How to relate complex multiplication to Cartesian products?

In summary, my teacher was trying to explain to us that complex numbers in ℝ×ℝ can be thought of as plane with imaginary and real parts, and that the product of two complex numbers is the same as the magnitude of the real part multiplied by the angle between the real and imaginary parts. He demonstrated this by working through an example. However, he was not able to explain how to create the Cartesian product of two complex numbers.
  • #1
Eclair_de_XII
1,083
91

Homework Statement


"ℝ×ℝ and ℂ are very similar in many ways. How do you realize ℂ as a Cartesian product of two sets? Consider how complex numbers are multiplied; by grouping real and imaginary parts, show how the pattern of complex multiplication can be used to define multiplication in ℝ×ℝ. Using this multiplication, find the multiplicative inverse of (1,1) in ℝ×ℝ."

Homework Equations


##ℝ×ℝ={(x,y): x, y ∈ ℝ}##
##ℂ={x+yi: x, y ∈ ℝ}##

The Attempt at a Solution


For the first part, my teacher demonstrated that the imaginary component has its own axis, as does the real component of a complex number. He told us that they formed a plane on the real-imaginary plane, and that's analogous to the Cartesian plane. So I figure that I should somehow relate ##x## to the real part, and ##y## to the imaginary.

Let ##a,b,c,d∈ℝ##.
Then ##(a+bi)(c+di)=(ac+adi)+(cbi-bd)=(ac-bd)+(cb+ad)i##.
Setting these equal to the given ##x## and ##y## values, I tried to fix the variables so that ##(a+bi)(c+di)=1+1i##, but I'm getting many possible solutions. I'm trying to reverse-engineer the process, in other words.

For example, it works when ##a=1,b=0,c=1,d=1##, but also when ##a=0,b=-1,c=-1,d=1##.

So I can't really form a Cartesian product from a complex product. And I'm just confused on the concept he is trying to explain to us.
 
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  • #2
Eclair_de_XII said:

Homework Statement


"ℝ×ℝ and ℂ are very similar in many ways. How do you realize ℂ as a Cartesian product of two sets? Consider how complex numbers are multiplied; by grouping real and imaginary parts, show how the pattern of complex multiplication can be used to define multiplication in ℝ×ℝ. Using this multiplication, find the multiplicative inverse of (1,1) in ℝ×ℝ."

Homework Equations


##ℝ×ℝ={(x,y): x, y ∈ ℝ}##
##ℂ={x+yi: x, y ∈ ℝ}##

The Attempt at a Solution


For the first part, my teacher demonstrated that the imaginary component has its own axis, as does the real component of a complex number. He told us that they formed a plane on the real-imaginary plane, and that's analogous to the Cartesian plane. So I figure that I should somehow relate ##x## to the real part, and ##y## to the imaginary.

Let ##a,b,c,d∈ℝ##.
Then ##(a+bi)(c+di)=(ac+adi)+(cbi-bd)=(ac-bd)+(cb+ad)i##.
Setting these equal to the given ##x## and ##y## values, I tried to fix the variables so that ##(a+bi)(c+di)=1+1i##, but I'm getting many possible solutions. I'm trying to reverse-engineer the process, in other words.
You're trying to find the multiplicative inverse of (1, 1). IOW, what must (a, b) be so that (1, 1) x (a, b) = (1, 0)? There is a unique solution.
Eclair_de_XII said:
For example, it works when ##a=1,b=0,c=1,d=1##, but also when ##a=0,b=-1,c=-1,d=1##.

So I can't really form a Cartesian product from a complex product. And I'm just confused on the concept he is trying to explain to us.
 
  • #3
Mark44 said:
IOW, what must (a, b) be so that (1, 1) x (a, b) = (1, 0)?

Let's see... ##a=\frac{1}{2},b=-\frac{1}{2}##, so ##(\frac{1}{2},-\frac{1}{2})##. I'll still need to explain the connection between the Cartesian product and complex multiplication, though.
 
  • #4
Eclair_de_XII said:
Let's see... ##a=\frac{1}{2},b=-\frac{1}{2}##, so ##(\frac{1}{2},-\frac{1}{2})##. I'll still need to explain the connection between the Cartesian product and complex multiplication, though.
Complex multiplication is easiest to understand by considering the complex numbers in polar form. If you have two complex numbers ##z_1 = r_1e^{i\theta_1}## and ##z_2 = r_2 e^{i \theta_2}##, then ##z_1z_2 = r_1r_2 e^{i (\theta_1 + \theta_2)}##. IOW, the magnitudes of the two complex numbers multiply to make the magnitude of the product, and the angles of the two complex numbers add to make the angle of the product. Note that ##z_1## could also be written as ##r_1(\cos \theta + i \sin \theta)##, and similar for ##z_2##. What I'm calling the "angle" is also called the argument, or arg, for short.

From your example (1, 1) as a complex number in polar form is ##\sqrt 2 e^{i \pi/4}##. (1/2, -1/2) as a complex number in polar form is ##\frac 1{\sqrt 2}e^{-i\pi/4}##. If you multiply these you will get a complex number whose magnitude is 1 and whose angle is 0; that is, the complex number 1 + 0i. This is why (1, 1) and (1/2, -1/2) are multiplicative inverses of each other.

I'm not sure whether this is what you were looking for.
 
  • #5
Eclair_de_XII said:

The Attempt at a Solution


Let ##a,b,c,d∈ℝ##.
Then ##(a+bi)(c+di)=(ac+adi)+(cbi-bd)=(ac-bd)+(cb+ad)i##.
Yes. Now put this definition of multiplication in terms of (a,b) x (c,d) = (?,?) and you have multiplication in RxR.
Setting these equal to the given ##x## and ##y## values, I tried to fix the variables so that ##(a+bi)(c+di)=1+1i##,
Before you can talk about finding the multiplicative inverse of (1,1), you need to know what the multiplicative identity is. What values of (Ix,Iy) will give (Ix,Iy)x(c,d) = (c,d) for all values of c and d in R? (Hint: 1 is the multiplicative identity in R.)
Once you know what the multiplicative identity, (Ix,Iy) is in RxR, you can then find the multiplicative inverse, of (1,1). It is (a,b) where (a,b)x(1,1) = (Ix,Iy). Solve that for a and b.

I can not really say more in a homework question.

PS. I see that I am way too slow at writing these. There have been several posts since I began.
 

FAQ: How to relate complex multiplication to Cartesian products?

How is complex multiplication related to Cartesian products?

The connection between complex multiplication and Cartesian products lies in the fact that both operations involve the combination of two numbers to create a new number. In complex multiplication, two complex numbers are multiplied to produce a new complex number. In Cartesian products, two sets are combined to create a new set. This relationship can be visualized using the properties of complex numbers and Cartesian coordinates.

What are the properties of complex multiplication?

Complex multiplication follows the commutative, associative, and distributive properties, just like regular multiplication. However, it also has an additional property called the closure property, which means that the product of two complex numbers is always a complex number.

How can the Cartesian product be represented in terms of complex numbers?

The Cartesian product of two sets A and B can be represented as A x B, where x represents the Cartesian product symbol. In terms of complex numbers, this can be written as (a + bi) x (c + di) = (ac - bd) + (ad + bc)i, where a, b, c, and d are real numbers and i is the imaginary unit.

What is the geometric interpretation of complex multiplication?

Geometrically, complex multiplication can be understood as a combination of scaling and rotation in the complex plane. The magnitude of the product is equal to the product of the magnitudes of the two complex numbers, while the angle of the product is equal to the sum of the angles of the two complex numbers.

How is the concept of complex multiplication used in real-world applications?

Complex multiplication is used extensively in fields such as engineering, physics, and mathematics for various applications. For example, it is used in electrical engineering to analyze AC circuits, in signal processing to manipulate signals, and in quantum mechanics to describe quantum states. It is also used in computer graphics to transform and rotate objects in 3D space.

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