Taken from Practical Statistics for Medical Research where Douglas Altman writes in page 285: 


  ...for any two quantities X and Y, X will be correlated with X-Y.
  Indeed, even if X and Y are samples of random numbers we would expect
  the correlation of X and X-Y to be 0.7


I tried this in R and it seems to be the case:

x <- rnorm(1000000, 10, 2)
y <- rnorm(1000000, 10, 2)
cor(x, x-y)

xu <- sample(1:100, size = 1000000, replace = T)
yu <- sample(1:100, size = 1000000, replace = T)
cor(xu, xu-yu)


Why is that? What is the theory behind this?

If $X$ and $Y$ are uncorrelated random variables with equal variance $\sigma^2$, then we have that
$$\begin{align}
\operatorname{var}(X-Y) &= \operatorname{var}(X) + \operatorname{var}(-Y)\\
&= \operatorname{var}(X) + \operatorname{var}(Y)\\
&=2\sigma^2,\\
\operatorname{cov}(X, X-Y) &= \operatorname{cov}(X,X) - \operatorname{cov}(X,Y)
& \text{bilinearity of covariance operator}\\
&= \operatorname{var}(X) - 0 & 0 ~\text{because}~X ~\text{and}~ Y ~\text{are 
uncorrelated}\\
&= \sigma^2.
\end{align}$$
Consequently, $$\rho_{X,X-Y} = \frac{\operatorname{cov}(X, X-Y)}{\sqrt{\operatorname{var}(X)\operatorname{var}(X-Y)}}= \frac{\sigma^2}{\sqrt{\sigma^2\cdot2\sigma^2}} = \frac{1}{\sqrt{2}}.$$
So, when you find 
$$\frac{\sum_{i=1}^n\left(x_i - \bar{x}\right)
\left((x_i-y_i) - (\bar{x}-\bar{y})\right)}{
\sqrt{\sum_{i=1}^n\left(x_i - \bar{x}\right)^2
\sum_{i=1}^n\left((x_i-y_i) - (\bar{x}-\bar{y})\right)^2}} $$
the sample correlation of $x$ and $x-y$ for a large data set $\{(x_i,y_i)\colon 1 \leq i \leq n\}$ drawn from a population with these properties, 
which includes "random numbers" as a special case, the result tends to 
be close to the population correlation value $\frac{1}{\sqrt{2}} \approx 0.7071\ldots$