Is C > R? Complex vs Real Set Size

  • Thread starter samkolb
  • Start date
In summary, the conversation is about whether the set of complex numbers is bigger than the set of real numbers. One participant believes that the cardinality of the set of complex numbers is equal to the cardinality of the set of real numbers, while another participant provides a bijection from the set of complex numbers to the set of real numbers, showing that they have the same cardinality. The discussion also touches on the topic of decimal expansions and their use in creating bijections.
  • #1
samkolb
37
0
Is it true that the set of complex number is bigger than the set of real numbers?

I know that card C = card (R x R) and I think that card (R x R) > card R. Is this true, and if so, why?
 
Physics news on Phys.org
  • #2
samkolb said:
Is it true that the set of complex number is bigger than the set of real numbers?

I know that card C = card (R x R) and I think that card (R x R) > card R. Is this true, and if so, why?

I think card (RxR) = card R

I would show this by setting up a one-to-one map between RxR and R

I will just show you a one-to-one between the unit square [0,1]x[0,1] and the unit interval [0,1]
You just look at the two decimal expansions and merge

(0.abcdefg..., 0.mnopqrs...) -> 0.ambncodpeq...
 
Last edited:
  • #3
C is with cardinality c, or aleph if you want, the same as R.

The simple bijection is a+ib |-> (a,b) into RxR.

If you want a bijection from C to R, then z=x+iy|->Im(z)/Re(z) it's a bijection to [-infinity,infinity] which is RU{infininity,-infinity} this cardinality is aleph+2=aleph.

QED
 
Last edited:
  • #4
loop quantum gravity said:
If you want a bijection from C to R, then z=x+iy|->Im(z)/Re(z) it's a bijection to [-infinity,infinity] which is RU{infininity,-infinity} this cardinality is aleph+2=aleph.
How could that possibly be a bijection? Obviously, [tex]z_1=a+ib[/tex] is mapped to the same point as [tex]z_2=a z_1[/tex], so it is not an injection.

Marcus has already provided a valid bijection, his "decimal merging" is the classical example of this. Notice how it is also valid in [tex]\mathbb{R}^n[/tex].
 
  • #5
Correct Big-T, but at least it's onto.
(-:
 
  • #6
|C| = |R2| = |R|.

There's some discussion about that in this thread.

Minor point: marcus's function isn't even well-defined; consider decimal expansions with infinite trailing "9"s. (For example, 0.0999... = 0.1000..., but (0.0999..., 0.0000...) maps to 0.00909090..., and (0.1000..., 0.0000) maps to 0.10000000... .) However, the mapping from 0.abcdefgh... to (0.acef..., 0.bdfh...) is a well-defined surjection from [0, 1) to [0, 1)2, and that's all you need.
 
  • #7
Marcus' function would be well defined if we agreed to use trailing nines wherever the decimal expansion is terminating, this should of course have been specified.
 

1. Is C > R?

The answer to this question depends on the context in which C and R are being compared. In mathematics, C typically refers to the set of complex numbers and R refers to the set of real numbers. In this case, it can be said that the size of the complex set C is greater than the size of the real set R, since the complex numbers include both real and imaginary numbers.

2. What is the difference between the complex set C and the real set R?

The main difference between the complex set C and the real set R is that the complex numbers include both real and imaginary numbers, while the real numbers only include numbers on the number line. Additionally, complex numbers can be represented in the form a + bi, where a and b are real numbers and i is the imaginary unit, while real numbers are represented as just a single value on the number line.

3. Why is the set of complex numbers larger than the set of real numbers?

The set of complex numbers is larger than the set of real numbers because it includes both real and imaginary numbers, while the real set only includes numbers on the number line. This means that the complex set has more elements than the real set, making it a larger set.

4. How are complex numbers and real numbers used in science?

Both complex and real numbers are used in science to represent quantities and measurements. Real numbers are used for quantities that can be measured on a number line, such as distance, temperature, and time. Complex numbers are used to represent quantities that involve both real and imaginary components, such as electric current and sound waves.

5. Can complex numbers and real numbers be used interchangeably in scientific calculations?

No, complex numbers and real numbers cannot be used interchangeably in scientific calculations. Real numbers are used for quantities that are measured on a number line, while complex numbers are used for quantities that involve both real and imaginary components. This means that they have different properties and cannot be used interchangeably in calculations.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
2K
Replies
2
Views
647
  • Set Theory, Logic, Probability, Statistics
Replies
13
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
945
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
1K
  • Engineering and Comp Sci Homework Help
Replies
11
Views
966
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
1K
Back
Top