# Trying to show that rationals exist on the + real number line field K

1. Jul 7, 2013

### CubicFlunky77

This is the first 'problem' in my Linear Algebra/Geometry text book. I just need to know if I am doing it correctly. Any hints? I also need to know if I am using correct notation/presentation.

Question: $\mathbb R^+ \leftrightarrow \mathbb Q$?

What I've done:

Suppose: $(ε_1,...,ε_n) \in \mathbb R^+ \rightarrow \mathbb K$ and
$(c_1,...,c_n) \in \mathbb Q \rightarrow \mathbb K$

Assuming: $\mathbb R^+ ⊂ \mathbb K$ and $\mathbb Q ⊂ \mathbb K$ where $\mathbb K$ is a numerical/object field; we can say that

$\forall (ε \in \mathbb R^+, c \in \mathbb Q) \in \mathbb K \exists (ε \cap c) \in (\mathbb R^+ \bigcap \mathbb Q)$ |$\mathbb R^+ \leftrightarrow \mathbb Q$

2. Jul 7, 2013

### Bacle2

I'm not sure I understood what you meant, but you can do this:

i)1 is in ℝ , 1 as the identity, since ℝ is a field.

ii) 1+1=2 is in ℝ , by closure of operations

iii) (1+1)(-1)=1/2 is in ℝ , since every element in a field is invertible

iv)..... Can you take it from here ( if this is what you meant)

3. Jul 11, 2013

### CubicFlunky77

I get it, thanks!