The half-life of Radium-226 is 1590 years. If a sample contains $400 \mathrm{mg}$, how many $\mathrm{mg}$ will remain after 1000 years?

Step 1 ：The half-life of a substance is the time it takes for half of the substance to decay. Therefore, after one half-life, half of the original substance will remain. After two half-lives, half of the remaining half will remain, and so on. This is an exponential decay process.

Step 2 ：In this case, we know the half-life of Radium-226 is 1590 years. We want to know how much will remain after 1000 years.

Step 3 ：We can use the formula for exponential decay to solve this problem: \(N = N_0 * (1/2)^(t/T)\) where: \(N\) is the final amount of the substance, \(N_0\) is the initial amount of the substance, \(t\) is the time that has passed, \(T\) is the half-life of the substance.

Step 4 ：In this case, \(N_0 = 400mg\), \(t = 1000 years\), and \(T = 1590 years\). We can plug these values into the formula to find \(N\).

Step 5 ：Calculating the above expression, we get \(N = 258.66218985773327\)

Step 6 ：Final Answer: After 1000 years, approximately \(\boxed{258.66 \, \text{mg}}\) of Radium-226 will remain.