Real equivalent of the surreal number {0.5|} I've been reading up on Surreal numbers, but have some questions.
Some equivalent real and surreal numbers.
2.5 =
    {2|3} =
    {{{0|}|}|{{{0|}|}|}} =
    {{{{{|}}|{}}|{}}|{{{{{|}}|{}}|{}}|{}}}

0 =
    {-1|1} = {-2|1} = {-2,-1|1} =
    {{|0}|{0|}} = {{|{|0}},{|0}|{0|}} =
    {{{}|{{|}}}|{{{|}}|{}}}

-3/8 = {-0.5|-0.25} = {{-1|0}|{{-1|0}|0}}
    {{{{}|{{|}}}|{{{}|{{|}}}|{{|}}}}|{{{}|{{|}}}|{{|}}}}

What about the real number for {0.5|}?
 A: In the surreal numbers, $$\{0.5|\}=1=\{0|\}=\{\{|\}|\}.$$  
In general, if $a$ is real and $a\ge 0$, $$\{a\mid\}=\{\lfloor a\rfloor\mid\}=\lfloor a \rfloor+1,$$ where $\lfloor a \rfloor$ is the largest integer less than or equal to $a$.  If $a<0$, $$\{a\mid\}=\{\mid\}=0.$$
A: The number for {1/2|} is "1"...

a = {{{|}|{{|}|}}|}
numeric label for a = 1
a == "1" = True
Surreal {1/2|} represented by form {{{|}|{{|}|}}|}
 is equivalent to form {{|}|} represented by name "1"

...according to python code...

from surreal import creation, Surreal
s = creation(days=7)
a = Surreal([s[1/2]],[])
name = a.name_in(s)
equivelence = s[name]
print('a =',a)
print('numeric label for a =',name)
print('a == "{}" = {}'.format(name,a==equivelence))
print('Surreal {} represented by form {}'.format('{1/2|}',str(a)))
print(' is equivalent to form {} represented by name "{}"'.format(str(equivelence),name))

...using [PySurreal] (https://github.com/peawormsworth/PySurreal):  
A: Consider the games, which happen to be numbers, $G = \{ 0\,|\,\}$ and $H = \{ 0.5\,|\,\}$.
$G = 1$, directly from the simplicity rules. You know that $H = 1$, but that is not a direct implication. What you can do is show that $G \leq H \leq G$.
$H \leq G$, because $0.5$ is the only Left option of $H$, there are no Right options in $G$ and $0.5 < G = 1$.
$G \leq H$, because $0$ is the only Left option of $G$, there are no Right options in $H$ and $0 < H$. The last inequality, $0 < H$, is true because in the game $H$, Left wins no matter who starts, and this means the game is positive.
Now it is clear that $\{ 0\,|\,\} = \{ 0.5\,|\,\} =  1$. I hope that it helps you understanding why Polichinelle's answer is correct.
