?

Well i have worked with complex numbers before but not in great depth really. Oh i see, so the inverse is a-bi/(a^2 +b^2)?
 
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  • #6
Yes, so if a+bi is not 0+0i, you have a multiplicative inverse.
 
  • #7
Dick said:
Yes, so if a+bi is not 0+0i, you have a multiplicative inverse.

Brilliant, I've shown the group is abelian now pretty much, thanks again.
 

Related to Complex numbers as an abelian group

1. What are complex numbers?

Complex numbers are numbers that consist of a real part and an imaginary part, written in the form a + bi, where a and b are real numbers and i is the imaginary unit equal to the square root of -1.

2. How do you add and subtract complex numbers?

To add or subtract complex numbers, you simply add or subtract the real parts and the imaginary parts separately. For example, (2+3i) + (4+7i) = (2+4) + (3+7)i = 6 + 10i.

3. What is an abelian group?

An abelian group is a mathematical structure that consists of a set and an operation, such as addition or multiplication, that follows certain properties. These properties include closure, associativity, identity, and inverses. In an abelian group, the operation is commutative, meaning the order in which the elements are combined does not affect the result.

4. How are complex numbers an abelian group?

Complex numbers can be considered as an abelian group under addition. This means that the set of complex numbers, together with the operation of addition, satisfies the properties of an abelian group. For example, the sum of any two complex numbers is always a complex number, and the order in which the complex numbers are added does not affect the result.

5. What are some applications of complex numbers as an abelian group?

Complex numbers as an abelian group have various applications in mathematics and physics. They are used in electrical engineering to represent alternating currents, in signal processing to analyze signals, and in quantum mechanics to describe the wave function of particles. They also have applications in geometry, particularly in the study of transformations and rotations in the complex plane.

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