Cardinality of Complex vs. Real

In summary: There is no real number 4.9999...In summary, the complex numbers have the same cardinality as the reals.
  • #1
Parth Dave
299
0
Prove that the set of complex numbers has the same cardinality as the reals.

What I did was say that a + bi can be written as (a, b) where a, b belong to real. Which essentially means i have to create a bijection between (a, b) and z (where z belongs to real).

Suppose:
a = 0.a1a2a3a4a5...
b = 0.b1b2b3b4b5...

Then,

z = 0.a1b1a2b2a3b3...

Is there anything wrong with that?
 
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  • #2
You need to do a little more work. First, does that constitute a one to one correspondence between the reals and the complex numbers? Also, you need to deal with ambiguous representations of certain numbers such as 0.5 and 0.4999...
 
  • #3
Well essentially that shows that there is a one to one relationship between the reals and the cartesian plane (x, y). Also, there is a one to one relationship between complex and the cartesian plane. Thus, the cardinality for all 3 is the same.

Pertaining to the ambiguosity of certain number, I'm not sure if i see how they pose a problem because if x is 0.5 or 0.499999 you get a different result.

ps. pardon my ignorance. I just started set/number theory two days ago :-p .
 
  • #4
Parth Dave said:
Well essentially that shows that there is a one to one relationship between the reals and the cartesian plane (x, y). Also, there is a one to one relationship between complex and the cartesian plane. Thus, the cardinality for all 3 is the same.

Pertaining to the ambiguosity of certain number, I'm not sure if i see how they pose a problem because if x is 0.5 or 0.499999 you get a different result.

The problem is that [itex]0.5[/itex] and [itex]0.4\bar 9[/itex] are the same number and lead to different results!
 
  • #5
he only needs an injection from the complexes to the reals, since there is an inclusion the other way, and if he assumes his reals are finite or infinite decimals not ending in all 9's, then his map never sends any pair of reals to a decimal ending in all 9's. so he does get an injection.
 
  • #6
Tide said:
The problem is that [itex]0.5[/itex] and [itex]0.4\bar 9[/itex] are the same number and lead to different results!

There is no real number 4.9999...
4.99... and 5 are just two metha-variables to denote the number 5 in R .Here denote means "interpretation." .
 
Last edited:

FAQ: Cardinality of Complex vs. Real

What is the difference between the cardinality of complex numbers and real numbers?

The cardinality of a set refers to the number of elements in that set. In the case of complex numbers, the cardinality is infinite, while the cardinality of real numbers is also infinite but of a different type.

How can we compare the cardinality of complex numbers and real numbers?

The cardinality of two sets can be compared by using a one-to-one correspondence. If two sets have a one-to-one correspondence, then they have the same cardinality. However, there is no one-to-one correspondence between complex and real numbers, therefore their cardinalities cannot be compared in this way.

Does the cardinality of complex numbers include the cardinality of real numbers?

Yes, the set of real numbers is a subset of the set of complex numbers. This means that all real numbers are also complex numbers, and therefore the cardinality of real numbers is included in the cardinality of complex numbers.

Is the cardinality of complex numbers larger than the cardinality of real numbers?

Yes, the cardinality of complex numbers is larger than the cardinality of real numbers. This can be seen by considering that the set of real numbers is a subset of the set of complex numbers, meaning the set of complex numbers contains all the elements of the set of real numbers plus additional elements. Therefore, the cardinality of complex numbers must be larger.

Can the cardinality of complex numbers and real numbers be counted?

No, the cardinality of both complex numbers and real numbers is infinite, and infinite sets cannot be counted in the traditional sense. This is because there is no endpoint to the set and for every element you count, there will always be another element to count.

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