hello everyone I've got a really
surprising fact about numbers to tell
you about today it's so surprising that
you have mathematicians are surprised by
this
in fact they can use this then to detect
fraud and it's called Benford's law now
you can discover this for yourself
for example pick a newspaper and circle
or the numbers now I'm using the
Financial Times here because it has a
lot of numbers in it and I'm only
interested in numbers that can grow
naturally so I'm thinking of things like
prices and percentages and stock shares
I'm not interested in page numbers and
telephone numbers because that's a
little bit too constrained for our
purposes but if I do that there were 127
numbers on the front page of The
Financial Times today and my question is
this how many of them do you think start
with the number one now you might think
well that should be about one night so
numbers can start with one two three
four five six seven eight nine we're not
including zero here so if there's 127
numbers about 14 of them to start with
the number one when in fact it was a
huge 42 numbers started with a 1 and if
I keep going 21 started with a 2 12
numbers started with a 3 7 started with
a 4 6 numbers started with a five eight
numbers started with a 6 9 started with
a seven eleven started with an 8 and
another 11 started with a 9 if I keep
going I'll find something like this is a
surprising 30% of numbers start with in
number 1 this law was first discovered
in 1881 by the astronomer Simon Newcomb
but then was later we discovered over 50
years later by the physicist Frank
Benford while he was working for General
Electric but they both noticed it in the
same way they both noticed that pages
near the beginning of their log tables
were getting more worn and dirty than
pages near the end we suggested that
they were looking up statistics I
started with a 1 morphed and then with
higher numbers and then found Benford
did what we just did it will take
newspapers and he
look at the numbers inside and he
noticed the same thing that number
started with a 1 turn up about 30% of
the time and then he started to look at
other statistics he started to look at
populations the length of rivers even
things like mathematical and physical
constants even street addresses and they
showed the same thing at numbers
starting with a 1 turn up about 30% of
the time it doesn't matter what units
you use so if you want to measure the
length of a river you can do that in
kilometers miles feet or centimeters it
will still be the same the number 1 will
turn up more after and you can even mix
up your statistics so if you're taking
numbers from a newspaper those
statistics are coming from a variety of
sources and yet the law still holds now
this isn't true if your choice is to
random if you went to random.org and
generated a bunch of numbers you all
find that numbers starting with a 1 will
turn up about 1/9 of the time and it
doesn't work either if your choice is
too restrictive so I'm measuring
people's height using centimeters and
not a lot of numbers are going to start
with a 3 except for oh you got one guy
in China to show you my numbers starting
with a 1 turn up so often imagine I put
one pound in the bank and it's a very
generous bank I own 10% interest every
day so I start with 1 pound then the
next day I have 1 pound 10 and then I
have 1 pound 21 then I have 1 pound 33
and so on and you'll notice I spent a
long time around the load numbers but
then I started to skip through the
higher numbers and then I hit the teens
and it happens again I spent a long time
between 10 and 20 but then I start to
skip through the higher numbers and even
from this you can see that numbers
starting with a 1 turning up about 30%
of the time and that's the general idea
and it is used to detect fraud so if
you're cooking the books if you're
making up numbers people tend to maybe
spread out the numbers
evenly or maybe start picking the middle
in numbers like four or five and sixes
and if these numbers don't follow
Benford's law they may be committing
fraud now I'm going to show you why this
is true
but be warned this next bit is for
serious magnet ish ins
only
like I said this was discovered when
neuquén Benford noticed that pages at
the beginning of their log tables were
getting worn and tatty but some of you
may not know what log tables are in the
days before calculators this is what
they use the multiply large numbers now
I imagine you know how powers of 10 work
so if I give you a number n you can
return 10 to the power n so if I give
you 2 you return 10 squared 10 times 10
100 if I give you 3 you return 10 cubed
10 times 10 times 10 1,000 now as powers
go up you multiply by 10 so you get one
hundred one thousand ten thousand one
hundred thousand and so on as powers go
down you do the opposite you divide by
ten so you've got 1,100 10 and one so 10
to the power 0 is 1 now log or logarithm
to the base 10 is the reverse of that so
if you give me a number I can give you
the original power so for example log of
1,000 is 3 log of 100 is 2 log of 10 is
1 and log of 1 is 0 and you can connect
those points together and work out real
odd off things in between so log of 50
is one point seven lot tables we use to
multiply large numbers because the log
of x times y is equal to the log of X
plus the log of Y so multiplication now
just becomes the addition of logs and
you can reverse the process and get your
answer
now lock tables only needed to go
between 1 and 9 because if you had
something larger like the log of 273
then that was just the log of 2.73 plus
the log of 100 and the log of 100 is 2
now new command benefit notice that the
probability that a number starts with
the digit n was the log of n plus 1
minus the log of n but he couldn't
explain why I'm going to show you why
but the
the idea essentially is that if we
collect a lot of data we want the amount
of things that between say 1 and 2 to be
the same as the amount of data between
10 and 20 and I want that to be the same
as the amount of data between 100 and
200 and the only way you can do that is
if the probability that you start with a
number 1 is 30%
if Benford law exists and it's universal
it should be unaffected by which units
we choose the measure things with so I
could measure things with meters speeds
miles or pounds dollars ninnies whatever
and benfits law should still hold true
so let's imagine we have some data like
this and what this shows is I have a lot
of data between 25 and 40 but not so
much data between 1250 and 2000 now they
say I want to convert this data into
something else so I'm going to multiply
it by 50 and would provide everything by
50 it's some sort of conversion factor
and the data changes and I might get
something like this and what we have now
is I have a lot of data now between 1250
and 2000 if this was scale invariance
that blue gap would be the same as that
red gap so the blue gap represents data
that is 50 times larger than the red gap
now we took the log of this day 2
instead we would get this and the blue
gap now just represents data that is 1.7
higher than the red gap now remember if
I want this to be scale invariant
I want the size of the blue gap to be
the same as the size of the red gap so
all I'm saying is I want this to be
unaffected on the shifts in other words
the log of the data should be uniform
like this now all I need to do is
reverse this to find the original
distribution of something that is scale
invariant and this is what we get and
from this you can see if I doubled the
distribution the gap between 1 and 2 is
the same as the gap between 2 and 4
which is the same as the gap between 4
and 8 if I times byte
the gap between one and two is the same
as the gap between 10 and 20 which is
the same as the gap between 100 and 200
now we want to know the probability that
number starts with a particular digits
because the pattern repeats we only need
to consider the numbers between 1 and 9
so imagine I threw a dart at this was
the probability I'm going to hit a
number beginning with a 4 well it's
going to be the length of the section
between 4 & 5 divided by the total
length now the length of the section
between 4 & 5 is lon of Phi minus log of
4 and the total length will ask the log
of 10 which is 1 in general the
probability that a number starts with
the digit n is the log of n plus 1 minus
the log of n and in fact this works for
any string of digits imagine I wanted to
know the probability that a number
starts with the string 1 2 3 well that's
the log of 124 minus the log of 123
which is about 0.4% and using this you
can start to work out the probabilities
our digits appearing in other positions
like the second position the third
position although that quickly becomes
10 percent for each of the 10 digits 0
to 9 now I'm not saying that this is an
easy proof but it is a truly surprising
fact about numbers and if you have been
thanks for watching