All real numbers are complex numbers?And are I #'s orthogonal R#'s?

In summary, the conversation discusses the relationship between real and complex numbers, with the understanding that complex numbers can be represented as z= a+ib where a and b are real numbers. It is noted that real numbers are a subset of complex numbers, and it is questioned why we distinguish between real and complex numbers if this is the case. It is also asked how to determine which number is larger when comparing a pure imaginary number with a real number. The concept of orthogonality is brought up in relation to real and imaginary numbers, and it is clarified that they are indeed orthogonal. However, the dot product for complex numbers is defined differently, and two complex numbers can have a dot product of 0 even if they are not orthogonal.
  • #1
nabeel17
57
1
A) I understand that complex numbers come in the form z= a+ib where a and b are real numbers. In the special case that b = 0 you get pure real numbers which are a subset of complex numbers. I read that both real and imaginary numbers are complex numbers so I am a little confused with notations.

What then is the purpose of saying something is a member of the reals or a member of complex if real numbers are also part of the complex numbers. Also how do you determine what is a larger number when comparing say 4i with 3. Do I take the modulus of 4i (which is 2) and compare that?

B) Now going to the complex plane, we use something like vector addition to map a complex number using some amount of real and some amount of imaginary. The real and imaginary axis are orthogonal to each other so I am wondering if real and imaginary numbers are orthogonal? For example if I have two real orthogonal vectors u=(1,1) and v=(1,-1), the dot product gives 0. But two complex numbers that are "orthogonal" say s=1+i and t=1-i give a "dot product" of 2 (using i*-i=1). Also since a complex number uses real and imaginary components, why can I not do a dot product (or vector multiplication) and say s (dot) t =|s||t|cosα and since α= 90, then s (dot) t=0? I'm not sure if I am asking the right questions here or if I even understand what I'm asking but can someone try to clear this up?
 
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  • #2
nabeel17 said:
What then is the purpose of saying something is a member of the reals or a member of complex if real numbers are also part of the complex numbers.
It is true that the real numbers can be considered as a subset of the complex numbers. Saying that ##x## is a real number gives you more specific information than saying that ##x## is complex, just as saying that ##x## is a positive integer gives you more information than saying that ##x## is an integer.

Also how do you determine what is a larger number when comparing say 4i with 3. Do I take the modulus of 4i (which is 2) and compare that?
It's not too hard to show that there is no way to order the complex numbers, so it does not make sense to ask which is larger: ##4i## or ##3##. Indeed, it can be shown that, up to isomorphism, the only complete ordered field is ##\mathbb{R}##, the set of real numbers.

B) Now going to the complex plane, we use something like vector addition to map a complex number using some amount of real and some amount of imaginary. The real and imaginary axis are orthogonal to each other so I am wondering if real and imaginary numbers are orthogonal?
Yes, that's true. But be careful to define the dot product correctly for complex numbers. If ##z = a+bi## and ##w = x+yi##, then ##x \cdot y = ax + by = \text{Re}(z\overline{w})##.

For example if I have two real orthogonal vectors u=(1,1) and v=(1,-1), the dot product gives 0.
Correct. In general, the dot product of two vectors ##(a,b)## and ##(x,y)## in ##\mathbb{R}^2## is ##ax + by##.

But two complex numbers that are "orthogonal" say s=1+i and t=1-i give a "dot product" of 2 (using i*-i=1).
No, the dot product of ##s## and ##t## is defined to be consistent with the ##\mathbb{R}^2## definition. In this case, ##s\cdot t = \text{Re}(s\overline{t}) = \text{Re}((1+i)(1+i)) = \text{Re}(2i) = 0##.
Also since a complex number uses real and imaginary components, why can I not do a dot product (or vector multiplication) and say s (dot) t =|s||t|cosα and since α= 90, then s (dot) t=0?
Yes, that's correct.
 
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  • #3
Ahh, thanks for clearing that up!
 
  • #4
I don't know if this has already been stated but imaginary numbers (or complex numbers) are not real numbers, but all numbers are complex numbers because you can think of all numbers as having "+0i" after them.

For instance, 1 is a real number, but it is also a complex number because it is also "1 + 0i".
1i on the other hand is a complex number and not a real number because you cannot represent it on the real number line.
 
  • #5
Werkzeug said:
I don't know if this has already been stated but imaginary numbers (or complex numbers) are not real numbers, but all numbers are complex numbers because you can think of all numbers as having "+0i" after them.

"all numbers" is wrong, you mean "all real numbers".

For instance, 1 is a real number, but it is also a complex number because it is also "1 + 0i".

The correct way to phrase this is that the embedding ##x \mapsto x+0i## is a field isomorphism.

1i on the other hand is a complex number and not a real number because you cannot represent it on the real number line.

It is worth pointing out that this is only true as fields (or rings). As groups the real numbers is isomorphic to the imaginary numbers.
 
  • #6
jbunniii said:
It's not too hard to show that there is no way to order the complex numbers, so it does not make sense to ask which is larger: ##4i## or ##3##. Indeed, it can be shown that, up to isomorphism, the only complete ordered field is ##\mathbb{R}##, the set of real numbers.

Nitpick: the Complex Numbers cannot be made into an ordered field, i.e., so that there is a specific relationship between the field properties and the order properties,but, by the well-ordering principle, they can be well-ordered.
 
  • #7
WWGD said:
Nitpick: the Complex Numbers cannot be made into an ordered field, i.e., so that there is a specific relationship between the field properties and the order properties,but, by the well-ordering principle, they can be well-ordered.
You're right, my wording was careless.
 
  • #8
jbunniii said:
You're right, my wording was careless.

Join the (my) club.
 

1. What is the difference between real and complex numbers?

Real numbers are numbers that can be plotted on a number line and include all rational and irrational numbers. Complex numbers are numbers that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit. Real numbers are a subset of complex numbers, as they can be written as a +0i.

2. Why are all real numbers also considered complex numbers?

This is because all real numbers can be written in the form a + 0i, where a is a real number. In other words, the imaginary part of a complex number is 0, making it a real number. Therefore, all real numbers can also be classified as complex numbers.

3. Can complex numbers be plotted on a number line like real numbers?

No, complex numbers cannot be plotted on a number line as they have two components - a real part and an imaginary part. They are usually represented on a complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part.

4. Are imaginary numbers orthogonal to real numbers?

No, imaginary numbers are not orthogonal to real numbers. Orthogonality refers to the perpendicularity of two vectors. Since real and imaginary numbers are plotted on different axes, they cannot be considered orthogonal in the same way that two vectors on a Cartesian plane can be.

5. How are complex numbers used in science?

Complex numbers are used in a variety of scientific fields, including physics, engineering, and mathematics. They are particularly useful in describing and analyzing alternating current (AC) circuits, quantum mechanics, and fluid dynamics. They also have applications in signal processing and cryptography.

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