[cjslep]

Just because it is a long-lived radioactive isotope does not mean it is dangerous. I'm referring to U238. It decays by alpha radiation, so unless you are literally eating it or inhaling it, those alpha particles will not penetrate your skin. Additionally, U238's long half life is why it remains as the most plentiful naturally occurring (on Earth) radioactive isotope. I'm tired of people trying to fear-monger radiation by throwing big numbers around. It would be instead be helpful discussing how to get roughly 4 alpha particles to decay per day inside them someone needs to eat roughly 1,000,000,000,000g of pure U238 that day.

[the_solution]

U238 decays to (among others) Polonium which will kill your very fast from inside you body.

I think your estimate is off by a few orders of magnitude.

Edit: "pure U238" is another point where you go wrong. There is no such thing in nuclear waste, nor will there ever be when it decays. So if i grant you that pure U238 on it's own is not that dangerous, that does absolutely not make nuclear waste less dangerous. The U238 decays to other radioactive substances, then those decay and so on. The long half life of the U238 guarantees that you have a very toxic mix for tens of thousands of years.

[cjslep]

That's what I get trying to balance rough numbers in my head. I was using roughly the number of atoms, not grams. Here's the math laid bare:

    Frac Remain In 1 Day = 0.5 ^ ((1/365)/4,500,000,000 years)
    Frac Remain In 1 Day = 0.999999999999578
    
    Frac Decay In 1 Day = 1 - Frac Remain In 1 Day
    Frac Decay In 1 Day = 4.22 * 10^-13
    
    Have 4 Decays In 1 Day = Total U238 * Frac Decay In 1 Day
    4 = Total U238 * 4.22 * 10^-13
    4 / (4.22 * 10^-13) = Total U238
    Total U238 = 9.48 * 10 ^ 12 atoms

I stand corrected, thanks for catching that.