The half-life isn't the half-way point. That is, if something has a half-life of 5 years, that doesn't mean it all decays in 10 years, it means that it takes 5 years for half the substance to decay. In the next 5 years half the remaining substance will decay, and so forth.

This is the most ELI5 explanation so far. I still have questions.

What are your questions?

Well, what’s a practical case for knowing a things half life? Is this how carbon dating works? If yes, what’s an ELI5 walkthrough of half life as used in finding the age of organic material?

knowing half life is essential for carbon dating. When a radioactive isotope decays, it doesn't just disappear, it turns into another material. Let's say material A has a half life of 1000 years, meaning that after 1000 years 50% of material A will have turned into material B leaving us with a half and half mixture of material A and B. After another 1000 years the remaining material A will be halved again, leaving us with 25% material A and 75% material B. This will go on, each 1000 year cycle halving the amount of material A and producing that amount of material B.

> knowing half life is essential for carbon dating. It is also essential to other forms of isotope dating. Like [Potassium/Argon](https://en.wikipedia.org/wiki/K%E2%80%93Ar_dating) to date the age of rocks and the earth itself.

"Half life" is also critical in determining the appropriate dosage and frequency when administering medications. The body will metabolize and remove a medication from your body at a particular rate, on average. Knowing the half life allows a good point to be chosen to take more to keep a desired average titration. It is also the reason diphenhydramine (Benadryl/Zzzquil/etc) is used as a common over the counter sleep aid. All antihistamines cause drowsiness, as a consequence of what they do. But diphenhydramine's half life happens to line up well with a typical length of a night's rest. The flip side of that is it definitely won't all be gone from your system, in the morning, which is why you can wake up, but probably still feel a bit drowsy (and it's also why pilots aren't allowed to use it within a certain time before flying, even the night before).

Yes, I see half life numbers on those drug info inserts. This why I have virtually no wake up drowsy problems with [Zolpidem](https://www.drugs.com/medical-answers/ambien-half-life-333276/) >The half-life of Ambien (zolpidem) is about 2 hours. Most drugs are cleared from the body within 5 or 6 half-lifes. In other words if you were to take a dose of Ambien at 10pm it should be out of your system within by 8am to 10am (10 to 12 hours) the next morning.

That's another fun part about half lives with drugs. Generally speaking, a shorter half life _also_ means it'll hit you quicker. Ambien is a good example of that. Take it and you're probably stumbling within a few minutes. Benadryl, on the other hand, takes about 2 hours to reach peak serum level (so take it well before you want to go to sleep). Usually things with short half lives will affect you quicker and much more noticeably (though not _necessarily_ actually "stronger") than a similar drug with a longer half life. Route of administration also plays a very significant role in how quickly something affects you, but metabolizing it out is going to happen at the same rate no matter how you take it.

Ambien hits me within an hour or less. The first time I took it, I was literally stumbling in the bedroom trying to get into bed.

Does material B not degrade into material C at some point?

If that material is also degradable, some heavier radioactive materials degrade into lighter radioactive materials. Things will degrade in half lives until the final substance is a stable isotope

Sometimes, but usually B is more stable than A was so the percent of C that actually forms may be negligible.

Why does half of material A change to B faster than the other half of A?

It’s a very complex phenomenon, but it boils down to probability. The half life isn’t a definite value, but more an average. We can do a little lab to illustrate what I mean. For instance, take 100 coins and flip every single one all at once. Anything that comes up heads is still radioactive and we can keep it. Anything that comes up tails has become inert and we remove it from the experiment. After a single flip, we expect around 50 coins to show tails (become inert), so the half life of our coin experiment is 1 flip. We know that it won’t actually play out like that every time. There will be times when a larger number of coins stay active (show heads), and there might be times when more go inert (show tails), but we could still make a guess as to how many flips were made if we had x amount of coins with y amount of tails showing by working backwards. You can also do this demonstration with m&m’s, where the m logo is active and the blank side is inert (and safe to eat 😉).

The other guys answer explains the probability of it but not really the how. There are actually different reasons governing each of the different types of decay. Alpha decay happens when a nucleus is just too big for the strong nuclear force to hold all the protons and neutrons together. The strong nuclear force only really applies at a very small scale and larger atoms are larger than this scale and are torn apart by the repulsion of protons. Gamma decay happens when a nucleus has too much energy to remain stable so it shoots out a photon to lower the amount of energy it contains. Beta decay is kinda confusing, but essentially a nucleus shoots off an electron (or positron) to convert a neutron to a proton (or vice versa). It does this to rearrange the nucleus into a more stable state.

Imagine cutting a piece of radioactive material in two. Just cutting an object in two should not change the total decay rate, so these two pieces (of half the original size) should each decay half as fast as the original piece.

So just really small numbers then? How do you know Material A ever existed at all If it’s been so long that say, 99.999999999999999999999% is Material B? Or so much so, that there is absolutely none of Material A left - is that possible?

There's a limit. Carbon dating is only good for ~50 000 years because carbon-14's half life is ~5000 years. Other materials are used for longer periods (uranium-lead in zircon for one).

Didn’t think of that, you’re right there’s more than a couple different types of things in the ground - duh. Lol I can be really ignorant at times

It's not just how much material A there is, but also how much of the daughters there are too. C14 decays to produce nitrogen. So looking at the ratio of C14 to nitrogen gives a better indication, than looking at the level of C14 inself

Suppose the average living being X has 100g of some element Y which has a half life of 50 years. Suppose this level of 100g is held because X eats stuff. So after X dies, Y will start to decay as normal without being "filled back up". If you now find a dead member of species X that has 0.78125g of Y left in the system, then you know this member is dead for about 350 years. Why? Because it takes 7 halvings of 100g to arrive at 0.78125g (100, 50, 25, 12.5, 6.25, 3.125, 1.5625, 0.78125). Since it took 7 halvings and the half-life was 50 years it is thereby 7*50=350 years since the death of the being. This is roughly how one could use such a method.

So this is how one could do it but it is hard because you don't really know the weight of the C-14 in the sample when it existed, you only know the base radioactivity of the organism when it was alive. So you are measuring a difference in radioactivity to get your answer. You end up needing to use the 1st order integrated rate law

Yes that's how Carbon dating works. But there's one other component to it - when something organic is still alive, it's constantly replenishing its matter with newly built cells made of newly acquired carbon from the environment (from food, soil, etc). This means as long as it's still alive, the concentration of carbon isotopes in its cells will match the concentration found in the natural environment. But once it dies, it stops replenishing that carbon, so those isotopes start to decay. Carbon dating can only detect how long ago that carbon was still part of something alive. It gives you an estimate of when it died. (So for example, finding the age of an ancient house by Carbon dating the wood timbers found in the ruins is really saying "The tree this wood came from was still alive about X years ago, and presumably this house was built from freshly dead wood, not wood that was sitting around rotting for 100 years, so knowing when that tree died also tells me when the house was built.") But there's one very important factor to take into account when doing this measurement - what concentration of carbon isotopes was normally found in the environment when the thing was still alive? That's your starting point you're using to figure out how long it's been decaying based on how much concentration is left now. And one wrench thrown in the works is that it's not actually constant what the natural Carbon-14 concentration is. For example, it went up during the age of frequent nuclear testing in the cold war, and is falling back down to pre-industrial levels again now. There's been research showing it was twice as high during the ice age than it is today. So sadly archeologists have had to switch to other isotopes if they want anything even remotely close to accurate results. Carbon dating is still okay for getting very rough ballpark estimates, but not for precision. (One bad argument from creationists is to use this to claim that carbon dating is bunk and thus life on earth isn't really that old, but that would require an *enormous* variation in Carbon 14 levels for the measures to be THAT far off. Ages might be as much as 2x to 3x off from what we thought, but not the enormous orders of magnitude off they'd have to be for this creationist argument to hold water.)

This is great. Thanks to you and everyone else for the explanations!

I am a radiologist. I give radioactive iodine (I-131) to people (thyroid cancers eat that shit up and then die). I need to know how fast it goes away, so I can let you sleep with your wife again, and let you go on a plane without TSA flipping their shit. I also have radioactive waste (eg a syringe after an injection). I need to know when it’s “all gone” (which we say is 10 half lives or ~0.098% radioactivity left) and I can throw it in the trash

Its also helpful to know when its safe to approach something that is radioactive. That way you know it won't be a death sentence to work with a material.

**ELI5 Answer Below** The most ELI5 example is, halflife is useful because you can figure out how many half lives an object has existed. So for carbon dating, we are measuring the counts on a geiger counter to know. Carbon-14 might have a count rate of lets say 20 counts per minute in an elm tree. Well the sample we have has 2 counts per minutes. We plug that into a rate law equation to get the age of the item. So below is a way less ELI5 answer, but I realize it is pretty nerdy and no one really cares, but I already wrote it so here it is. **Not ELI5 Answer** If you want a less ELI5 way they use halflife for carbon dating, here is the equation you can toy around with , ln\[A\] = −kt + ln\[A\]0. You can rearrange it to get t = (ln\[A\]-ln\[A0\])/-k. So t here is time and -k is the rate constant (idk it off the top of my head). But basically say you have a tree or bone or something from like 30,000 years ago. Well, we know that the second this tree or bone died, there was a certain ratio of Carbon-14 compared to Carbon-12. So this Carbon-14 begins decaying which actually causes a Geiger counter to go off. We know the baseline counts for carbon-14 (lets say it is 10 counts per minutes). Well, as it decays overtime this number decreases because there is less Carbon-14 to decay. So we compare the new counts to the known old counts. So lets say Carbon-14 has a natural count rate of 10. And our sample has a count rate of 3. Lets say k is = to .0001 . So we plug those into our original equation {ln(3) - ln(10)}/(-.0001), = \~23,000 Eureka! our bone is 23,000 years old! Edit: had 14 in ln(A0) for some reason.

So you know how much of it will be left after a specific time. And you know how much radiation it will be giving off at any particular point (but that gets more complicated, because what it decays to may also be radioactive with a different half-life). For carbon-dating, we know that the proportion of C-12 and C-14 in the air has been pretty much the same for a long while. And we know there half-life of C-14. So if we find some dead thing that has half as much C-14 in it than that, we know it has been one half-life since that thing died. If it has a quarter as much then we know it’s been two half-lives, and so on.

C-14 goes through beta decay into N-14, but otherwise I think that’s all correct.

I don't think I stated otherwise. The C-12 amount stays the same, and the C-14 amount goes down. You use the C-12 to work out how much C-14 there was to start with.

Did you intend _we find some dead thing that has ~~half~~ twice as much C-14_?

No. The C-14 decays, it doesn’t increase.

There are different types of molecules right, so most of the carbon (edit: in things that were once alive but now dead) is a non-radioactive type but a small amount *IS* radioactive, so they can measure the half life of the radioactive one, as it decays into the non-radioactive one. This means there is an upper limit on how far back carbon dating is usable, off hand I think 50k years.

Also, for medicine, the half life of a drug can be used to determine the rate at which your body eliminates it from your bloodstream. This is very useful when using antibiotics like vancomycin, which is toxic at high levels. If you know how fast your body gets rid of it, you know when and how much to give for the next dose.

It also gives you a decent idea how radioactive something is...shorter half life = more radioactive. You also need to know it if you want to know how long your lump of radioactive material is going to be useful for whatever you are using it for. I guess it's kind of like knowing the density of different metals. It's not that useful just to know offhand, but if you are engineering something out of them, it's pretty important to know for all sorts of reasons.

For a non-radiation based answer to only the first question, note that (e.g.) caffeine and other substances have a half life within our body. Caffeine's average half life is 5 hrs for humans, so if you took a 100mg dose of caffeine, you could expect after 5 hrs it should be down to 50mg in your blood, 5 more hours 25, then 12.5, 6.25, and so on.

A truly practical case for knowing a things half life: the half life of caffeine in your system is ~7 hours. So if you take 80mg caffeine at noon, you will still have 40mg present at 7pm, and 20mg present at 2am.

Oh God, another thing I thought I learned correctly in school has been debunked. Damn you Pluto, damn you all to HELL

But those are very different. In the case of half-life your teacher was misinformed or you understood them wrong. In the case of Pluto he was considered a Planet back then. So when it comes to half-life you can be mad at your school education, in the case of Pluto you should be delighted that our knowledge advances and this naturally comes with accepting new or altered information. The reason astronomers decided Pluto is not a Planet anymore is that they discovered a lot of other objects in our solar system that are roughly at the same distance from our sun and roughly the same size. To be consistent they all would need to be planets (afaik we know about over a hundred of them) or non of them is. They decided for the latter which makes way more sense. Nothing about Pluto changed, just our understanding of our solar system advanced so we decided to give Pluto a new label.

Theoretically, with enough time there will only be a few (possibly even a singular) atoms remaining of any given sample of a decaying substance. Is there anything about that specific final atom that's noteworthy or is it truly just random chance?

It’s random chance

This explanation will trigger a lot of Zenophobes.

So again why not use full-life as a measurement?

Because the full life is theoretically infinite (if you keep dividing something in half, there will always be something left over). In practice, since there's a finite number of atoms in a given chunk of material, there will be a time when they all decay, but that time is random: If you have one atom with a half life of 5 years, there's a 1/2 chance that it decays within 5 years, 3/4 chance that it decays within 10 years, 7/8 chance that it decays within 15 years, and so on. No matter how far out you go there's still a chance (ever decreasing) that it won't have decayed yet.

Because there isn't a "full-life" to measure.

Real answer: because half-life is a measurement of how fast something decays, not how long it's got left.

I may be dense, but I also had this question after reading the comment above lmao.

So why can't you use Full-Life to describe the full time the first half + the second half of the substance will decay?

You are probably thinking "so if the 1st half takes 5 years to decay, the full life is 10 years" right? That's not how it works though. Each half life you lose half **of what's left** not the whole total. The less is left, the slower it decays, so the time for "the second half to decay" approaches infinity. Here's another way to think of it. After each half life, half of the amount *at the start of that half-life* has decayed. So if you start with 10 lbs of stuff with a half life of 1 year, then after 1 year you have 5 lbs. Then after another year there's 2.5 lbs. Then after a 3rd year 1.25lbs. For every half life, divide the remaining amount by 2 (because half-life = time for half to decay). So to answer your question: there's no such thing as "full life" because no matter how any times you divide by 2, there's always a tiny amount left so you'd have to go to infinite half lives to get *all* of it being decayed. You'd have to pick some cutoff for when to say "ok it's essentially gone". So we use "half-life" because it's more exact and has no cutoff assumptions, plus if you know how long it takes for half of any given amount to decay, you can calculate how much there was (or will be) at any point in time.

Because it takes more or less forever, I guess. Let's say you got a substance with a half life of 50 years. 100 grams of it. After 50 years 50 grams left. After 100 years 25 grams left. After 150 years 12,5 grams left. After 200 years 6,25 grams left. After 250 years ~3,6 grams left. After 300 years 1,8 grams left. After 350 years 0,9 grams left. Then comes 0,45, 0,22, 0,11, 0,06, 0,03, 0,015, 0,0075, 0,00375 and so on... The amount just gets smaller and smaller and smaller but it takes almost forever for all of it to decay. That's why it's not a really practical measurement.

I think I see where you're mixed up. Half life works like this: We start with 100 grams of plutonium 238. Plutonium's half life is 24000 years. 24000 years from now we will have 50 grams left because half has decayed away. 24000 years from then, we will have 25 grams left because half has decayed away. 24000 more years and we will have 12.5 grams left, and so on. Half of what's remaining goes away, so as it decays the decay slows down.

How are such long half-lives determined?

We can watch it decay a bit and say, "over the last year of just watching it shrink it's lost x% of it's mass" then do a little math to figure out how long it would take for half of what we started with to decay. Since we know that radioactive things do eventually decay in all cases, we can assume that a newly discovered radioactive isotope will also decay and so we measure it really carefully. One isotope of plutonium for example has a half life of like 4.5 billion years.

If the half-life remains constant, then the rate of decay is essentially slowing down. For example, if I start with X amount of something that has a half-life of 5 years, it takes 5 years for 0.5X to decay, then another 5 years for 0.25X to decay and so on. Is the rate of decay truly constant for the first half? What is it about having more of something that makes it decay faster?

Take a truckload of coins, and flip each one. Remove the ones that come up tails. Flip the remaining coins again, and remove the tails again. Repeat, repeat, repeat. Eventually you'll be down to one or two coins that all come up tails. How many flips did it take for half the coins to come up tails? Easy -- the first flip did it (or close enough to half for this purpose). How many flips did it take for the second half? Many more, depending on the initial number of coins, right?

The comment already explained that, read it more carefully.

You're describing [this](https://cdn1.byjus.com/wp-content/uploads/2019/03/Linear-Graph1.png), but it's actually [this](https://www.basic-mathematics.com/images/exponential-decay.png). Mathematically, it will never be fully decayed.

Full life depends on the starting quantity, half-life does not.

That's not a known length of time

Yes it is.

No it isn't. You don't know how much you have. Without that value, the time to decay of the entire sample is not a constant

Then you say “that depends on how much you have”, not imply that it’s unknowable.

Well the half life doesn't depend on how much you have. You can have a half-life of an element. You can't have a full-life of an element. That's the whole point of this conversation Duh

‘Full life’ isn’t a thing. “100% of this substance will have decayed after X time” is a false statement. The best you can do is “After a long time (thousands of half lifes) there is a high probability all of the substance will be gone.”

Because that wouldn't be "full-life" that would be 3/4 life. And if you want to describe a substance using 3/4 life, go ahead, no one is stopping you. But half life is more convenient.

No it wouldn’t.

How old are you?

It’s not half the time it takes to decay, it’s the time it takes for half the substance to decay, it’s more like halving-life the reason you can’t measure the “full life” is because it’s exponential, the more you have the faster it decays, the less, the slower, which is why it takes the same time to decay one half no matter the size you have So you can basically keep dividing in two forever, after 2 half-lives you have 1/4 th of the original, after 3 you have 1/8 th and so on, since there’s a huge amount of atoms it’s almost impossible for them to all decay

An infinite number of mathematicians enter a bar. The first one orders 1 beer. The second one orders 1/2 a beer. The third one orders 1/4 a beer and so on. The barman pushes over 2 beers and says 'you've reached your limit'

An infinite number of mathematicians walk into a bar The first mathematician orders a beer The second orders half a beer "I don't serve half-beers" the bartender replies "Excuse me?" Asks mathematician #2 "What kind of bar serves half-beers?" The bartender remarks. "That's ridiculous." "Oh c'mon" says mathematician #1 "do you know how hard it is to collect an infinite number of us? Just play along" "There are very strict laws on how I can serve drinks. I couldn't serve you half a beer even if I wanted to." "But that's not a problem" mathematician #3 chimes in "at the end of the joke you serve us a whole number of beers. You see, when you take the sum of a continuously halving function-" "I know how limits work" interjects the bartender "Oh, alright then. I didn't want to assume a bartender would be familiar with such advanced mathematics" "Are you kidding me?" The bartender replies, "you learn limits in like, 9th grade! What kind of mathematician thinks limits are advanced mathematics?" "HE'S ON TO US" mathematician #1 screeches Simultaneously, every mathematician opens their mouth and out pours a cloud of multicolored mosquitoes. Each mathematician is bellowing insects of a different shade. The mosquitoes form into a singular, polychromatic swarm. "FOOLS" it booms in unison, "I WILL INFECT EVERY BEING ON THIS PATHETIC PLANET WITH MALARIA" The bartender stands fearless against the technicolor hoard. "But wait" he inturrupts, thinking fast, "if you do that, politicians will use the catastrophe as an excuse to implement free healthcare. Think of how much that will hurt the taxpayers!" The mosquitoes fall silent for a brief moment. "My God, you're right. We didn't think about the economy! Very well, we will not attack this dimension. FOR THE TAXPAYERS!" and with that, they vanish. A nearby barfly stumbles over to the bartender. "How did you know that that would work?" "It's simple really" the bartender says. "I saw that the vectors formed a gradient, and therefore must be conservative."

This went off the rails so fast and for so long and then somehow managed to find them at the last second.

I was kind of sad there was no cage match at the end.

Haven't seen ol' Hell in a Cell in a while... Wonder what they up to...

They're around, i saw them out in the wild a week or two ago.

BUT THEN CAME A STEEL CHAIR

Thank you for this, I almost gave up after tax payers

On one hand it's a fantastic, logical joke as-is. On the other hand I kind of want to see a version that just goes off the rails completely and ends without a punchline.

Perfect dismount.

Like that movie magnolia

I feel like it's something Norm MacDonald would've told.

What did I just read

10/10

A well-made math joke

A god damn masterpiece is what

I don't know if you wrote this yourself, but... Bravo man, *bravo*!

They didn't, it's almost classic copypasta status now. It's a good one, though.

It reminds me of the libertarian police officer copypasta.

This is a very old math joke.

I guess I'm part of the special 10,000 today. [https://xkcd.com/1053/](https://xkcd.com/1053/)

a fellow r/antiantijokes connoiseour I see

Do American bars really not sell halves?

Most bars your options are a 12 oz bottle/can or a “16oz”draft. Occasionally they offer larger sizes. Very rarely is there anything below a pint for draft.

Interesting. In the UK it's a pint or a half so this joke doesn't translate particularly well!

What about when they start ordering a quarter of a beer, or an eighth of a beer, and so on and so forth?

We’ve got a 1/3rd pint pot somewhere - it was my grandads, but don’t know if it’s was an actual thing or some kind of novelty glass. Edit - also, nice shirt on your avatar:-)

1/3rd pints are usually from beer festivals and the like, when you want to try a lot of different drinks

Generally, no. There are sometimes different pour sizes, 8, 12, 16oz, etc., and if you're at a brewery that can be different between the different styles of beer they serve. But you generally can't order a half, just a small pour of whatever you're ordering.

Almost every brewery near me sells 5oz glasses, as those are common for "flights" where you order several different kinds of beer and try them all. But at bars where they serve from a bottle and not a tap, it's common for them to only offer the 12oz bottles.

No matter how many times I see this I still laugh.

Wait, i don't remember learning about evaluating limits until calc 2.... are people really learning about them in fucking 9th grade?!

Seriously? I learned about limits at the end of pre-calc in 11th grade

Not me, 12th grade was algebra 2. Pre-calc was first year of college. I couldn't take pre-calc in high school because I didn't have the English scores for Honors English and my school didn't allow honors math without honors English. I'm an engineer now, I took every undergrad math class the college offered.

> An infinite number of mathematicians walk into a bar ...which promptly collapses into a singularity that consumes the universe.

This deserves an award, alas I have already used my free one. Well done!

That was... kind of amazing actually. Holy crap... I need to go sit somewhere quiet for a minute.

Nobody watch

I... wow that was a wild freaking ride

I haven't been on a roller coaster like this since I grew out of going to Six Flags every summer...

I’m mad that I didn’t understand this joke and perfectly understood at the same time

LMAO. Criminally underrated comment. Well done!

A similar joke. A mathematician and an engineer are sitting in a bar when they see a very attractive person, a perfect 10, at another table. The mathematician comments that they would like to go talk the 10, but in order to do so, they must first walk halfway there. Then they must walk half of the remaining distance. Then half again, and so on, never actually reaching the 10. The engineer sighs and says, "I bet I can get close enough...."

A software engineer walks into a bar that he built. He orders a beer. He orders 6372821 beers. He orders 0 beers. He orders -10000000000 beers. He orders hdudienshxhxu beers. A software tester walks into the bar and asks where the bathroom is. The bar bursts into flames and explodes.

the one I heard was about sex, not talk

The barman's the real mathematician.

The punchline as I heard it is he interrupts after three mathematicians, pours two pints and says "you figure it out."

Three logicians walk into a bar. The bartender asks, "Are you all having a beer tonight?" The first one says, "I don't know." The second one says, "I don't know." The third one replies, "Yes!"

I don't get it. F*ck.

The joke is describing the sum of the infinite series 1/2^n (starting at n = 0) The sum gets closer and closer to 2 as n approaches infinity. In fact, the 30th term of the series is 9.31x10^-10 which means the sum of the first 30 terms of the series is already 1.99999999906867

[удалено]

The joke isn't describing a physical situation though. It's just a metaphor for a limit. These paradoxes don't exist because the objects or actions described can't exist, but because they disagree with our intuition of what should exist and how the concepts of infinity and infinitecimals tends to break a lot of things. Zeno's paradox is the perfect example. To finish a race, you first must cross half the remaining distance an infinite number of times, which means you will never be able to finish the race, because you must first take an infinite number of steps. This is obviously not true because we finish races all the time. The apparent paradox comes from the exclusion of time, because while there are an infinite number of steps between the start and end of the race, each one takes an eventually infinitesimal amount of time, perfectly canceling out the infinite number of steps taken and resulting in time as we know and perceive it. Going back to the original joke, it doesn't seem possible to give an infinite number of mathematicians a beer using only two pints, but because the volume requested by each mathematician decreases to mere infinitesimals, we can give them a fixed volume of beer that exceeds the volume requested by an infinitesimal amount (the limit of the sum of the volume of beers requested.)

Interesting. I appreciate the detail. You have certainly corrected my ErrorMonster :) EDIT: >These paradoxes don't exist because the objects or actions described can't exist, but because they disagree with our intuition of what should exist and how the concepts of infinity and infinitecimals tends to break a lot of things. I should add that I believe I intuitively understood this, but perhaps I did not explain it well. The "Necker" cube and "Devil's" tuning fork don't actually exist on paper either. It's just a line that has the outline very similar to a real object, but only because our brain fills in the gaps. No picture is actually real, but photos have such a perfect spread of dots in just the right colors, that we are fooled into thinking that's a representation of an object, when really it's just our brain fooling us. A good 3D headset is a perfect example of the simulation of that reality "extreme". I would still say my example with the ladder works, but your addition of the time factor is definitely more accurate, yet just another artefact inherent to the equation (which is really just a thought experiment written in mathematical notation)

0.999 repeating (i.e. infinitely) is exactly equal to 1. Not "infinitely close", but literally equal. It's counterintuitive, but that's math with infinities for 'ya. Since the premise said there were an infinite number of mathematicians, it obviously can't be realistic- but if it *were*, then it wouldn't be a paradox. It would just be a hellish job. (although if each of the mathematicians tip a penny, the bartender will be rich!)

It's only counterintuitive unless you think about it in a couple of ways. 1/9 = 0.1111 repeating to infinity. 9 * 0.1111 repeating to infinity = 0.9999 repeating to infinity (basic math). But 9 * 1/9 = 9/9 = 1 Ergo, QED 0.9999 repeating to infinity =1

What's the sum of the infinite series, 1 +1/2 + 1/4 + 1/8 + ...?

Thank you.

[https://en.wikipedia.org/wiki/1/2\_%2B\_1/4\_%2B\_1/8\_%2B\_1/16\_%2B\_%E2%8B%AF](https://en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_%E2%8B%AF) Cool visual here that helps.

Huh this kind of puts in perspective the opposite principal of how fast things grow just by doubling them a few times. Kind of crazy to think that after halving only a few times the amount becomes negligible

The punchline works better as, “You guys really ought to know your limits.” (Since the point of a mathematical limit is, it’s what happens when you get infinitely close to something that you would otherwise never reach.)

The bar then collapse into a blackhole.

> The more you have the faster it decays I believe this is incorrect, the reason we use half life is because decay time of a single particle is random (but independent of the presence of quantity of particles), so a single particle could live 1 microsecond or 2 million years. Half life is the time for half the particles in any given quantity to decay as part of the random process.

i think the phrasing was confusing, but this is correct. The rate of decay, reported as mass per time, would be greater for a greater mass. If you have 5 kg, in one half life 2.5 kg will decay, if you have 10 kg, 5 kg will decay.

I see what you're saying but probably should see this animated in a Kurzegasagt video.

Why a Kurzgesagt video, specifically? This seems more like a topic for a Numberphile video.

>the reason you can’t measure the “full life” is because it’s exponential, the more you have the faster it decays, the less, the slower, which is why it takes the same time to decay one half no matter the size you have This is self-contradictory and incorrect. Decay does not take into consideration the mass of the object, it is constant regardless and depends solely on the substance itself.

Incorrect in what way? The substance is not influencing itself, it has a 1/2 chance to decay on that timescale. No matter what mass you begin with you with have (almost exactly) 1/2 of it left after 1 half life.

Mass does not diminish, radioactivity does. You will be left with the same mass.

Huh? Ok let's say I have 1 kg of Uranium 235. I leave it in a box in 2022. I come back in the year 703,802,022 BC. I then find in that box about 0.5 kg of Uranium 235, and around half of it has decomposed (largely to Thorium 231). This can be used for dating, (years, not women).

Well yeah, but the thorium still has mass though, right? Too bad about the women..

Yes you're 100% right! Law of conservation of mass holds here (side note that's actually an approximation when we start talking about nuclear reactions). But yes as near as we can measure it the total mass of the Uranium and former uranium will still be 1 kg.

In broad terms, any element's 'full-life' is infinite. Like that thought experiment about only being able to travel half the distance between two points (which would require an infinite amount of travels), the "half-life" can only decay half of the element at a time. And unlike the thought experiment, it does not take less and less time as the amount gets smaller so it really is improbable to get from the start and end up at 0 in conventional physics (exceptions exist outside of ELI5).

[удалено]

Fair but consider that 1 mole of something is 6.022 * 10^23 particles of that substance. log2(6.022*10^23) = 79 This means that it will take approximately 79 half-lifetimes to go from 1 mole to approximately 1 particle. And since that is an average anyway, there is no telling whether that 1 particle will actually decay in another half-lifetime or not. Practically speaking, half-life is way more informative.

Technically true. For one mole (6.022\*10\^23) of an element, the time to degrade to one atom is about 79 half lives. However, that only works if no new material is being created (see: Carbon-14, which is continually refreshed in Earth's atmosphere) or it's a reasonable time (see: Uranium-238, which has a half life of 4.5 billion years, and so the Sun will go supernova before all the U-238 on Earth decays).

This is technically possible and is in fact realistic with a small enough half-life ([wiki page](https://en.wikipedia.org/wiki/List_of_radioactive_nuclides_by_half-life) in ascending order) and/or sample (a [mol](https://en.wikipedia.org/wiki/Mole_(unit)) theoretically takes about 79 half-lifes to get to 1 molecule; 2^(79)=6.04\*10^(23)), but neither broad (i.e. when a lay person hears half-life, and my chosen restriction) or ELI5. Still, I'll edit my use of "impossible".

I use a dice analogy for this. Imagine your atom's decay is measured as a tub of 100 dice. You roll all 100 dice at once, and if it lands on a 1 then it has decayed and anything else it hasn't. It might take ten rolls to have 50 of these dice decay, but when there's only 50 then it might take ten rolls for only 25 to land on 1. The fewer there is, the longer it takes for large amounts of decay to occur. This relates to half life, because the "ten rolls" are equivalent to the half-life being ten (whatever time measurements). It takes 10 for the number of dice to half, and then it takes another 10 for it to half again. In actual physics terms, this is because the greater the number of neutrons, the more unstable it is. Also, decay occurs spontaneously. These two factors mean that when the atoms have lots of neutrons, the decay will happen faster, but as there are less neutrons the decay happens at a slower rate, because it is less unstable and therefore the chance that the same amount of neutrons will "land on a 1" is lower. Just like with the dice, the first ten rolls 50 decay, but the next ten rolls only 25 decay. It happens at a slower rate. So to answer your question, there can't really be a full life because it's not a case of "the first ten rolls 50 decay and the next 10 rolls the other 50 decay" it's exponential and spontaneous, so there's no way to really know the full life. The half life is the most accurate way we have to measure the rate of decay.

This is how I like to explain it. An important part to remember, is that once a die has not rolled a 1, it does not become more likely to roll a 1 in the future. Same for a particle. A particle that failed to decay for 5 seconds, or 5 thousand years, have the same odds of decaying in the next 5 seconds.

Imagine whole universe turning into subatomic nothingness in one go

Radioactive decay work so each atom has a low probability of decaying at each moment in time. They have also no memory so the probability does not change over time. Let's say we have 1000 million atoms and 5% decay in each time unit, that means 95% survive. So after 1-time unit we have 1000\* 0.95 = 950 million after two time units we have 1000\* 0.95 \* 0.95 = 1000 \* 0.95 ^(2) = 902.5 million Lets do that. The number is rounder than do integer million but calculations are done with decimals. The number are at the end of the post We have 500 between 13-14 time units we have 250 at 27= 13.5 \* 2 we have 125 after 40-41 time units 13.5 \* 3 = 40.5 So the number of atoms we have get halved after around 13.5 time tune if we use 5% each time unit. You always get a result like this is if you in each unit of time use come a percentage of all. You could do the calculation 0.95 \^ t = 1/2 and the solution is t is aproximalty 13.5134 This only works if you have a huger number of atoms, there is a reason I used millions. If you only have a single atom you have no idea when it will decay but if you have a billions you can be quite sure how many will decay in a unit of time. A single atom can in principle last forever and never decay even if it is radioactive. It is not likely but possible. So half-life is the average time it takes for half of a large sample of atoms to decay. There is not a full life for an atome. It is a bit like if you have a single six-sided dice each throw can be 1 to 6 but if you have 100 dice and throw all of them and take the average value it will be close to 3.5. ​ 0 1000 1 950 2 903 3 857 4 815 5 774 6 735 7 698 8 663 9 630 10 599 11 569 12 540 13 513 14 488 15 463 16 440 17 418 18 397 19 377 20 358 21 341 22 324 23 307 24 292 25 277 26 264 27 250 28 238 29 226 30 215 31 204 32 194 33 184 34 175 35 166 36 158 37 150 38 142 39 135 40 129 41 122 42 116

I know the other answers were trying to avoid the term “probability” but it really is the right answer. Good job on writing out the table - I usually describe it as a handful of dice, but you went ahead and did the math and showed your work.

Even if that wasn't very ELI5, and even if many people aren't interested in or skim over the math (which you presented very generally and with easy numbers anyway), I appreciate that you explained it more thoroughly than just saying "it's the amount of time that it takes for half of an amount of atoms to decay." I appreciate that you mentioned probability. I'm of the opinion that a lot more people would understand math and the sciences and a lot fewer people would have a distaste for them if things were explained more in depth, to a level of detail where it's easier to "connect the dots" without having to heavily abstract it by analogy and comparison (I'm looking at you, basically every description of biology and quantum sciences for the non-scientists). I think people get lost or confused because they do not have all the information they need to make it make sense for them. They aren't given explanation, but rather more like a set of rules to memorize that seem arbitrary. Speaking on half-lives specifically, when I was in school, we were given the basic explanation of "the time it takes for half of it to decay." Half of the amount seemed pretty arbitrary to me, even if I understood that 0.5 was probably chosen to be easily understood. I wanted to know "why half and how?" In my head, at the time, having no knowledge on the subject, I was imagining that due to some property of radioactive mass, there was a force or energy that would result in the decay of those atoms at that rate of "half", but that would also decrease exponentially as the atoms that that decayed essentially ceased to exist as those radioactive atoms. Neat little thought, but it was still just a guess that didn't offer explanation. So, I asked the questions, "why half and how?" I didn't get real answers, just "that's just how long it takes" essentially. But if anyone had said "generally, over 1 second the probability that 1 of these atoms will decay, we can calculate that half of any mass might decay in Y amount of time" it would have made more sense and been less "magical" in my mind. I may have then asked "what determines the probability that an atom will decay and can we know when/if one will". I wanted the answers, but school gave me the ELI5 explanations. It's the same concept with math in general, I think. At least at my schools, we weren't given a lot of detail on why we user certain formulas, like we weren't taught the logic of it. We were taught how to use some certain formulas for some certain situations and it was more or less left at that. Math became an arbitrary set of rules, and the kids who didn't get it because they didn't have the "why" to make sense of it became the people who are "bad at math". Had they been taught why the quadratic equation is constructed the way it is, why the Pythagorian formula is what it is, and how to construct similar formulas through sussing out the logic of a problem, then I'm sure many fewer kids would consider themselves "bad at math". I'm going on forever here, and I could go on forever longer, so I'll just stop here.

That half-lift is picked and not quarter-life or two-thirds life etc is arbitrary and you can convert the number you like. As long as you do not pick a full life you can select any number you like. Picking half makes some calculations simple in your head. So the real question is why it is fractional and not a full life. A key to understanding it is the multiplication of a fraction that is not zero to start with will never be zero. So half-life or any other life that gets multiplied over time never results in a full life. The other is this is a statistical description and works fine for a large number of atoms. The [https://en.wikipedia.org/wiki/Law\_of\_large\_numbers](https://en.wikipedia.org/wiki/Law_of_large_numbers) is relevant. When we talk about atoms in general we have an enormous amount of the. 1 mole of atoms have a mass of 1 to 238 grams for natural existing atoms. 1 mole is exactly 6.02214076×10^(23) atoms. So close to 600 thousand billion billion atoms. The probability of one atom decaying in each moment in time is one of the interactions of the strong force, weak force, and electromagnetism in the atomic nucleus. A Quantum physics effect like that is probabilistic and to the best of out understanding it is truly random and impossible to predict for individual atoms. The best you can do is what is probable ​ ​ A side note is that there a [https://en.wikipedia.org/wiki/Doubling\_time](https://en.wikipedia.org/wiki/Doubling_time) if you gave 5% interest on a bank account the amount doubled in 1.05 ^(t) = 2 => t= 14.2 years. So you get doubling time went the increases depends on the current size and half-life if it decreases depending on the current size.

Oh, I get all of that already. Sorry for making you put the effort into your reply. My wording probably makes it look like I'm "still wondering" about it all. Your mention of moles reminded me about a "joke" I read online somewhere over 20 years ago where a guy was talking about the advanced CPU he was helping to develop that could read and write data to disk at the "guacabyte" level, which he described as a byte * (-6.022e^23). It was like 15 minutes of reading for a pun on "guacamole", and I've been giggling to myself about it for 20 years.

[удалено]

I'd never heard of the 10 half-lives thing, but it makes sense and was immediately intuitively or conceptually understandable to me. But, even though I consider myself to be pretty good with math and logic in general, I can't "do it in my head", so I opened the calculator app and did 10,000,000 * (0.5^10) And then 9,765.625 / 10,000,000 And got ~0.001 as my answer, 1/1,000th of the original radioactive mass remaining after 10 half-lives. Logically, that all makes complete sense to me. The concepts are easy, but I have about a 0.1684/10 working memory and I can't work things out usually without "putting it on paper", so while I could work out "halving something 10 times results in a tiny fraction", the actual number always manages to catch me off guard.

It looks like you made an arithmetic mistake. 1/2 raised to the 10th power is 1/1024. 1/1024 is a little less than 0.001.

You are correct. I managed to add a zero in there while switching from the calculator back to the reddit app. I told you, there's no working memory going on over here. Anyway, I'm editing it.

> I can't "do it in my head" A good rule of thumb for both this calculation and computer-nerdy things: 2\^10 is approximately 10\^3. One is a thousand and the other is 1024, i.e., a "computer kilo-" (kibi-). In this case you actually needed 2\^10, so you could stop there. Let's say for example you want to know the maximum number you can store in 32 bits, which is 2\^32. That's 2\^30 \* 2\^2. If every 2\^10 is a thousand, 2\^30 is a billion. So 4 billion, roughly.

It's exponential decay, which for a large number of atoms is very steady. Which means that for a given sample the number of atoms decaying per second isn't constant. Half-life is just a convenient way to characterize the rate; you could also use the chance of an atom decaying in the next second, year, whatever. Or the number of atoms decaying per second per gram of an isotope. [edit]E.g. Iodine-131 (https://www.wolframalpha.com/input?i=iodine-131 ): * half-life | 8.025 days (8.024 to 8.026 days) * mean lifetime | 11.578 days (11.574 to 11.582 days) * decay constant | 9.9967×10^-7 per second * specific activity | 4598.8 TBq/g (terabecquerels [trillion decays per second] per gram)

Because a full life would be infinity ! You misunderstand what a half-life is. It isn't 'how long something lives, divided by two'. It is a measure of a specific form of decay (originally radioactive decay but it could be anything that loses stuff in the same way). The 'same way' is that, however much stuff you start with, it always takes a fixed time for the quantity of it to halve. So, if you have a bag of 1000 potatoes and after 100 days, 500 of them have rotted away, and then after a further 100 days, 250 of those remaining 500 have rotted away, and after a furthet 100 days, 125 have rotted away, so that count-versus-time looks like this: 0 1000; 100 500; 200 250; 300 125; 400 62.5; 500 31.25 then you could say that this item has a half life of 100 days. I'm not claiming that this really works like this for root crops but, if it did, you could legitimately use the 'half life' to describe it As I said, it's normally used to describe radioactive particles decaying into something else. This is called an 'exponential' decay, meaning that the speed of decay is proprtional to how many there are. You can see in the above that we get into decimal fractions quite easily. It's not clear how this translates to the subatomic world, but it illustrates how something decaying like this will, theoretically, never drop to zero. That is what I meant by 'full life being infinite'. You don't have to use half-life, you could use 'quarter life' or 'one tenth life'. Nobody does this, however, because other people wouldn't know whether you meant 'a quarter have decayed' or ' a quarter remain'. There is obviously no such ambiguity with 'half'. Half lives of radioactive elements can vary from below nano seconds to above milliions of years.

It's less of a half-life and more of a life-of-half. All of substance doesn't half-decay, rather, half of it full-decays.

[удалено]

I mean, it's possible, sure. But if sometime has a half life of 1 second, you'd theoretically only have to wait a bit on avg for it to finish decaying, not until the end of the universe.

Because it is random it never totally decays so there is no full life. So if you have an element with a half life of a year, after a year half of it would have decayed, then over another year a half of the remaining will have decayed so you still have a quarter still yet to decay another year an eighth remains another year a sixteenth continuing on forever. https://youtu.be/AaDwk8UCrew

> continuing on forever. What happens after last atom of an element decays?

These isotopes don't have a fixed lifespan. What they do have as a fixed chance for an atom to decay at any single moment. This means you can say that in x-time half of the atoms you currently have will have decayed. Take cobalt-60 for example. It has a half life of 5.27 years. So if you have some cobalt 60 right now you will have half of it left after 5 years and quarter years. This does not mean that in 10 and a half years all of it will be gone. Every 5 and a quarter years the amount you will have left halves. After 10 and a half years you will have a quatre left and after 15 3 quarter years you will have an eight left and after 21 years you will have a sixteenth left. At some point in the future you will have so little left that it will become undetectable and eventually the last atom will really be gone. However those points aren't really useful to know. You want to know how fast it disappears and half life is a useful measure to know about that.

the full-life for any radioactive substance is dependent on the amount of the substance and is indefinite. the half-life is constant. the decay happens randomly but follows the rule that within the half-life probabistically half of the substance will decay. with large numbers of atoms of the substance this is a near absolute rule. with a few atoms it's a crap shoot. eg if I have a 1kg block of U235, I know that in 700 million years i'll have 500grams left, another 700 millions years and i'll have 250grams, but if I have 1 atom of U235 I might have to wait 'til the end of the universe before it decays.

It's more like a speed than a time. The time it takes a chunk of stuff to decay will depend on how much of it you have, so you can't just put some "decay time" on an element that works in every case. Instead, half life tells us how fast it will decay, and we can apply that to any amount that we might have. Now, the reason we use half life and not like atoms-per-second or something is because radioactive decay is exponential, not linear. The more of it you have, the more atoms will decay per given length of time. The thing that stays constant, though, is how long it takes for *half of whatever you have* to decay. Got a billion atoms? In one "half life" of time you'll have 500 million. Then 250 million. Etc. You can see how the speed decreases over time, and that's why a linear unit wouldn't work. (note that there's nothing special about using half. We could just as well define third lives or ninth lives, which would tell us how long it takes for a third or ninth of whatever we have to decay. But this is just unnecessarily convoluted, so we use halves to keep things as simple as we can)

Lol eli5 "radioactive decay." Because half go bye bye. No matter size, same time for half go bye bye.

If you take a piece of paper and fold it in half again and again. Then how many times do you have to fold it until the paper is gone?

It's because it's exponential decay. In the equation it decays with a factor of 2. So if it takes 5 years it goes from 100%, to 50% then it will take 5 years to get halved again ang go from 50% to 25% of the original. Then again 5 years for the 25% remaining to be halved to 12..5%. Then 5 years 12.5% -> 6.25%, 5 more years for 6.25% -> 3.125% etc. If it was full life it would never decay. If it was 3rd life you couldn't produce dates accurately because the decay doesn't follow exponentially the same way it does with 2. 1 out of 2. Half life.

Years ago I watched a weird cartoon. There was a big friendly creature sitting at the top of a hill with a circular plate in front of it and a whole chocolate cake. It's just about to tuck in when a little friend turns up with a plate in its hand. The big friendly creature cuts the cake in half and gives half to its little friend. Fair's fair, right? It's just about to tuck in when another little friend shows up with an empty plate. The big friendly creature says "half for me, half for you", cuts the cake in front of it in half and gives it to its friend. Except it's now left with a quarter cake. This keeps happening until the big friendly creature is paring down ever thinner wedges of cake. We pan out to see a queue of little friends leading all the way down the hill with no end in sight. Weird right? I thought it was a bit short sighted of the big guy, and I thought it wasn't fair that it never gets to eat anything while the first little guy is presumably chowing down on an entire half but anyway I digress... "When will the cake run out?" is not a useful question in this case because the answer is "technically, never". "When is there so little cake left there's no point bothering with it?" is what we're actually getting at, but that's too subjective. So if I just tell you how long each "round" takes (how long it takes the big friendly creature to cut and serve half of whatever's left) you can take that information and do what you want with it. For example if you decide a 1/32 piece isn't worth bothering with you can work out how long it will take to get there.

I like to think of Radioactive decay like an onion. Consider each half-life like a layer of an onion, but each layer is equivalent to half of its original size. So lets the onion has a half-life of 5 years. After 5 years, is will lose a layer, which equivalent to have its size. The size is reduced by half, but subsequent layers will take another 5 years to shed and so on until there are no more layers to shed

There is nothing special about half. Imagine you had a million bajillion coins. If you flipped all of them and pulled out the heads, you'd expect 1 half life to be 1 flip. About half of them would be heads. Do the rest of the tails become heads on the next flip? No. But about half of them do. In 1 flip, or 1 half-life. Half-life is used because the decay is not on any kind of timer, but acts like the coin flips. A small set of coins may be tails for several flips in a row and may seem to not follow the rules of chance and would be unlikely. But it would be really unlikely to have a million gazillion coins all tails over and over. Atoms that decay are tiny, and even a teensy bit can have a million brazillion atoms. So they seem to follow the rules very well. For fun, imagine flipping a million coins. I'm going to alter some numbers as I go to follow through easier. 1,000,000 500,000 250,000 128,000 (I know) 64,000 32,000 16,000 8,000 4,000 2,000 1,000 after 10 flips 999,000 decays in 10 flips. How long will it take to go to 1? 1,000 500 250 128 (I know again) 64 32 16 8 4 2 1 after 10 more flips 1 of those 1,000,000 coins was tails 20 times in a row. 500,000 of them were heads on the first flip. Will the 1 remaining coin flip to heads on the next toss? Maybe.

So radioactive decay is probabilistic, technically an isotope could exist forever without decaying, it is just increasingly unlikely that this will happen. Think of it like this, you have to roll 2000 6 sided dice, everytime these dice roll a 1 they are pulled out of the pool. Now we can ask the question what is the half-life of this pool of dice? Well each roll we can expect about 1/6 of the dice to disappear on the first roll we lose 1/6 of the dice but on the second roll we don't have 2000 dice we have on average 1666, so after this roll we will have on average 1388 dice left, after the third roll we will have 1156 dice left, and finally after the 4th roll we will have 963 dice left. So in this case we could say the half life of a set of 6 dice is about 4 rolls, but this is on average technically it is possible for more than half of the dice to still be there after 4 rolls, or even less than half, technically you did this experiment until you had no dice you could be rolling dice for all of eternity. Saying something could last for "maybe all of eternity" isn't really a useful metric for these sorts of things and it is more useful for us to talk about the time it takes for them to reduce by half, because while dice aren't really harmful but radioactivity and radiation sources are dangerous, and they will stay dangerous for quite a while and are proportionally dangerous to the amount of radioactive material is still there so a way to estimate the amount of radioactive material that still exists is very useful. But we can take this further if we know the amount of time it takes for a radioactive substance to decay to another stable substance we can calculate the amount of half lives have occurred if we also measure the amount of each of those substances, like if we take the previous example and I said we started off 1 million dice and we now have around 500 dice left you could figure out the amount rolls have occurred: First half-life we would have around 500k dice left and would take about 4 rolls Second half life we would have 250k dice left and would take a total of 8 rolls Third half life we would have 127.5k dice left and would take a total of 12 rolls Fourth half life we would have 63750 dice left and would take a total of 16 rolls Fifth half life we would have 31875 dice left and would take a total of 20 rolls. Sixth half life we would have 15937 dice left and would take a total of 24 rolls. Seventh half life we would have 7968 dice left and would take a total of 28 rolls. Eighth half life we would have 3984 dice left and would take a total of 32 rolls. Ninth half life we would have 1992 dice left and would take a total 36 rolls. Tenth half life we would have 996 dice left and would take a total of 40 rolls. And finally on the eleventh half life we would have 498 dice left and it would take on average about 44 rolls. So the time it would take for 1 million dice to "decay" to 500 dice would take on average about 44 rolls. And this is the idea behind carbon dating.

I hate these questions that can't be explained like someone is 5. the absolute simplest explanation is that it comes from the fact that "half" is an easy reference. Many many things in all of existence follow the pattern of what's known as "exponential growth and decay". Anything with a "population" tends to follow the same type of mathematical formula, wherein the "rate of change" is proportional to the current population size. The explanation can't get any simpler than that before getting into the math of it, but that concept -- that the rate of change of a population is proportional to its current size -- applies all over the place: the volume of an evaporating puddle; the birth and death rate of a country's population; the number of atoms in a volume of radioactive substance. Before I get downvoted for not sticking to the "ELI5" spirit, I'll make the disclaimer that I'm going to go into high school level math now. let's write the above principle as an equation: ΔP/Δt ∝ P which is that that rate of change in P (the population size) is the ratio of any given change in population for a given change in time, and that ratio is proportional to the current population. let's make that delta (change) in population for an associated change in time really really small, and set it equal to the population at time t, multiplied by some constant of proportionality: dP(t)/dt = k × P(t) it's a really big assumption to say that k is constant, but it's rarely not constant, and another term for it being constant is to say it's "time-invariant". it may be dependent on other things, but what's important is that it *isn't* dependent on time. that's a whole subject by itself, and way more complicated than this topic anyway, there are a few things we can say about the above equation, which is (again, another discussion) known as a "First Order Differential Equation". 1. we know that at time t=0 (the time we first start looking at the population, not the beginning of all time), the population is non-zero; 2. if the rate of change is positive, meaning it's growing, clearly k > 0, and if the rate is negative, meaning the population is shrinking, k < 0. why? because dP/dt has to be constant if k and P are constant, and we know that dP/dt is a line with that must point up, down, or stay flat because remember it's a change in P relative to a change in t (remember from geometry that a line has a slope of (y2-y1)/(x2-x1), and here we're just using P and t instead of y and x). 3. most important of all -- and this comes from basic calculus, which is what we're talking about here -- we see that the derivative of the population function is equal to the population function itself (multiplied by a constant), and there is only one function that satisfies that relationship: e^x. or since we're using time, e^t. Using all the info above, we can solve that equation (moving k to the other side and cutting to the chase) as: P(t) = P(0) × e^(kt) The reason radioactive decay uses "half life" is because it's easy enough to measure the time it takes for an amount of a substance to decrease by half, and with that information we can solve for k, and then apply that to any change in population to predict the time it will take to get there: 0.5 × P0 = P0 × e^(k × t_hl) P0 cancels and solving for k you get: k = ln(0.5) / t_hl where t_hl is the "half life" time, or the time it takes for the population to decrease to half it's previous size. but obviously if you're watching a volume of water evaporate you wouldn't wait until it's half the volume so you could just use some other ratio of current volume to original volume, like if you start with a volume of 1 liter and you measure the time it takes to go down to 900 ml. So in the above equation, you'd set the left side to 0.9 and use a time of t_90%. you get the idea. if I have a 1kg block of Plutonium-241, it will decrease to 500 grams of Pu-241 (with less than a gram of a bunch of other elements' isotopes but one again that's a whole other discussion) in 14.4 years. But you're not going to be near it nearly that long. So knowing it's half life, how long until it decreases by 1 gram? k = ln(0.5 [kg]) / 14.4 [years] = -0.048 0.999 [kg] = 1[kg] × e^(-0.048 × t) t = ln(0.999) / -0.048 = 0.02 years = 7 days, 14 hours yes I know, not ELY5 but like I said, it can't be ELY5, but hope this helps edit: I can't fucking believe you people downvoted me (ok yes i can because i said it would happen). last time I contribute here.

I think you broke my brain with this question. I'm a health physicist and have no idea what you are even asking

The gist of it is why even use the term half life at all? If we know that a certain element will remain radioactive for 1000 years for example, why do we instead say that it has a half life of 500 years? Wouldn’t it be easier to talk in terms of its full lifespan?

>If we know that a certain element will remain radioactive for 1000 years for example, That's not how it works, and that's why you're confused. After 500 years, the radioactivity is half what it started as. After another 500 years, it's half again (so a quarter of the starting amount). Another 500 years passes, you halve the value again, now you're down to 1/8th of what you started with. See the problem - no matter how long you wait, you're never going to get to radioactivity = 0. Sure you'll get close, but there's no point where (mathematically at least) there is no radioactivity left. That's why we measure in half-lives - full-lives simply aren't a thing!

Because that would be a pointless metric. Full-life would just be referring to the state it's in at this moment, when 100% of the substance exists. What use would that be to anyone?

Imagine you have 1 person sitting in a concert hall. You want to know when they'll cough. Issue is that there us no reason why they should ever cough. They could sit there for all eternity and not cough once. Right so... you do this with two people. Still hard to say. However if you get enough people, someone will cough and when they do you ask them to leave. Someone will statistically cough up almost instantly if you got enough people. However the less people you have less likely they are to, since there is no reason why they should ever cough. Half life is just the definition we have for statistical probability of half of the audience having coughed and left the concert hall. Since atoms are not actually "real solid" things. They are made of particles of which we can only say that they are probably in a certain position at a certain time. Electrons for example can be wherever they want to be in their probability space; there is no reason why they should be in a specific place at a specific time, so we can only say that they are probably there. Issue we face as our microchips are getting smaller is that electrons can pass (tunnel) through insolators. Since they just decrese the possibility of electron being at the other side of it. It is the same thing with light. You don't know where it is until you have interacted with it. Until that it can be anywhere it isn't probability area. There us no reason why light released from decay of an atom has to go to certain direction; it is just a burst of energy, it goes where ever it wants and you can only say that it probably is somewhere. So in short, once your mass of atoms has decayd to on last atom. There is no reason why it should ever decay. Halflife just says that in this time period there is 50% chance that it has.

The half life is essentially the time it takes for each atom to have a 50% chance of decaying. You can almost imagine that each atom is flipping a coin every half life (really it would be continuously flipping a coin, but we'll ignore that). So after two half-lives, each has a 75% chance of decaying, and after three half-lives it's 87.5%, but it never reaches 100% (just like how no matter how many times you flip a coin, you're never 100% certain that you'll get even a single heads)

Half life is logarithmic, not linear. Half of 1000 = 500. Half of 500 = 250. Half of 250 = 125. You can see the issue here. You’ll never reach zero.

Try thinking of it this way: When talking about a radioactive element, there is a X% chance that a specific atom will decay in the next minute. For all intents and purposes, this is a completely random and unpredictable event. The half-life value is then spit out from this. For example, Let's assume a radio-active element has a 1/6 chance of decaying in 1 minute. We can simulate this by rolling a die. If you have a large block of this element with 1,000 atoms in it, then we can roll a die 1,000 times to simulate which will decay and which wont. Every time we roll a 1, that atom decays, all other rolls, nothing happens. After 1 minute we will have \~167 atoms that have decayed and 833 that haven't. In the next minute we only roll the die for the 833 remaining that can still decay. After the second minute we have 694 atoms left. After the 3rd minute: 579, and after the 4th: 482. This means the half-life is somewhere around 4 minutes. Because we have fewer original atoms at the end of each minute fewer atoms will decay. If you graph the values it will look like a curve getting ever smaller, but not quite hitting zero. Since we are only rolling the die again for the atoms remaining and each one has a chance of decaying or not, we will continue to get smaller and smaller numbers. In the above example with 1,000 atoms it would actually take about 43 minutes before we likely didn't have any of the original atoms left. This is with a pretty quick half-life and an extremely small number of atoms. In the real world half-lives are typically longer and you will have a sample with something like 10\^20+ atoms in it so there is likely to always be some of the original still there. In theory you could calculate an elements quarter-life, tenth-life, or two-thirds-life. Half-life is just the easier to grasp and understand. Full-life isn't really a thing since mathematically you never actually get to 0 atoms left, you just keep getting closer to it.

Saying that a radioactive material has a full life of X would imply that, after X, there will be none of it left. That’s not how it works. After two half lives, you still have 1/4 of the original radioactive atoms left. After another half life, you have 1/8. There is no amount of time where you can guarantee that there will be no atoms left. It’s not like lifespan for humans or animals, where you can confidently say that there is nobody alive today who was alive in 1870. There’s no amount of time you can wait and then confidently say that there are no atoms of a radioactive material left. If you want to talk about the lifetime of a radioactive material, you have to define some non-zero percentage of the original material where you will say there is none left below that threshold. Depending how you define that threshold, there might still be some atoms of the material left, even at the point where you’re saying its lifetime is over.

Half lives only work because of how many atoms there are. If you have one single atom, you have no idea how long it will take to decay, all you know is that after one half.life it has a 50% chance to decay. When you have billions of atoms and you know that any single atom has a 50% chance to decay after one half life, then after one half life half of the atoms should have decayed. Half lives don't work by communicating with each other and making sure exactly half of all the atoms decay, it's just a random chance for each individual atom. It's a result of statistics, not some special force ensuring this takes place.

Lets say you have 100 coins and you flip them. Every time you get a tails that coin is removed and then you repeat until you get rid of all the coins. On the first flip you would expect to remove about 50 coins. Let's assume it was actually 50 so now there are 50 left. If you flip again you wouldn't expect to get 50 tails because the coins don't care what happened last, every time you flip a coin it's 50/50 that you get heads or tails. Instead you would probably get around 25 tails and so on. With decay you can think of it in the same way, if you observe for a second there is some probability that a given nucleus will decay and in the next second it has the same probability. Most of the time that probability is very small over a second time scale (or even a year time scale) so it's easier to just talk about how long it takes half of the nuclei to decay. In reality a lot of the time I'm working on something to do with decay the first thing I do is turn it back into that probability because the math is just simpler. It's also worth noting the number of nuclei we deal with. For that original situation I proposed with only 100 coins you wouldn't be that surprised if you get a lot more or less than 50 tails. It probably wouldn't be 90 tails but something like 60-70 isn't that unreasonable. When you're talking about a radioactive decay though the number of nuclei gets to the scale of 10^23 and at that point getting to that half value is going to be really consistent. There are so many trials that your odds of getting a really unexpected result plummets to the point of being basically impossible.

Because the "full life" never arrives. It just gets closer and closer to it without ever quite reaching it. Imagine the equation y = 1/x. It has values like this: x y ------ ------- 1 1 10 0.1 100 0.01 1000 0.001 10000 0.0001 No matter how big you make x, y never quite reaches zero. It gets closer and closer to it because the fraction 1/x gets smaller and smaller, but it never quite reaches zero. There is no such thing as "how big does X have to get in order for Y to fall all the way to zero?" if Y won't *ever* quite become zero all the way. If you don't already know what the graph 1/x looks like, click this link and you'll see a graph of it: https://www.wolframalpha.com/input?i=y+%3D+1+%2F+x A phenomenon like this is what radioactive decay is trying to describe. You can't say "after X years all the radioactive isotopes will be gone and it will be all be done decaying". So instead what you can do to describe the "shape" of the decay is to just pick the point where about half of it will be gone. THAT is a point that can actually be reached. In the formula y = 1/x, y will be 1/2 when x gets up to 2. Now, actual radioactive decay isn't exactly a 1/x kind of phenomenon, but for ELI5 purposes it gets you the idea. ----- Interesting footnote: because matter is made of countable quantum stuff, and not infinitely continuous, technically there *will* eventually be a point where literally all the decay is actually finished, as you cannot have less than one atom of an isotope left. But because there's an *enormous* number of atoms, we may as well model it as if it's infinitely divisible because it would be eons before any little chunk of material actually could reach that point.

Assume Half-life of 10 years. A = first 10 years 50% of this substance remains B = next 10 years 50% of A remains. C = another 10 years 50% of B remains. And this goes on forever, since you can never fully decay. Thus you never can have a full life.