As an IT engineer I'd say the fact you're watching this video and commenting on it without giving it a second thought is a prime example of things that "just work" and are taken for granted, somewhere there are a whole bunch of people making youtube work and keeping it working everyday and the only time people notice them is when it doesn't work.
When studying aboriginal cultures when asked how much larger a 2 is then 1 and 3 an 2 etc they draw a logarithmic curve, so do children before they've been taught that it's linear
It's not profound at all. When you are 2, 1 year is 50%of your life. When you are 10, 1 year is only 10% of your life. When you are my age, one year is an insignificant amount of my life.
@@user-Orkb186-3 I don't think that just because something seems obvious in retrospect, it's not profound. It's been said that simplicity is the hallmark of brilliance, and this seems to me to be a perfect example of that.
Yeah in my hunt for finding a perfect equalizer setting to get more bass out of the music, I did find that simply amplifying bass frequencies didn't quite do the trick. What mattered is how much difference was there between the bass frequency and treble frequency. In a way, I was feeling more bass (there's no such thing as too much bass XD) when I lowered other frequencies than when i was simply amplifying bass frequencies.
Sybilantly distracting - is there a logarithmic scale involved in how much hissy s’s jar like scratching nails down a blackboard? Why does one person track sybilance as an annoyance while someone else doesn’t at all and describes the overall utterance as perfect?
Yay ! New Hannah Fry featuring videos ! Great ! Also I wanted to mention the Mel-scale which is a scale where each step in pitch is "judged" to be equivalent but when you compare it to the actual frequency scale you can see it's a logarithm.
This explains so much. Not even kidding. A lot of these videos are just fun, like, "oh how would you evenly divide a ham sandwich" or "in an infinite vineyard of evenly spaced trees how many trees could you see" which is great and interesting, but this one actually fills in holes in the logic I had built in my mind over the years I've lived. Thank you Hannah.
I think a good example of how marketing uses this is that percentage sales are used for cheap products, whereas raw numbers are used for large ones. They'll say you're saving up to $3000 dollars on a car, but you're getting 20% off your chocolate bar. That's because usually in the former case, the percentage is comparably small ;)
I always say how a year feels like such a long time when you're 5 years old vs. when you're 20 because when you're 5, a year is 1/5 of your life, whereas when you're 20, a year is 1/20 of your life, so it feels shorter. Didn't know there was a term for this, though- cool!
Actually, this probably isn't the reason. Perception of the passage of time is very closely related to the formation of new memories. This is why it feels like time passes slowly when you're in danger: you're remembering a lot of what happened, so time feels slow. Conversely, as you get older, you're not forming as many new memories because you're not having as many new experiences, so time feels like it's going faster. Your brain's basically saying, "I've not formed many new memories since last year, so last year can't have been very long ago."
@@beeble2003 Yeah I actually read an article maybe a couple of years ago showing that the reason why the time appears to "shrink" as you get old, is because when you are younger you can actually process visual information faster and better. Thus you form more memories and consequently, as you said, times seems to go by slower. And when we get older, our ability to process visual information reduces and consequently times go by faster. Don't really quote me on that though. I am just paraphrasing what I thinks was in the article that I read a couple of years ago haha.
Awesome! I remember watching something about how some tribes not exposed to "our" number systems count logarithmically--that they have numbers for 1, 2, 3, but then skip to 5, and then to 9 (or whatever). Which really speaks to this idea...
I wonder what the equation is for determining the ratio of comments on a Numberphile video featuring either Hannah Fry or Holly Krieger that are about the women or the math?
I find it amazing that as a child, one Christmas to the next seemed like an eternity, yet now in my 70s, I honestly ask myself why I should bother to take the decorations down as I will be putting them back up again shortly.
I immediately starting thinking about fractional and percental differences vs absolute differences. Sure 20g is 20g. But in the first example it is 20% and in the second it is only 10%. It is a matter of tolerance.
This is exactly what they're talking about, the fact that perception tends to inherently logarithmically transform quantities. Linear reasoning is very difficult except in small quantities, and requires a lot of training to do.
This applies to many things, including most of our senses, but not to everything. We don't experience differences in pressure logarithmically, for instance (that is, we could tolerate a 100 kPa atmosphere or a 90 kPa atmosphere, but we could not tolerate the winds from a 105 kPa atmosphere meeting a 100 kPa atmosphere, because a 5 kPa difference in pressure is too high). In many cases, our natural tendency to treat situations logarithmically is not rational. For example, we may spend a few extra minutes to save $3 on gas, but we will not spend a few extra hours to save $300 on a house, even though you would be getting more for your money in that case. It's absolutely something worth pointing out.
And it correlates with a study done on tribes which showed that people without education tend to naturally place numbers logarithmically on a line (at least a good amount of them do) It's a very interesting study you should check it out
I searched "uneducated tribes logarithm" and probably found your study: "Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures"
I'm in marketing and product development. We used an anecdote to help explain this phenomena and use as a cautionary tale. The scenario is a cost savings for a chocolate chip cookie (or biscuit for you). A cookie had around 20 chips. You do testing and find that people have a hard time perceiving the difference between 19 chips and 20 chips. So you remove some chocolate and save production costs. The next year we look to save money again. "People have a hard time distinguishing 18 chips from 19 chips in a biscuit". At this point I'd draw a plot with error bars and such. Fast forward a few years and: it's very easy for people to discern 1 vs 2 chocolate chips. Of course they would have noticed far earlier! As I mentioned, we used this as a cautionary tale to help marketers understand how changes to a product might be perceived and what might be valid and important points of reference that a consumer may use. To follow your chocolate bar example - they may not notice a 10% reduction, but they will likely notice when the bar is significantly smaller in their hand than in the past (hand is a constant). Terrific show!
All makes sense, especially with how time “feels” quicker as you get older. When you’re 5 years old and it feels like it’s been ages between christmases, that’s because it’s been 1/5th of your life. The older you get, the smaller fraction a year of your life becomes.
yeah, audio volume sliders also use a log10 in order to work nicely if you keep them linear they have way too massive perceived gain steps in the bottom, and nearly nothing on the top. decibels = 20 * log10(amplitude)
Just a note -- the decibel unit is not an absolute unit. Your equation is technically wrong because it needs a reference point. Your equation is referring to decibels-acoustic, which has some commonly-agreed-upon reference point. The equation for decibels is different. The actual dB equation should be Bels = log10 ( p / pREF ) --> deciBels = 10 log10 ( p / pREF ) ... although there are further nitpicks to be made about p in this one.
It is more complicated than this. How so? Because the percent error between 10 and 9 is 10%, but the percent error between 10^10 and 10^9 is 90%; however, the absolute difference in the latter case is MUCH larger and the ratio between the two numbers is the same in the second case as in the first, which underlined the fact that the perception of differences is not a rational function, but an actual logarithm.
@@angelmendez-rivera351 How ratio between 9 and 10 is the same as between 10^9 and 10^10 ? According to my mental mathematical calculations: 9 divided by 10 is simply 9/10 or 0.9. When 10^9 / 10^10 = 1/10 or 0.1. Can you explain what do you mean by saying 'ratio', because I struggle to find any other meaning of this word than what I have just did.
Sigma 1 I admit I made mistake. I definitely meant to use a different word other than "ratio" in that particular sentence, but as this was 4 months ago, I have no clue what is it that I truly meant to say. I have to rethink it.
i think this comes down to how we have a tendency to compare things proportionally rather than with absolute differences, if a product costs 2 and compare it with the same but it costs 4, it makes much more impact in our perception than comparing some other product whose price ranges between 304 and 306, since in the former example it has a 100% increase, whereas the latter...the percentage difference is negligible
If P is the variable that represents the quantity describing how much one perceives another particular quantity or some particular stimulus S to change, then dP = k dS/S. This implies that P(S) - P(S0) = k ln(S/S0) = A log S + B => P(S) = A log S + C.
Importantly, behavioral economics shows this applies to prices as well as weights. We can't 'see the difference' between $20 off of a $400 object but can sense a $20 off of a $30 object.
This explains the myth of shaved beards growing faster. It's effective just saying we can perceive difference as a percentage much better than absolutes (in this instance 20%). I'M GUESSING you can tell a 10% difference like 100g vs 110g about the same as 1kg vs 1.1kg. But 1kg vs 1.01kg.....well that's only 1%. That's why when I look at difference I always do it as a percentage first and consider
This was the starting point, typically attributed to Gustavo Fechner in the 19th century, of the area in psychology known as “psychophysics”. It basically looks at the performance of humans as measuring instruments using any of the sensory systems or strength of opinions. Nowadays, many psychologists don’t follow the areas as it is rather formally mathematical. It largely remains in the specialty of psychology that is called mathematical psychology”. However, it shows how formal and rigorous studies of scientific subjects always result in its mathematisation. The language of science always, it seems, ends up being mathematics!
Or to me, why people later in life can't deal well with big changes - why old dogs can't learn new tricks, why people die of broken hearts after years living together. The change in intensity is a huge leap that they hadn't been accustomed to since their youth.
Our currency is logarithmic: 'parts' of a one (1p, 2p, 5p, 10p, 20p, 50p or in USD the cents, nickels and dimes), then £1, £2, £5, £10, £20...£50. The value of each item approximately doubles each time. To a small RU-vid channel, being featured on some blog and getting an extra 1000 views and 50 subscribers can be an enormous boost but by the time you are the size of, say, Numberphile, that same boost becomes less significant even though it represents the same time investment and commitment from people. I think CGP Grey even mentioned once that the value of each individual piece of feedback goes down as more and more people give feedback. Our brains certainly seem to naturally analyze situations using a logarithmic scale, where we value the proportional relation between values more than the net difference.
When applied to wealth, Weber's law is equivalent to risk aversion: the pain of losing 10% of your money is greater than the joy of a 10% wealth gain. This is why people pay more than the expected loss amount for insurance policies.
So many people responding that it is obvious that the difference between1 and 2 is more noticeable than 10001 and 10002 miss the point. The point was not that the former is more noticeable than the latter.. The point was that this phenomenon always follows the same basic arithmological structure that can be described mathematically. Or, in other words, this aspect of human psychology can be modeled mathematically. The notion that mathematics can be used to describe such aspects of our subjective experience of reality is far from obvious.
Small point. The initial experiment with the weights is unscientific because the weights in the hands vary is number and size. This could certainly affect the perception of weight. That should have been done with the weights in tiny buckets placed in the hands or picking up the handles. Doesn't change the outcome but could easily change an experiment to determine more precisely what differences are noticeable.
This is truly fascinating, thx! I will just point out a little flaw of your videos in general: I couldn't find this looking for "numberphile logarithm"...
well, when i think about it, even love is logarithmic. When you first meet your lover, every date is exciting and stuff, but after a year or two such dates get "normal"...
Reducing the size of the problem enough so a small change is noticeable - a different color shirt might matter - the spelling of a word - what language you speak - a Weber corollary ?
This is also why, if you have one truck that gets 10 miles per gallon and another that gets 20 miles per gallon, that's a huge improvement in efficiency that will have a massive impact on the price of gas for a given distance... but if you have one motorcycle or hybrid vehicle that gets 80 mpg and another that gets 90 mpg, the amount they spend on gas will be nearly the same for the same distance.
I'm sure someone has mentioned it in the comments already, but in case someone hasn't: there is a behavioral economics model called Hyperbolic Discounting that is precisely Weber's Law applied to how people's preferences adapt with respect to time. This law shows up EVERYWHERE!!!
I wish many more concepts in high school math classes were introduced with some interesting application or appearance of said concept in everyday life like this; it could keep many people interested in the subject or at least see it's not completely useless.
As far as time goes, I think there have been studies that have shown that our perception of passing of time has more to do with the number of “frames” per second drops significantly and in per portion to your age, but we still count them internally the same.
At the end, Hannah seems to imply that giving jail sentences logarithmically is a bad thing because "3 months is 3 months". But the point of jail isn't to quarantine individuals for a certain amount of time, but rather to reform them, correct? So if a 20 year term and a 20 year and 3 month term both FEEL like the same amount of time, then aren't they essentially the same punishment? Maybe prison terms make you feel time differently than you normal do?
Ryan Wilson Your last sentence nails it, but also, you did miss an important point: a lot of people in the West (well, mostly just the U.S.A, but the U.S.A comprises a very significant percentage of what is called “the West”) do think of prison as quarantining process, and they do not agree with using the system as a reform.
time has noticeably sped up after my 18th, 20th and 22nd birthday, but not after my 19th or 21st, strangely. I will try to check in to update you after my 23rd to see if my experience of time changes (judging by the pattern, I expect not)
iPhone torch is my new word of the week. Edit: I wonder if you could do this with calories? What would the scale be? How many calories could you cut in a day without your body noticing and feeling hungry?
That's pretty tricky to measure. Calorie information on most foods is an estimate only. And I've heard any given thing you eat that lists them can have variances exceeding 20% based on unpredictable factors such as variation in ingredients and how any given batch was prepared. Actually pretty important to know when you think about it. Because if you're actively trying to count calories and intended to drop from say 3500 to 3000 calories... Well, 20% of 3000 is +-600 So in any event you'd need more reliable calorie measurements to meaningfully test this, I suspect.
Well not really. You just need a suficiently large sample size. Those calorie measurements are based on the average of the test batch and for most products, those batches are large enough to be representative. Even if for a specific product they are not, you have 2 things in your advantage: first, you eat a large variety of products. So even if the information on one of them is wrong by 30%, it doesn't really affect your daily intake, because it's just 1 product. Second, you can always compare the calories from one product to the other. Like take noodles for example. You know how many calories they have, even if one product claims a value that is somewhat different from the other products, it's still in that ballpark range. They're noodles after all. So you just need to make sure the average calorie intake goes from 3500 to 3000. Yeah in the short term you might get "unlucky" but it'll even out at some point.
In the short term, hunger is more of a learned response a lot of the time. This is why if you say fast for a day, you will most likely feel hungry at the times you would usually eat.
This law applies to conscious minds. “Your body” is subconscious and “you” are conscious. Numbers do not mean anything to your body and your body does not perceive differences between anything. The difference between 1 and 2 calories is the same as the difference between 999 and 1000. Just my own thoughts
titaniumonkey it might seem slower at first with comparatively few things to do but then over time it also might seem to go faster because of the monotony and increasing number of repetitions of whatever there is to do.
Actually that might also be due to Weber's Law because a day seems much longer compared to the time you are on that deserted island, than compared to the time you have been living a normal live. So a day seems to be longer on the deserted island. Of course there are other psychological effects with time perception as well.
Even if Weber's relationship seems obvious to you, it is still critically important because it implies that most people are making suboptimal decisions most of the time. Weber's relationship implies that a person perceives a price doubling for eggs as the same as a price doubling for their rent. This is clearly irrational. We need to study and understand these failures of reasoning in order to make better decisions at personal, professional, and policy levels.
Surely the law actually implies that they would more negatively perceive a doubling of the price of eggs than a doubling of their rent? Or perhaps we should swap out rent for, say, a telly or something (because rent already has extra negative perception around it than eggs before any numbers get changed). If they perceived them as the same the perception would be linear rather than logarythmic.
Eggs: $1 TV: $1000 Now imagine an increase in price. Eggs: $2 TV: $1001 If our perception of cost were linear, then we would be equally unhappy with both price increases. After all, the extra dollar does not know whether it is being saved on eggs or on the TV: a dollar is a dollar. But that is not the case. We see the eggs doubling in price, while it "feels like" the price of the TV barely creeps up. Under the logarithmic relationship of perception, in order to feel an equivalent sadness, the price of the TV would need to increase to $2000. We do not perceive the change's magnitude, but rather the change's scale. This failure in rational decision making is captured by Weber's relationship.
1000 eggs: 1 TV 1 TV: 1000 eggs. That's how money works. BTW average lifespan of a TV is six years and average consumation of eggs over six years isn't far away from a TV's price which is a nice coincidence and by the same time destroying irrational ideas regarding prices and rationality.
heyandy x It would be terrible decision making to use linearity, much in the same way that it is terrible economist analysis to determine income rates by means and not by medians. Also, no, according to Weber’s law, the perception in difference between the doubling in rent is definitely not the same as the doubling in egg prices. dP/P = k dT, and if the boundaries of integration are T = T1 and T = 2T1, then ln P(2T1)/P(T1) = k(T1)
Interesting! I derived this when I was bored in 9th grade chemistry class. It's always fun to learn you deduced an equation that has a name. Makes me feel slightly less hopeless about my mathematical ability!
This law can be applied everywhere, because relative difference is very natural. For example, in radiocommunications the higher a frequency is, the better must filters/oscillators be to work with the same bandwith. And the amount of scailing is equal to this law.
This holds true if and only if the "just noticeable difference" is taken as a measurement unit for "sensation". If, on the other hand, you accept the so called Ekman's Law , the relationship between physical intensity and the magnitude of the "feeling" will follow a power function (not a logarithmic one; Steven's Law). Granted, for many things it does not really make much of a difference - but, for stuff like pain, physical exertion and electric shocks the logarythimic relation falls short of our actual sensations
Exactly, because the slope of the curve is very steep in the beginning. That's why we tend to stay on the same path and reject change, because the slope gets more and more flat the further you go that path, so it takes less effort to move forward even more.
Not even sure if we need courage. After the steep beginning, it evens out a lot. We just need the courage to start a new... oh I see it now (sorry I'm drunk).
Actually, this process is described by both log and exponential scales except that you can change the parameters of each and negate any differences between them with respect to how well they fit actual data. Most people in psychophysics apply a power function because your r^2 value is slightly higher than the log and exp functions.
@numberphile the perception of time moving faster with age is not just due to Webers law but also a number of other factors, predominantly exposure to new stimuli. Creation of new profound memories gives us the impression that there is a lot happening within a given time frame, but as we age, these new stimuli decrease in number, making it seem that not a lot has occurred within a given time frame, ergo giving us the perception that now the time has moved faster. Please make note of this.
i had to see this 4 times before i could concentrate what they were talking about….something made my mind go to a place of happiness….Wonder what it could be..?
Why? Its just a fact of life. Honestly it wouldn't really be much different if physics and biology happened to work on a geometric scale, as long as all aspects of reality were equivalently shifted. Basically it just means instead of drawing 1,2,3,4,... on your Y axis when you graph something, you draw 1,10,100,1000,...
It's quite depressing when this is the only problem in your life. To make it seem less depressing, you should surround yourself with more negative things so that this fact becomes proportionally less significant. :) But really, it's just the way brains work. You've been using Weber's Law all your life without ever noticing, the only difference is that now you know a little bit more about yourself and the way you perceive the world.
Also this is the reason why, when asked for example what is the number halfway between 1 and 9, some people will answer 3. Because 1 * 3 = 3 and 3 * 3 = 9
I discovered this law for myself when I was a student... long time ago. I didn't knew by this time that it has a name. Hannah talks about logarithmic experience and the prices of some goods. I think that such or similar law applies to perceiving of prices when a person lives in a state that suffers from hyperinflation. Surely this Weber's law will be in the core of human economic behaviour in that kind of hard times. Speaking of hard times, what about the WAR?... or any other stressful event that may occur suddenly? Then the time seems to stop or slowing down severely, which, of course, is entirely on psychological grounds.
proud moment, i came up with the "years" theory when i was about 25 when i asked myself why it seemed to speed up so much and concluded it could only be a classic case of "absolute vs relative" values.
There is also an alternative explanation to this, that there are less novel experiences when we get older, so Time actually slows down because there is more neural activity. if time is only as long as we perceive it is, then it ultimately boils down to neural activity. That is why when you drive to a certain location, travel time always seems faster when you return, simply because your brain is processing less information from the trip because it already knows what its seeing.
I've read somewhere that logarithmic thinking might actually be an evolutionary advantage. Because whether you're facing one lion or two lions makes a _huge_ difference. But whether you're facing 20 lions or 21 really doesn't matter, all you need to know is that it's _a shitton of lions._ It's definitely not that easy, but it's an appealing simplification.
The chocolate bar thing is true. It’s a form of inflation. People notice when prices go up, but they don’t notice when the size goes down and the price stays the same. Watch tissue paper in particular. The tubes get narrower and don’t fit in the tissue holder anymore, but keep the same number of “squares” and the same price.
OMG - that explains also a psychological problem. If you are not very much loved (perceived or in reality) you are fine even with ridicules little changes. But if you feel loved (perceived or in reality) you are more stable to some discussions and problems.
I once heard a really good explanation for WHY people would have evolved to think this way. It really makes sense if you think about it because if you're looking at one tiger trying to eat you and then suddenly there's two tigers trying to eat you, that's a pretty big deal, but if you've got 7 tigers trying to eat you and another shows up to make 8 tigers trying to eat you, you're honestly probably not that much more concerned. 😝
from this you can figure out that if we assume the average human life to be about 80 years, by the time you're around 17 you've experienced half of your life.
