# What's the probability of R in C?

1. May 30, 2005

### Sabine

R ]-∞;+∞[ Є C (complex numbers) so what's the probability to find R in C?

i think it is barely equal to more 0+

2. May 30, 2005

### Moo Of Doom

The probability of finding R in C is equal to the probability of finding 5.29 in R (or any number, for that matter). Infinitessimal (actually, transfinitessimal :P)

3. May 31, 2005

### matt grime

In order to define probability you need to assign a measure to the sets in the sample space. There isn't really a way to do that for R in C, but by any reasonable measure (ie not a probability one) then the measure of R in C is zero.

4. May 31, 2005

### Sabine

i've been thinking of it and i think it's indetermend

5. May 31, 2005

### matt grime

Well, it's not defined, that's different.

You need (*you*) what you mean by "picking a number at random" or "finding R in C".

If we take the set of complex numbers of modulus at most K, then the probabilty that a number selected uniformly at random will be real with probabilty 0 since the lesbegue measurable sets are the "events", and the reals have measure 0 in that set.

6. May 31, 2005

### Sabine

in that case the probabilty is not 0 it is 0+ bcz this number may be found (sorry for my vocab i am french educated)

7. May 31, 2005

### matt grime

What on earth is 0+? is that even a real number? Don't think it's one i've heard of.

Given any lebesgue measurable subset of C with finite measure, eg the disc of radius K, there is a probability measure on it. If u is lebesgue measure, and T is the measure of the set, then u/T is a probability measure, and the events we can describe are what is the probability that a number picked at random is in S where S is a lebesge measurable subset is u(S)/T

Since the strictly real numbers will form a set of measure zero, the probability of picking a real from this set is zero. Not zero+ whatever that may be.

This is a consequence of how we *define* probability for such discussions.

8. May 31, 2005

### Sabine

0+ is the number that comes right after the 0 (0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000...1) it is the smallest number after zero

9. May 31, 2005

### matt grime

Right, ok, doesnt the fact that there is no such thing as "smallest nonzero positive real number" bother you? after all( 0+)/2 is another number, bigger than zero and smaller than the one you just wrote, which, by the way, isnt a real number, or at least isnt' a decimal expansion of any real number.

10. May 31, 2005

### master_coda

The fact that it's possible for an event to occur does not imply that the probability of that event occuring is greater than zero.

11. May 31, 2005

### Sabine

but the occurence of this event is barely negligeable compering with the univers in which i am studying the probability .

for the 0+ this is its definition that's why i can't give u the exact number but u can find the 0+ maths books so i don't understand why u are discussing it it is a FACT it is universally known.

12. May 31, 2005

### matt grime

Sabine, I assure the notion that there is a real number "just greater than 0" (ie bigeer than zero but smaller than any other real number) is not a sound one.

13. May 31, 2005

### Hurkyl

Staff Emeritus
Thread closed. Not only is it in the wrong place (this is not a number theoretic issue), but the original poster seems to be more interested in preaching than listening.