# Cardinality of Complex vs. Real

## Main Question or Discussion Point

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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Tide
Homework Helper
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....

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 :tongue2: .

Tide
Homework Helper
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 $0.5$ and $0.4\bar 9$ are the same number and lead to different results!

mathwonk
The problem is that $0.5$ and $0.4\bar 9$ are the same number and lead to different results!