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## Algebra (all content)

### Unit 11: Lesson 31

Logarithmic scale (Algebra 2 level)# Benford's law (with Vi Hart, 1 of 2)

Vi Hart visits Khan Academy and talks about the mysteries of Benford's Law with Sal. Created by Sal Khan.

## Want to join the conversation?

- What brought Benford to his discover of this pattern? Was it just curiousity? Did Benford apply this law to any use in science, finance, or engineering?(50 votes)
- Actually, it was Simon Newcomb who noticed the wear on the book and originally stated the law. Though he published in 1881, the paper didn't have much impact. The law was rediscovered by physicist Frank Benford, who published in 1936. This time people took notice, and his name is now attached to the phenomenon.(39 votes)

- Just two minutes ago I went to a site with the first 100000 digits of pi, and jotted down the number of times my browser said it found "1" on the page, then "2" on the page, etc etc. Expecting Benford's law, this made my results much more astounding. The number of times each digit appeared in comparison to all the others was stunningly regular, as each tally was between 9800 and 10200. I know it's kinda not related to this video, but why is this happening? What is going on?(28 votes)
- Pi is suspected to be a normal number. A normal number is a number that has a equal frequency for all it's digits, so five would be as common as six. However we do not know if Pi is actually a normal number.(33 votes)

- Would this work in other number systems like octal, hexadecimal, binary. base 7 etc.?(21 votes)
- Yes, it works in any number system. However, in binary it's rather trivial - Benford's law predicts, accurately, that a random number will surely begin with 1.(44 votes)

- Montenegro IS a country! It is on the Adriatic in Eastern Europe. But he was thinking of Monaco, which is the little country on the eastern coast of France.(27 votes)
- Yes,it is Monaco, which has the feat of the world's most densely populated country in the world,with 36,000 residents in about one sq. mile.

Montenegro is a country,but it was made in 2006, and was part of Serbia and Montenegro,so it may have not yet have become a country when Sal and Vi made the video.

Good catch!(4 votes)

- Who is Vi hart?(10 votes)
- She has an amazing series on "Math for Fun and Glory". Search it up on khan. Well worth your time!(5 votes)

- In a more general sense, isn't this because every time you cross a numerical "threshold" the numbers all start with 1? as we go from 1's to 10's to 100's to 1000's, wherever these numbers stop there will be more numbers in the earlier part of that group than the later ones. (so for example, if you just took some random sample of a lot of numbers from 1 to 60, obviously more numbers would start with 1, 2, 3, 4, 5, or 6. if you did it with 1 to 3000, a lot would start with 1, 2, or 3)(7 votes)
- I am thinking it is probably a property of exponential functions. Most of the examples given are exponential in nature. If you look at an exponential function it is constantly getting stepper. Hence within any given range the smaller numbers will take a larger amount of the x axis. So, if you pick a random assortment of numbers from the x axis and then find the exponential value, for any given magnitude, it is more likely to be from the lower end (ie closest to 0).(7 votes)

- What is the distribution of Benford's law in base two? How would one go about finding that?(4 votes)
- In base two the first significant digit is always one.(8 votes)

- About the odds and evens thing, doesn't that make the most sense? because there are only 4 evens besides 0 and 5 odds.(4 votes)
- I wonder if Bernard's law behaves the same within other count systems, say like the Mayan or Roman count systems?(2 votes)
- Yes; the effect is independent of the counting system you use. If you count in base 60 like the Mayans then you will still see a 1 crop up as the most common first digit. The Roman example is slightly different because the Romans counted in base 10 like we do; they just wrote their numbers differently. In Roman numerals the concept of the "first digit" has no meaning because the numbers are written according to absolute value, not place value. So Benford's Law has nothing to apply to in the case of Roman numerals.(2 votes)

- Surely this couldn't be telling us something fundamentally about the universe though? isn't it just a side effect of the fact that we use a base ten number system? i mean would this still be true if you used a base Sexagesimal like the Babylonians?(2 votes)
- This is independent of what base number you use. The effect is observed in exponential functions, because those show linear distribution on a logarithmic scale (because they are linear on a logarithmic scale). If you've ever seen logarithmic graph paper, you will understand why ones are more likely to pop up.(2 votes)

## Video transcript

SAL KHAN: I am very
excited to have Vi Hart visiting the
office over here. And we were just having a
very mathematical conversation earlier today. And she mentioned something
that is fascinating. VI HART: Yeah, I
was just telling Sal about a cool thing
called Benford's Law. SAL KHAN: Benford's Law. And what is Benford's law? VI HART: It's this
weird phenomenon that you get when you're looking
at numbers in the real world. So for example, we've
got some graphs here. If you take the populations
of all the countries and you, say all right, what
is the first digit of the population
of the country? Whether it's 1 million
or 1,000 or 100,000, we'll say, OK,
that starts with 1. So we'll count up all of the
countries that start with 1. And I guess here
we've got 27 of them. SAL KHAN: Yes, about 27. So literally anything
that starts with a 1 here. So it could be a country
that has a population of 1, a population of 17, or a
population of 1 billion, 300 million, blah, blah, blah. They would all fall into
this bucket right over here. VI HART: Right. And then if you
start with 2, you fall in the second bucket,
and so on and so forth. SAL KHAN: Better color. Oh yeah, go ahead. VI HART: Yeah,
definitely better color. SAL KHAN: Better color. VI HART: Oh, that's great. SAL KHAN: Yeah, that's
the blue, better contrast. VI HART: So the
question is, all right, you'd think you'd have kind
of random numbers here. SAL KHAN: Yes. For that first digit,
it's kind of random. VI HART: Yeah. I mean, there's huge differences
in populations of countries. Some have billions
and some have-- I don't know what the
smallest population is. SAL KHAN: Yes. It's, like, Montenegro
or something like that. VI HART: Yeah. So-- SAL KHAN: Montenegro
is not a country. What am I thinking? I'm thinking of-- what's the one
that's on the French Riviera? Anyway, we can edit that out. [LAUGHTER] VI HART: I don't know. I would be interested
to see for populations of states and everything. SAL KHAN: The Vatican is the
smallest country, I believe. VI HART: Yeah, it is. Does that still count? I guess-- SAL KHAN: I think
the Vatican counts. They have their own-- yes. VI HART: I don't know exactly
what the requirements were to be. [INAUDIBLE]. SAL KHAN: But it would
include the Vatican, which I think would be in,
like, the thousands. VI HART: Yeah. And so why would this happen? Why would you see
more 1's than 2's? Like, what is going on? SAL KHAN: Yeah. And it's not some small chance. I mean, we were talking
also about the idea that it is more likely to
have odd-numbered addresses than even-numbered. We were talking
about that earlier. VI HART: Yeah, I just
learned about that recently. And that makes sense. SAL KHAN: That makes sense,
because every house will have a number 1 on it, or a 10. VI HART: Right, every street,
if your street starts, you know, with house
number 1, house number 2, house number 3, if you have
an odd number of houses, then your street has more odd
numbers than even numbers. SAL KHAN: Exactly. And if you have an even number,
you have the same amount. VI HART: Right. SAL KHAN: Right. VI HART: But that's starting
with 1, which is odd. Whereas here, populations
don't start with 1. SAL KHAN: Exactly. And that phenomenon that
we're talking about the street numbers, it's not an
extreme phenomenon. It's like, 50 point
some 0-- you have a slightly higher probability
of having an odd-numbered house, or I guess a 1 house,
than everything else. VI HART: Yeah, it's kind of
exactly what you would expect. SAL KHAN: It's exactly
what you'd expect. But here, it's a significantly
higher probability, of a random
country's population, that its first digit is a 1
versus its first digit as an 8 or a 9. I mean, it seems a
little bit strange. VI HART: Yeah. And this isn't
just in countries. You see this if you're looking
at a lot of financial stuff. Like, how much money
does a company make. SAL KHAN: Yes. The 1's just show up as a first
digit much more frequently. VI HART: Much more
frequently, yeah. And here, we have
another fancy graph, which is completely crazy. It's the first digit
of physical constants. So what would be examples
of some physical constants? SAL KHAN: I'm
assuming that-- and we weren't able to figure out
exactly what they applied here-- but I'm assuming it's
things like the Gravitational constant, Planck's constant. And this seems kind
of crazy to me, because it depends on the
units that you're using, it depends on a
whole bunch of things that you have to
assume about it. But even when you use these
kind of arbitrary physical constants, which I'm
assuming they're doing here, the most significant digit
in these physical constants is still much more
likely to be 1. It almost exactly
follows Benford's Law. And it kind of gives
you goose pimples. VI HART: Yeah. So the challenge here
is to-- oh, by the way, Benford is the guy
with the glasses. SAL KHAN: Oh, yes. Yes, you might be wondering. These aren't-- yes. These are-- VI HART: They're
not both Benford. SAL KHAN: These are not Benford
pre-shave and post-shave. No. This right there is Benford. And obviously it was named
after him, Benford's Law. But we put this gentleman,
who didn't shave-- VI HART: Yeah, the cool guy
with the beard is Simon Newcomb. SAL KHAN: Simon Newcomb. VI HART: Not Duke Nukem. SAL KHAN: Not Duke Nukem. And we put him here
because he's actually the first person who
stated Benford's Law. He obviously did not
call it Benford's Law. VI HART: And he had
the better beard. SAL KHAN: And he
had, yes, the most-- he was overall a more
imposing character. At least to me. This guy looks a little
bit like Harry Truman. Maybe this is Harry. I don't know. Yeah, maybe I've got
the wrong picture. Anyway, let's just-- VI HART: So the question, what
is kind of a pure case of this? I mean, when you've
this random data, you see some fluctuations. Like, in a country,
right, there are more-- SAL KHAN: It's
pretty close, though. It's pretty close to that curve. VI HART: Yeah, and our
sample size is pretty low. I mean-- SAL KHAN: Right. VI HART: --when
you've only got-- SAL KHAN: There's,
like, 200 countries or something like that. VI HART: Yeah. SAL KHAN: It went
up by, like, 50-- VI HART: It doesn't
perfectly follow it. SAL KHAN: --after the Soviet
Union fell and all that. But yeah, it's not a
huge number of countries. And even physical
constants, I don't know how many physical constants
they randomly sampled over here, but it is shockingly
close to Benford's law curve. But there's kind of a more
pure way of studying it. VI HART: Right. So when we look at
this other graph, this is kind of like
the pure Benford's Law. SAL KHAN: Pure Benford's Law. So that's this curve
that we're kind of fitting to that other,
more rough data. And what's amazing
here is that if we take kind of pure
mathematical constructs, like the powers of 2, or-- VI HART: Or the
Fibonacci series. You'd think that in
the Fibonacci series, you're adding all this
stuff up, and why-- SAL KHAN: And then you
just take the first digit-- VI HART: Just the first digit. SAL KHAN: --and put
them in these piles, it would actually exactly
match Benford's Law. Like, no deviation. It is exactly-- VI HART: Right,
mathematically [INAUDIBLE]. SAL KHAN: Mathematically. So let's just be clear,
because this is fascinating. So if you were to take
the powers of 2's, you get 1, 2, 4, 8, 16, 32, 64. I'm gonna go pretty
high so you can start to see how
we're doing this. 128, 256, 512. And you just keep
going on and on forever with every power of 2. And you say, OK, how many
of these start with a 1? And you would go
and you would say, well, this starts with
a 1, this starts with-- or the most
significant digit is 1. That starts with a 1,
that starts with a 1. And you would just find the
percentage that start with a 1. And you would plot
it on-- you would give the 1's that credit
for that percentage. And then you'd say, OK, the
ones that start with a 2. And you'd say, OK,
that starts with a 2, and that starts with 2. But we want to
keep going on this. And we could probably do
this with a computer program or something, where we'd
go really high powers of 2. And then you would
say, what percentage of all of these powers
of 2 start with a 2? You would, say, get this
percentage right over here. The numbers that start with
a 9, you would get this. And you would perfectly
match Benford's Law. This seems magical. VI HART: It does
seem pretty magical. SAL KHAN: And this isn't
just for powers of 2. This is powers of any number. VI HART: Almost any number. There's special cases. SAL KHAN: I believe it
might be every number. Oh, yes. No.
VI HART: Well, powers of 1. SAL KHAN: 1 would
not work, right. VI HART: And then there's,
like, powers of 10. SAL KHAN: Powers of 10
would not work as well, yes. But every other-- numbers that
kind of mix it up a little bit. VI HART: Powers of 0. SAL KHAN: Yes. You are correct. Every number that would
mix it up a little bit. VI HART: Yeah. It'd have to have a
certain amount of mixing it up a little bit. SAL KHAN: Right, right. But every other, yes, they would
exhibit Benford's distribution. And so we want to challenge
you to think about why that is. And maybe you could even
put your own explanations in our little message board,
on either YouTube or our page, if you're curious. But we'll challenge you
to think why that is. And then we'll offer at least a
decent shot at an explanation. Maybe we'll have
other explanations. VI HART: Yeah. SAL KHAN: So actually, we'll
leave you there in this video. And the next video
will explain why we think-- an intuitive
reason why it works.