Kinda.
Ionizing (“nuclear”) radiation is dangerous because each particle has enough energy to slam into random molecules and send electrons flying.
This is particularly dangerous, because when molecules suddenly lose electrons, they become extra-reactive. And a lot of parts of your cells, really don’t like unexpected chemical reactions. Especially DNA. Your body can repair some DNA damage, but it can only handle so much at a time, and it always risks making a mistake. If a cell makes too many mistakes, or just can’t repair its DNA fast enough, the cell can die. If too many cells die at once, that’s a radiation burn. (“acute radiation sickness” is just a radiation burn affecting specific internal organs.). If the cells live, the mistakes increase the risk that they become cancerous.
Let’s use electromagnetic radiation as an example. The highest-energy electromagnetic rays, are generally referred to as either “x-rays” or “gamma rays”. These are both ionizing, and can both pass through enough of your body to reach your internal organs. (This is why carefully controlled amounts of x-rays can be used for medical imaging.) Lower the energy, and we have UV. This can trigger specific unwanted chemical reactions, but can’t blast apart random molecules. UV is also too weak to get very deep into your body, so the worst it’ll do is a sunburn (which can still be serious, and still increases your risk of cancer). Then, light. Light is mostly harmless. It can generally only affect molecules specifically adapted to respond to it – we can see light, but it’s not very dangerous. Then infra-red, then radio waves (including microwaves and WiFi).
The thing is, even if single particles can’t do much, they still deliver heat. Put a kilowatt into a beam of even the lowest-energy non-ionizing radiation, and anything absorbing that beam will still gain a kilowatt of energy. This can be useful. It can transmit information, like the light hitting our eyes or the radio waves hitting your phone’s antenna. It can transmit energy, like the wireless charging dock your phone might use.
Any energy that isn’t actively used, turns into heat. This is also useful. We can use it to cook food (this is how microwaves work), or heat rooms. But it’s also dangerous. Any kind of radiation, in sufficient quantity, can cause so much heat that it burns you. Visible light from the sun can burn your retinas. The flash from a sufficiently energetic explosion (e.x. a nuke) can burn exposed skin. Infra-red from a fire can burn you. Radio waves from a sufficiently powerful communications transmitter or broken microwave oven can burn you. (Your WiFi antenna, is not strong enough to do this.)
What people mean by “nuclear radiation” is usually “ionizing radiation” — radiation that has enough energy to knock electrons off of atoms, which changes their chemistry. Changing chemistry in cells can kill them, or — if you’re very unlucky — change their DNA (which is biological information stored with chemistry) in a way that causes the cell to become cancerous.
For electromagnetic radiation, whether it is ionizing or not depends on its frequency. [Here’s a simple diagram that illustrates the frequency spectrum of electromagnetic radiation](https://cdn.britannica.com/75/95275-050-5FC96002/Radio-waves-rays-light-gamma-ultraviolet-electromagnetic.jpg). Radiation that passes through walls and matter easily is on the low-end of the spectrum. WiFi, cell phones, and microwaves are all sort of in the same part of the spectrum. It is not ionizing. That does not mean it can’t have biological effects — microwaves certainly can! — but they don’t ionize.
Ionizing radiation starts pretty much after the ultraviolet, with what we call X-rays and gamma rays.
As others have noted there are other kinds of ionizing radiation beyond electromagnetic radiation. But to answer your question, WiFi, no matter how powerful, will never be ionizing radiation — _the definition of WiFi is of a specific frequency of electromagnetic radiation, and by definition that frequency is non-ionizing._ That does not mean that there couldn’t ever be harm from something like WiFi; microwave radiation does have biological implications (you wouldn’t want to be put inside a microwave oven), but it is not the same category of problems as ionizing radiation.
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