I've read about safe doses of radiation, briefly. Why is there such a thing a safe dose? Can't any radiation give you cancer? Why is it more and more dangerous to take in larger and larger doses? Just because the body can't recover as easily? Is there something fundamentally more damaging about larger doses of radiation, as opposed to smaller more frequent doses? I suppose this could go to a health forum, but I don't know of one, and I came about this question via thinking about Physics.
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Great question. It is unfortunate that more people don't understand what "radiation" is and how it affects biology.
Firstly, you must distinguish between ionizing radiation and non-ionizing radiation. Wikipedia describes the former rather well:
X-rays, alpha, beta, and gamma radiation are all ionizing forms of radiation. Non-ionizing radiation, as you might guess, is radiation of a sort which lacks enough energy to to strip an electron from an atom. All light is actually "radiation."
The reason why ionizing radiation can be so damaging is because the molecular underpinnings of our cells depend on many very specific chemical bonds. Changing those bonds can damage or even break anything from proteins to strands of DNA itself. Our bodies naturally try to repair damage, but the amount of damage is a crucial factor in how successful our bodies can be. More radiation means more damage; also greater amounts of radiation increases the chances of serious genetic damage to one or more cells. Damaging a cell's rulebook in just the right ways can disable its repair processes and even its fail-safes that keep it from becoming a hazard to the rest of the body (such as growing uncontrollably into a cancerous tumor).
A more in depth explanation of radiation effects can be found at http://www.epa.gov/radiation/understand/health_effects.html#q1
Another thing to understand about why the dose matters is that we are surrounded by ionizing radiation. We always have been. Our bodies have evolved needing to successfully deal with ionizing radiation. You can find a very informative and fun visualization of this fact at https://xkcd.com/radiation/.
First of all: Technically speaking there is no such thing as a safe dose in radiation like you can say for drugs or toxins. This is because every single event of radiation can cause damage that can cause cancer.
Besides that, there is no need to panic about low radiation, since you define certain values which are deemed safe, because the likelihood of getting damage from this radiation is so small that it can be neglected. What we use in working with radioactive substances is the so called ALARA principle ("As Low As Reasonably Achievable"), so you try to keep the dose as small as possible, even if you are below the limits.
Taking up larger doses of radiation (or even worse taking up radioactive material) will raise your chances of receiving a harmful dose and damaging cells. A single large dose can cause a damage to your DNA, but this is also true for continued small doses. In both cases there is a certain chance of damage, which is bigger for the larger dose. But since there are more events possible for the continuous small dose, this can be as dangerous as the larger single dose.