If I know the variance of two numbers, is it possible to find their difference?
As in, the only information I have is the variance and that there are only two numbers.
How could I work backwards to find out what the difference between the two numbers is?
On
For perspective (make it an extended comment), the formulas come from:
$$\begin{align}\text{Var}(X) &= \mathbb E\left(X^2 \right)\quad-\quad \left[\;\mathbb E\left(X\right)\;\right]^2\\[2ex] &=\frac{\displaystyle\sum_{i=1}^n x_i^2}{n} \quad-\quad \left[ \frac{\displaystyle\sum_{i=1}^n x_i}{n}\right]^2 \end{align}.$$
When working with a sample we apply the Bessel's correction:
$$\begin{align}S^2 &= \frac{n}{n-1}\,\left(\, \frac{\sum_{1}^n x_i^2}{n} -\left[ \frac{\sum_{1}^n x_i}{n}\right]^2\right)\\[2ex] &= \,\frac{n\displaystyle \sum_{i=1}^n x_i^2 -\left[ \displaystyle \sum_{i=1}^n x_i\right]^2}{(n-1)\,n} \end{align}$$
Let the two numbers be $a$ and $b$. Then the variance is $$\sigma^2=\frac{(a^2+b^2)-\frac{(a+b)^2}{2}}{2-1}=\frac{(a-b)^2}2$$ so the difference is $$a-b=\sqrt{2\sigma^2}$$