Whether you love 'em or hate 'em, chances are you rely on numbers every day of your life. Where do they come from, and what do they really do for us? This hour: stories of how numbers confuse us, connect us, and even reveal secrets about us.

## Innate Numbers?

In "25 Minutes to Go," Johnny Cash counts down the minutes to his hanging. This precipitates an argument between **Robert** and **Jad** about whether you could live without numbers. Jad introduces his newborn son, **Amil**, and insists that he has no concept of numbers whatsoever. Like father, like son? Producer ...

## From Benford to Erdös

Mark Nigrini shares the story of physicist **Frank Benford**, a man whose curiosity about a book inspired a bizarre discovery. Benford's Law, as it is now known, reveals a cosmic preference for certain numbers. Then **Darrell D. Dorrell**, a forensic accountant, describes how he uses Benford's Law to bust ...

## Calculove

Producer **Soren Wheeler** brings us a story about a friendship between Steve Strogatz and his high school math teacher, **Don Joffray**. Steve explains how numbers can connect you and where they fall short.

## Comments [87]

Benford's Law is really only accurate when you're drawing from a distribution spanning many orders of magnitude (hence the appearance of it in Frank Benford's log tables) or drawing from multiple unrelated distributions. How then, is it rightly applicable in the case of phony tax returns or other cases of accounting fraud? Obviously, a business citing $1,000,000 in annual revenue for one year isn't going to report $10,000,000+ the next year and you can't just add tax return numbers to some other unrelated data set and expect to squeeze out incriminating data from that. It seems to me that the use of the law for forensic accounting purposes would spawn some fishy conclusions.

What is the song after the first section where the choir counts and sings, crazy bass line in the background...anyone?

I have heard this broadcast twice. Before, I considered myself too poor in 'math sense'to latch on to its meaning.

Currently, I am amazed with the power and possibilities that the #2 represents both symbolically and mathematically. Symbolically, it represents duality as a function of what we perceive as polarity, ie Black vs White, positive versus negative, left vs right.

This has lead me to what maybe some rather naive assumptions which perhaps I would enjoy going into further here, in this forum.

So just to throw this out there; if all humans enter the world with math skills, there is a 'oneness' that argues against the polarity that we experience in societies throughout the world.

If a 2 can be successfully entered in to the binary code, not as a way to calculate faster, but slower, it will slow things down and thereby speed thing up simultaneously. Less heat, more light.

I believe what all humans experience as polarity is as artificial as our conception of 'time'. One Love, One Heart, And Justice For All.

Jakzen

So, can I use Bedford to predict lottery numbers? I have read conflicting accounts about whether or not such numbers conform to Bedford. This source (lotterypostdotcomslashthreadslash148025)concludes that they do, but fails to provide a prediction strategy.

EXCELLENT PROGRAM!!

I just heard this. Excellent program.

It is not surprising that our "unlearned" concept of numbers is logarithmic based. All of our senses are based on logarithmic values. Linear values are useful for measuring and creating physical items but that seems to be about the limit. Most recently it has become obvious that our monetary systems and national finances would more easily grasped if we used a logarithmic representation.

wonderful program

So this show is 3 years old but I still want to post...

Logarithmic counting:

We don't lose this ability entirely. Ask a 5 year old person "What is an old age?" and they will answer perhaps 10 or 15 years old. Ask a 20 year old and they will answer perhaps 40. A 40 year old will answer 70 or 80.

At the same time as we age each year that passes will appear to be shorter and shorter. "Where did the year go? Seems like a month ago I was writing 2011 on my checks..."

This is simply because the number of years we have lived increases in a linear fashion, but how long a year represents compared to our lifetime is exponentially reduced each interval. For a two year old a year is 50% of their lifetime. A 10 year old a year is 10% their lifetime. A 50 year old a year is 2% of their lifetime. As we age this will continue to get smaller and smaller (feel shorter and shorter) until death.

Could it be that baby"Emiel" is the original analog to digital converter? Does he exchange powers of ten additions for for multiple increments? This show was absolutely fabulous.I'm 70 and I was thrilled.... had to blab it to everybody I know. Thanks

I get it now. On the question of concentric ellipses, the foot-wide concrete walkway around the perimeter of a swimming pool, is an ellipse in only one case, when the ellipse is a circle.

If an ellipse is defined as PF1+PF2=2a, then (PF1+1)+(PF2+1)=2(a+1) is also an ellipse. However, a concrete walkway around the perimeter of a swimming pool, which has a a constant width, is an ellipse only in the case of a circle, where F1 and F2 lie on the same point. The width between the inner and outer circumference defined by the concrete walkway will vary from one to one half the cosine of the angle between PF1 and PF2 times the distance between the concentric ellipses (which is the width of the walkway).

So, it is not impossible for the concrete walkway around the perimeter of a swimming pool to be an ellipse, but it is possible in only one case, a circle. In all other cases, it is impossible.

The song at 19:40, bjorn in orlando, is the Pi Song by Hard 'n' Phirm.

Regarding the comment that Benford's law applies to Fibonacci numbers, factorials, powers of 2 and the powers of almost any other number. But, Benford's law does not apply to square roots?

If Benford's law applies to Fibonacci numbers, and to the squares of Fibonacci numbers, wouldn't Benford's law apply to the square root of Fibonacci numbers squared?

Just curious.

The question of concentric ellipses, the foot-wide concrete walkway around the perimeter of a swimming pool, is an ellipse.

If an ellipse is defined as PF1+PF2=2a, then (PF1+1)+(PF2+1)=2(a+1) is also an ellipse. The foot-wide concrete walkway around the perimeter of a swimming pool can't be anything but an ellipse.

Where, PF is the distance from a point on the circumference to a focal point, F1 is focal point one and F2 is focal point two, and a is the distance from the center point between F1 and F2 and a point on the circumference that lies on a line through F1 and F2 bisecting the ellipse.

Mystified by the question of concentric ellipses? (The foot-wide concrete walkway around the perimeter of a swimming pool.) Why impossible?

How would Benford's Law be observed in base-6? I'd enjoy seeing a comparison of the same measurements recorded in base-6 and base-10.

Nearly three years after it was first broadcast, I was delighted to listen to your "Numbers" episode today. The section on Benford's law was great, but I have a small quibble with the assertion that using it for forensic purposes is a recent phenomenon. Fritz Scheuren, past President of the American Statistical Association, told me that the Office of Naval Research used Benford's Law for many years to check data from research contracts. You could check with him about how long ago that was, and what other earlier uses he might be aware of. It has also been used to check on election outcome data.

What is that song with the woman singing a series of random numbers at 19:40? it's amazing

Xavier, it's definitely a version of Ebb Tide (that's the organ at least). The Ken Griffin version was also featured in the season 5 premiere of Mad Men.

Does anyone know what the music is at 19:51 with the cymbals and organ? Thanks for any info!

The section near the beginning of this show where they studied how babies have the ability to view changes in quantity of the ratio of the change is large enough was very fascinating. Question: what happened in civilized society if we teach math so differently from how our brains are equipped to learn it. I struggled with alogarythims,but it appears that our brains are not meant to struggle against them.

Thanks for the show!

I'm willing to suppose there may be a valid connection between our inability to grasp and adequately problem solve with large numbers (7 billion people, 16 trillion debt, and so on) and the natural logarithmic understanding we have lost, to which our minds may naturally tend towards. (Presumptions, I know. But it is an undeniably curious thought) We may have developed an integer based system that really doesn't illustrate these critical mathematical patterns in ways we can truly digest.

Might logarithmic or, perhaps more so, equational expressions of the values we observe be a more tangible and impressionable idea? That is, however, if these numeric taboos had been nurtured early in life as valid skills. Doubling, exponential growth and other remarkable and simple mathematical facts are all but lost in the public problem solving skill set.

Thoughts. So many good thoughts on Radiolab. Thank you forever.

There seems to be a lot of confusion on Benford's Law. People start counting, reach 19, see that a lot of the numbers start with 1, and say "Benford's makes sense."

This is not Benford's Law.

Benford's Law applies to things that have a uniform logarithmic distribution (for example, things that grow exponentially). Compound interest is one of the best illustrations.

Say you have $100 in an account that gets 5% interest. If you left it there for 94 years, you would have balances that began with 1 29 times!

"But wait!" you say, "You started with $100. Of course a lot will start with 1."

Fair enough. Start with $200. Now you won't see a number starting with 1 until you have a balance in the $1000's. However, 94 years later, you still have had 29 balances that begin with 1. (Benford's actually says you should get 1 about 30% of the time.)

The reason for this is that the interest is compounding by this function:

New balance = last year's balance * 1.05.

If your last year's balance starts with a 1, it won't grow as fast as if it started with a 2 (and much slower than if it started with a 9).

In response to Connor, I remember using logarithmic tables in college...that was around 2004. So they are still in use and still pretty handy, I think.

*Benford's.

Bemford's Law makes sense because every power of ten has ten times more numbers that begin with one than the last power... as in, like, within the millions, there are 9 million fewer ones than there are within the ten millions. So as you count up, you're always going to be getting more that begin with 1.

I gotta be honest,this episode was damn boring to me

Seriously, how can make a show about numbers without mentioning prime numbers? You could easily make ten shows about prime numbers.

I wonder if the intuitive reason why Benford's law is true simply boils down to the fact that the universe prefers lower energy levels, and higher energy levels are less frequent because energy/resource are limited- for anything. So, numbers are just counting quantities, that in the end, are measuring an energy level; hence, why low numbers are more frequent. Bank accounts are a perfect example of what I mean. To make the count of money in the account go up requires that you expend more energy to acquire more money. However, there is finite amount of money and with high competition and limited available resources, it requires one to expend a lot of energy and intelligence to get more money than others. So, it is much more likely that the lowest energy (money) level will be populated the most, and each subsequent energy (money count) level will be progessively less populated. This principle could be applied to anything being counted because everything requires some energy/Intelligence be applied to it for it to achieve the unique state of being that we attribute a name to.

Anyone ever heard of a theory like this? Please email me if so at ari_groups@comcast.net

Anyone know any books for laymen explaining Benford's Law?

I'm an engineer and I can't understand music at all, but I'm interested that our natural view of numbers is related to harmonics, our view of sound is related to sound frequency, our nerves are sensitive to pulse frequency, our eyes are sensitive to light frequency and our taste/smell is sensitive to the frequency response of molecules. Crazy stuff. I wonder how we quantify smell? Linear or log?

The idea of people thinking logarithmically about numbers by nature doesn't seem to be a stretch if you consider sound. If you look at an equilizer each fader represents a particular frequency. The faders on either side of the first will be the first frequency doubled or halved depending on the direction from the first fader. Or consider a piano...Middle C may be near the center of the keyboard, but nowhere near the middle of the frequencies represented on that same keyboard.

I beleive that most people would feel that they have a better understanding of music than they do math, but I think this show has implied a link to the way our brains percieve music and the way we naturally think about math.(?)

Regarding Benford's law. The way that I have made sense of it (it seemed counter-intuitive to me for a long time) and the way that I explain it now to people is to ask them to think about a quantity. Whatever quantity you pick to discuss. Then think about it being 10% larger. With a little more explaining, it becomes clear that 10% more than 80 is 90. And 10% more than 90 is 100. And 10% more than 100 is still 110, 120, 130 ...etc. etc. The portion of answers to 10% starting with 1 as a leading number are self-evidently more common.

can someone tell me wether im on the right track here...

in my opinion benfords law is just pure logic and im not seeing why so many people doubt it

ofcourse most numbers will start with 1's or 2's cause we start counting from 1

I mean if I open up a saving acoun I will not start with 99,99 I will start with 100 it's just common sense...

*wanted to place something here but lost my train of toughts*

only thing that didn't make sense is that they where talking about benfords law and random numbers

I mean if I did a random 1-100 each one got a equal chance.. but according to benfords law 1,2,10,12 etc. would all have higher chance...

This was a neat episode, but I can't believe you totally glossed over the connection between logarithms and Benford's law.

The first piece (natural counting is logarithmic) and the second (Benford's law) were saying the exact same thing: logarithms are natural. If you can wrap your brain around half of 9 being 3, then Benford's law is an easy step. If you asked the Amazonians from the first part to make up random numbers, they would probably follow Benford's law.

Just started listening to RadioLab after hearing you guys on This American Life. Fantastic. Really really fantastic. This has literally turned my work commute to one of the best parts of my day.

This episode especially is almost life changing for me. I'm so in love with numbers in general.

This episode makes me want to go back to school and study math.

Thanks so much.

It must be a slow night for me , because I just read most of the posts for this episode...

It must be fantastic to have a brain that thinks math is interesting. I don't. I'm not stupid, I just don't find math that intriguing. I could sit down and try to dissect a math concept I don't fully understand, but I'd rather cook a big plate of cheese enchiladas and have my friends over to devour them. Just a matter of what makes each of us happy.

I listen to RadioLab because it makes the things I might not find interesting just that - INTERESTING.

So, this very excellent radio show is not meant to be an upper level college class that is unequivocally proving something. It is meant to make us think (I think.)

You science dudes need to take a deep breath. Bless you, but the rest of us simply like to be smartly entertained.

I made my husband sit down and listen to this program, I've sent the link to my kids. A few weeks from now we will talk about it at the dinner table. This is not the first time this has happened. Robert and Jad have created moments in non-scientific lives where science (and math) were discussed and enjoyed.

Thank you!

A new NYTimes series: Steve Strogatz "writing about the elements of mathematics, from pre-school to grad school, for anyone out there who’d like to have a second chance at the subject..."

http://opinionator.blogs.nytimes.com/2010/01/31/from-fish-to-infinity/

Benford's Law is very intuitive when you consider that even if numbers are random, they are random according to a distribution, and that distribution has a RANGE that is NOT random. If the range is all within the same number of digits, then the starting number will be random, but once the range ticks up an order of magnitude, you get as many again numbers that all begin with a "1," then as many again as begin with a "2," etc. Take a random sample of bowling league scores: lots of 1s, some 2s. Or MLB batting averages: lots of 2s, some 3s, a few 1s (at least for part of a season). Physicians' salaries, hotel maids' salaries: lots of 1s, some 2s. Very intuitive.

I agree with DU, this started off great but I was disappointed overall. "Numbers" were covered for the first quarter of the podcast, then the episode spiraled into story time with barely any mention of numbers or math. I felt the same way about the "Randomness" episode. Let's get back on track guys!

I can't help but think the logarithmic understanding of quantity is built into our visual filters as well. Consider this optical illusion, where the weight of the white squares is increased by small, nearby elements, causing the squares to have more "weight" in the field of vision and distorting the perception of space.

http://www.youtube.com/watch?v=QKCSBkdEUXQ

Numbers are indispensable in a material world that sustains conscious and creative beings since you need to be able to quantify accurately in order to also create something accurately. Ask any architect if they would be able to design anything sensible without using numbers, or, for that matter, software that uses no numbers. The obvious benefits creep up in virtually any preoccupation of humanity.

The case about babies having a logarithmic intuition of numbers defines, on a prima facie level, an interesting conundrum but is very well explainable if one is willing to accept the idea that babies think visually about numbers. With that I mean that babies have a visual conception of quantities of things. The show brings up 8 ducks in the babies' visual field. Now doubling that number obviously leaves a very salient change in the perceived image by the baby. We are then at 16 ducks. Doubling it again, same story, same registration of change. We are now at 32 ducks. Now add one duck. The change from 32 to 33 leaves only a minor change on the babies perceived visual field. So consequently, the baby's mind also notices a small change. With logarithmic perception of numbers is meant that factors (2,4,8,16,...) matter, not so much increments (2,3,4,5...).

Logarithmic counting is quite primitive because it needs the number of quantities in the visual field to increase by a factor in order to properly register in the baby's mind. You know, going from 2 ducks to 4 ducks, to 8 etc... The baby is only able to count because the numbers increase very crudely and visually obviously, namely in factors.

When the baby gets older, the perception of quantities (numbers) gets more sophisticated. Now the baby learns to be able to detect small one-by-one increments in numbers.

The more primitive logarithmic intuition of numbers should be retrievable again by picturing numbers and so think in terms of factor increases (or decreases) again.

Hope this helps.... :)

I see Damon already beat me to it! But here's a pic of the Paul Erdos icon: http://is.gd/5oYG3

I was particularly moved by the story of Paul Erdos - as I was one of the folks asking "Why is he up there?"

I just love your weblog! Very nice post! Still you can do many things to improve it.

Then not only custom, but also nature affirms that to do is more disgraceful than to suffer injustice, and that justice is equality.

Quotation of Plato

There was some interesting stuff in this episode, but overall I found it pretty disappointing. You have a pro-intellectual, science-y program and yet you found the need to perpetuate several anti-math stereotypes. Several times during the program Jad and Robert both said things like "I could never understand this stuff" or "who needs numbers". Then the long story at the end reinforced the "math is the opposite of humanity" stereotype.

Also, there are so many topics you could cover in math and yet you lumped them all into a single "Numbers" episode. First, numbers are only a small part of math. What of topology, for instance? Second, throwing all of mathematics together in a single episode is like throwing biology, physics and geology under a single episode labeled "Natural Science".

I think you guys can do better and I look forward to hearing the result.

Poetry is nearer to vital truth than history.

Quotation of Plato

I am definitely bookmarking this page and sharing it with my friends.

:)

I've just listened back to this episode, and was reminded how funny I found it when you explained what Logarithmic Tables are – as often used in the first half of the 20th century I think. We had then when I was in primary school in Ireland… and that was in the 80s!

Usually we used the old ones belonging to our older siblings and even our parents – those ones had all the text in Irish to boot.

The were indeed worn more at the front that the back, so the new ones you'd buy in the village store looked sparklingly different.

And then we'd go home from school and watch Ferris Bewler on TV :-)

There are alot of examples to show that people never lose their logrithmic reasoning. One great one is to think about value comparisons; a person will cut a coupon and carry it for a week to save 40 cents on a can of fruit, but may not be willing to do the same for a car, under the idea that 1 dolar is "farther" from 60 cents than 10000 dolars from 9999.60 dolars, even though they share and equal absolute difference.

Also, just because I can't resist the urge to join the meta-discussion of Radiolab itself, does anyone else think Robert sounds a little like Shake from Aquateen Hunger Force? Sorry Robert :-)

I really do love this show and think it's one of the best things out there. It's so refreshing to hear people exploring the universality of the natural world instead of just trying to keep up with the machine of human information culture. Especially when you put so much creativity and style into your production value.

I don't think that there is anything spooky about Benford's law. When we count things, we always start with 1, then go to 2, and so on. That's what counting is. If your chance to stop on any number is completely random, but you can't get to 9 without going through 1, 2, 3... first, then of course your chances to stop on 9 are going to be the smallest, because you had 8 chances to stop before you got there. The reason why sets of counted numbers fall into logarithmic patterns is just because our numeral system is base 10, meaning that every time you get to a quantity of ten in the place you are currently counting at, you move up one place. That place always starts with a 1.

If, however, you counted in hexadecimal, you would create a similarly shaped Benford's curve, but it would be distributed between 1-16 (or 1-F) rather than 1-9. This might be a hard concept to grasp, simply because unlike babies and aborigines, we've been programmed to think in the base ten counting system because that's what our language has evolved as. Unless you know other numeral systems like hexadecimal, or understand why time is base 60, it might seem like magic. Really though, it's all in built into the number systems that we all agree to commonly use. This is where numbers are really amazing, because they show the human preference to standardization in communication, because every language (that I know of) uses a base 10 system.

If you've ever compared metric, which is all base 10, to the English system of measurement, which is all random and crazy and based on the sizes of various body parts, you know how convenient base 10 is. If you can grasp how other base number systems work though, it opens your mind to a lot of possibilities. Thinking like this is what allowed us to understand things like binary and hex, that are more friendly to electrons and computers. These are examples of newer number systems with the computer age, but time, and the minutes and seconds of latitude and longitude remind us that even base 10 was invented to simplify things down from base 60 counting language. I would love to hear Radiolab do a show on how language evolves.

The thing that cleared up Benford's for me was understanding that it only occurs in number sets that have logarithmic distribution, and not in every data set. These types of sets do exist all over nature, but we encounter data sets with non logarithmic distribution all the time. For instance social security numbers, zip codes, and phone numbers because they are assigned artificially and often contain digit distributions that signify other things (eg: the area code and exchange in phone numbers).

The thing that's really puzzling is, why does our natural world favor logarithmic distribution so heavily?

I find it interesting that I feel more comfortable thinking about numbers in the supposed "native" logarithmic way.

I have been diagnosed with dyscalculia which makes dealing with numbers a bit of a problem and makes my head all swimmy when I try. The logarithmic grouping does no such thing for me. I may have to proposed this to some folks...

http://www.youtube.com/watch?v=O8N26edbqLM

Below are how often 1-9 show-up in various counts. But is this the only influence? I thought the law reflected growth patterns described in numbers?

[you may have to c/p this for alignment sake]:

1 2 3 4 5 6 7 8 9

200 111 12 11 11 11 11 11 11 11

500 111 111 111 111 12 11 11 11 11

1000 112 111 111 111 111 111 111 111 111

2500 1111 612 111 111 111 111 111 111 111

ttl 1445 846 344 344 245 244 244 244 244

I am fascinated with the biblical implications of logarithmic logic. 2 fish and 5 loaves of bread, and 900 year lifespans take on new meaning when realizing that people living in this time probably had more in common with the Amazonian tribes than with us. The effort that goes into "tricking" us to believe in the decimal/numeric system that the program showed, was probably not present until modern times.

Radiolab does wonders for late night musings. Robert and Jad do more for stimulating new ideas than anyone else. Thank you so much.

My favourite Erdős story is about the Erdős graph. Vertices represent people, and an edge between two vertices (people) means that they have written a paper together.

Mathematical papers have been written about this graph that look at all of the properties of the graph (number of spanning trees, colourings, etc). In some cases, mathematicians wrote papers together expressly for the purpose of giving the Erdős graph a certain property!

I know it's a nerdy thing, but I'm pretty sure most of the people who listen to Radiolab are nerds, so maybe you guys like that story.

http://www.oakland.edu/enp/research/ is where to go to find out more about research related to Erdős.

To all of those leaving comments about Jad's negative attitude toward math, I'd like to point out that this is an excellent radio show featuring storytelling, one tool of which is to illustrate ideas by showing contrasts. I don't think Jad is truly anti-mathematical. He was humorously playing the part of an irrationally number-resistant person so that Robert could make counterpoints to illustrate that numbers are truly pervasive in our lives. It should be fairly obvious that Jab has no malicious intention toward math-lovers.

Anyway, another excellent show. I love you, Radiolab.

I agree with Casey about Jad's (apparent) tone throughout the show. I don't think he was simply expressing a personal discomfort with math, he was expressing it with an almost sneering condescension. Doesn't he realize that his prestige and salary as a radio/podcast host rests upon an unsung army of support workers that either have to understand math or, god forbid, actually *enjoy* working with numbers? Jad's voice would have carried just as well across the airwaves if he hadn't been looking down his nose the whole time.

I admit I was impressed, but confused by Benford's Law, until I wikipedia'ed it and found that the key is that it applies only to the first digits of the numbers from collection of measurements of a natural system. Not to all the digits of any set of numbers. I won't repeat the explaination, but it was pretty straight forward and keys on the first digit and measured system requirements, which I don't at all remember being in the radiolab description. Maybe I missed it in the broadcast, but without those restrictions it is wrong and no surprise people don't understand it.

A second point I would make is as an engineer, I always think in ratios and I think most people who aren't afraid of numbers do also. Calling it logrithmic thinking makes it sound much more complex than it really is and I think creates more confusion than understanding. If you call it ratios or percent (ratios of 100) then many more people would nod their heads that this is what they do even as non-babies. Either way, I go crazy when a news report says something like 'deaths from hulu hoops decreased by 412 last year.' Rather than saying the both the decrease and the base, or at least the percentage. Just giving the number without the context is pretty useless but so commonly done. Sadly that may indicate that too many journalist “don’t get numbers they’re so hard.”

Loved the show.

You know, I've had a theory for a while now about something very similar to the idea discussed in the first section of this podcast. Lulu is speaking about how infants see the difference between 1 and 2 as huge, and the difference between 9 and 8 as tiny. I heard this and it struck a cord with me because I've been trying to explain this idea to people for years now!

Imagine instead of numbers, we're thinking in terms of time and speed. I remember being younger and listening to music with my father. I would put on a punk rock album, something fast and slurred, and he would get so frustrated with me. "I can't understand a word they're saying! It's too fast!" he would always tell me. And I simply couldn't understand what he meant, as I could listen and enjoy and comprehend. I would play the same album for my mother, my grandmother, my sister, teachers, classmates, anyone who would listen and give me feed back, and I found that it was almost always the same results. The older you are, the harder it was to understand what was being sung.

(I should point out that I'm sure that this isn't ALWAYS true... At the time it sure did seem like it was fact.)

So I began, in my little 12 year old mind, to piece together reasons why this seemed to be so universally true. The conclusion I finally came to was that it had to do with our relationship with time. I thought about when I was 1, and how incredibly long my first year of life must have seemed without any point of reference to compare it to. I imagined that 2 must have seemed long as well, but I imagined my 2 year old brain saying, "Well, at least this isn't last year, that took forever!" The years go by, and I just knew that time must have felt like it was speeding up even if I wasn't fully aware of it.

And if this is true for time (just think about how many times you've heard someone say "This year has gone so quickly!") it only makes since that this true for how we hear music! A young mind actually HEARS things slower than someone older.

Anyway, enough of my rambles. Thanks for another great podcast guys! The part about Erdős really struck a cord with me :)

J.W.

p.s. kids tend drive fast, old folks tend to drive slow. I wonder if that has to do with this as well? Blah!

The logarithmic number issue with babies and other cultures reminded me of the miles per gallon (MPG) issue. When most people are comparing three cars that get 10, 20 and 30 MPG respectively, they see each jump as equal. But, the jump from 10 to 20 is much bigger than 20 to 30. If we reported the efficiency of cars in gallons per mile, we could make these comparisons using the integer counting instead of logarithmic.

See

http://www.mpgillusion.com/2009/02/overveiw-of-gpm.html

Casey, I think that Jad, like some people, have a serious disdain for numbers. It is not "a myth" to them. Although I feel he was acting it out some I didn't find him obnoxious. In fact there are some out there (not me) that I could see being drawn into a show that without Jad's comments would feel left out on a subject like numbers. I am sure they wanted to keep them listening for a future show they might be more interested in.

Also I agree with Paul on Chris' explanation on Benford's, all his sets of numbers would be represented as integers. I mean how often are bank accounts counted in that fashion? 1,2,3...no I would argue the phenomenon is seen in single digits as integers, ie $1500 or $2300.

The only thing I could take from Chris' sets of numbers is that these integers start at 1 if several digits long (1000 or 10000) and will revert back to a low integer before they will a higher one (7000) which could be a very small subset of explanation.

Johnny Cash is great and all, but I was more impressed with the Moondog music. The convergence of the mathematical nature of Moondog's music, his transient lifestyle and outsider status as a musician was a beautiful parallel to Paul Erdös' story.

I enjoyed this episode quite a bit, with the except of Jad's attitude towards numbers/mathematics. It's a common approach to be all, "oh I don't get numbers they're so hard", and I find it quite annoying. It perpetuates the myth that numbers and math *are* hard and it's ok to be ignorant of them.

Numbers -- integers, anyway -- are not that difficult to grasp. Neither is basic multiplication, division, addition, or subtraction. These basic operations cover almost all common experiences we have with numbers. To proudly declare that you don't understand them or that you don't like them is absurd. To put this in context, think of someone's response to you if you kept talking about how hard reading is, or how your mind just doesn't get reading, or that people who read are just so weird and unusual, all while trying to have this strange, artsy air about you.

People have hesitations towards numbers mainly because they don't interact with them enough to make parsing an easy experience. That doesn't mean it's innately difficult, or that only special people can know them well. Just be aware of them more, and you'll chill out about them and not need to be obnoxious when the subject is broached.

To be honest, I hate math. I'm horrid at it, however ever since I was a child I've had an obsession with numbers. Not how they add, or how they divide. Simply their existence. After a few years of training myself to think of someting else other than the millions of numbers I was able to focus on other things. Listening to this podcast made that obsession resurface. And I have to say, now that I'm older, I finally can have fun with the Fibonacci Squence Hahaha. I love this Podcast!

Does anyone know what song starts at 3:55? I realize this is the least interesting part of the show, but I really like the way it sounds.

Don't talk smack on Johnny Cash, Jad. "Is that Johnny Cash?" Come on.

Wow. I'm still trying to get over the "25 Minutes to Go," Johnny Cash song!

Benford's law would not be unintuitive if we continued to think of numbers according to logarithms. If you think of the number 9 as two instances of the number 3, then it only makes sense that the distribution of 3 would be more than 9. How could it not, since you can't have a 9 without two 3's

This was previously posted on October 9th of this year:

http://www.wnyc.org/shows/radiolab/episodes/2009/10/09

Here's one way to think of Benford's law. Let's say that you look at a bunch of people's bank accounts (in $), and see the distribution of the first digit. Then you convert the money to British pounds, or Yen, or some other currency - would you expect the distribution of the first digit to change? No, you wouldn't - there's nothing special about U.S. dollars.

But that means that the distribution of first digits can't be uniform. Consider a currency where $1 = 2 of this currency. When you convert to this currency, every dollar amount that starts in 5, 6, 7, 8, or 9 would start with 1 in this new currency ($5 becomes 10, $60 becomes 120, $999 becomes 1998, etc.). So if the distribution in dollars had each digit represented equally often, then in this new currency half of the bank account balances would start with 1 - nowhere close to uniform.

For converting to a different currency not to change the distribution of first digits, you can see that there have to be a lot of 1's and fewer of the higher digits. There's only one distribution where changing the scale doesn't change the expected distribution of first digits, and that's Benford's distribution. (This is called scale invariance.) So anything that's expressed in arbitrary units - money, distances, physical constants, etc. - should follow Benford's law.

Where I go to church, we have a giant mural surrounding our rotunda, of "Dancing Saints", painted in the style of Orthodox icons--including one Paul Erdos. I'm glad to hear we're not so unique in considering him a saint. I remember pointing out his icon to someone, once, who was appalled--"Erdos?! Why's *he* up there?!"--for pretty much the same reason that Robert expressed in the show: he had apparently been (or known) one (or some) of the people on whose couch Erdos had crashed. Despite Erdos' mathematical graciousness, he was not, I gather, always the most welcome of houseguests.

I'm not much for proselytizing, so if you're interested in reading more about the Dancing Saints mural, hey, that's fine, but don't pin it on me: http://www.saintgregorys.org/worship/art_section/243/

What's interesting is that the reason Benford's Law work is that the numbers are essentially uniformly distributed on a logarithmic scale. . .and that ties in with how babies and the more primitive people tended to think of numbers (e.g. 3 is halfway between 1 and 9).

That is, we tend to think that 1,2,3,4,5. . . is uniform, but babies think 1,10,100,1000. . .is uniform.

Mull that one over!

I get Benford's Law, but I don't see how Christopher's diagram naturally proves it. Benford's Law does not describe a phenomenon observed in "sets" of numbers, but instead describes a phenomenon observed in integers.

Translating Christopher's sets to integers, would look like this:

1

2

3

4

5

6

7

8

9

10

11

Thanks, Christopher! I was also puzzled about why anyone would find Benford's so miraculous.

As you note, our number system [which is merely a construct to describe some kind of phenomena] has to begin somewhere, and it makes intuitive sense that lower numbers are most common.

Furthermore, with something like the numbers in bank accounts, it seems much more to the point to look at average salaries.

Benford's law isn't too hard to grasp.

Think of sets of natural numbers of evenly distributed random size. For example:

{1}

{1,2}

{1,2,3}

{1,2,3,4}

{1,2,3,4,5}

{1,2,3,4,5,6}

{1,2,3,4,5,6,7}

{1,2,3,4,5,6,7,8}

{1,2,3,4,5,6,7,8,9}

{1,2,3,4,5,6,7,8,9,10}

{1,2,3,4,5,6,7,8,9,10,11}

etc.

It's easy to see that most of these numbers start with lower digits. If you continue to any maximum size (11 here), the same thing happens.

Hi Jarno,

Sorry for the confusion. Here's the deal:

The Numbers show BROADCAST a while back (every station has it's own schedule for broadcasting us, so it happens over a broad time range).

But the PODCAST stream follows it's own schedule (with a new full show every 6 weeks, and Shorts in the slots between).

As a consequence, sometimes an episode will show up on the podcast after it has already broadcast (and sometimes it shows up before).

Not trying to slip one past you ... just a consequence of putting the shows out in two different ways.

Soren.

Great episode. Listened to it a while back though, and don't really get why it's posted as a new post, dated at November 30th... Trying to sneak past making a new episode by reposting an old one, and hoping nobody'll notice? ;)

Anyway, great job, as always. Radiolab's become my favorite podcast, the one who's episodes I look forward to the most. Hoping for a new one soon. :)

Ack. It's killing me! I haven't been able to listen to any Radiolab since "The New Normal" episode. iTunes craps out after downloading only a few megs, the built in flash player craps out after only 20 seconds. Is it because i'm in Canada?

"The Pixar of radio". Perfect analogy.

Another great show. Two things: Radio Lab is like the Pixar of radio (one hit after another). Thing two: If it's possible to fall in love with a stranger's voice, then i'm in love with Lulu's (in a non-stalkery, non-weirdo way) :)

Great shows.

Great production quality

I subscribe and listen to all.

Cool graphic.

Listening to you about baby and tribe handling numbers logarithmically I thought about my radio experience with sound.

1db is about the smallest increase in loudness we notice.

3db is an apparent doubling of loudness while actually being a 10 fold increase in the power needed to generate that impression.

I think our impression of quantity and our measurement of quantity are at odds.

Sale!

10% off, who cares?

50% off, thats a deal!

http://www.youtube.com/watch?v=zJDu5D_IXbc

I finally understand this!http://www.youtube.com/watch?v=qjOZtWZ56lc

Johnny Cash on Radiolab. I'm happy about that, Jad's barbs notwithstanding...

## Leave a Comment

Email addresses are required but never displayed.