Advice for research mathematicians

The Bernoulli numbers are an intriguing family of rational numbers that arise in many areas of analysis. We introduce them in the context of J. Faulhaber's formulas for sums of powers of natural numbers, which in fact give us another important family of polynomials or polynumbers.

These numbers will also figure importantly in the family of Bernoulli polynomials that we will have a look at in a subsequent video.

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0:02

Hello everyone i'm Norman Wildberger thanks for joining me on this adventure we're working towards expanding our number theory from the zero dimensional case to the one-dimensional

0:11

case to the two-dimensional world of maxels and today we're going to look at a very interesting

0:18

family of numbers the Bernoulli numbers which lead ultimately to the Bernoulli polynomials

0:24

which very interesting family of polynomials that have a big role in analysis especially with regard

0:30

to the Euler Mclaren sort of summation formula but they're also involved in lots of different aspects

0:36

of mathematics and i want to introduce them to you today in the context of the foul hopper

0:42

investigations for sums of powers and that will lead us to an interesting family of polynomials

0:48

associated with the name of foul harbor so the territory that we're going through here is

0:54

is similar to some things that we've done in the algebraic calculus one course and we'll touch

1:00

base with some of that today okay so the Bernoulli numbers are a famous sequence of rational numbers

1:05

actually not not integers but rational numbers they were introduced by Jacob Bernoulli one of the

1:11

many famous Bernoulli brothers uh in proposed humor's work of 1713

1:18

and he was following the work of Johann Foulhaber who had previously created formulas for the sums

1:25

of of powers of numbers and here is a something from his his work this is a chapter called semi

1:34

protest statum written in Latin in his work ours conjectandi and what we see here is a bunch of

1:41

polynomials and these polynomials are actually the uh the follow hopper polynomials um that that uh

1:48

file however discovered and and the final uh sort of coefficients of every sort of second one of these things are the Bernoulli numbers and you can see here down this bottom row he's

1:57

identifying some of the first ones 1 6 minus 130 1 over 42 minus 1 over 30. so the Bernoulli

2:05

numbers are quite unpredictable and it's it's quite an interesting story so let's start by

2:14

looking at a so-called exponential generating function for the bernoulli numbers which is a

2:19

nice way of introducing them so i have to remind you about the exponential on series

2:25

okay so we're going to use the term x of t to denote this series in t whose kth term is t to the

2:38

k over k factorial so there's a 1 over k factorial in front and then increasing powers of t and we're

2:45

thinking of this as a like it's a polynomial or a poly number that just happens to keep on going

2:51

and we're not going to think about this crucially as a function so in ordinary standard analysis we

2:58

we talk about the function e to the t which rests on this this number e which is some infinite

3:06

decimal which we actually can't access okay so we're not able to to calculate e we can create a

3:14

program which will generate more and more digits in a certain direction but to say that there's

3:20

this number e that's at the end of the rainbow that's the result of completing this ongoing

3:27

algorithm or process is is wrong it's logically wrong okay so we don't have e and so therefore

3:33

we don't have e to the t either so the number e is unfortunately a fiction in pure mathematics

3:39

and ultimately that means that anything built up from it is also inherently fictional okay so what

3:45

we're we're doing here is i want to promote this idea that we should move away from the function

3:51

approach to things where we think of an expression like e to the t or x of t s something which inputs

3:57

a number you know t and outputs another number given by this infinite sum we want to move away

4:03

from that to thinking about this thing as a basically a list of coefficients okay it's it's

4:10

really the the discrete list of coefficients which is at the heart of what this thing is

4:16

okay and evaluation of such a thing for you know a given input t is not something that we're

4:22

interested in in doing okay because we can't do it because there's an unbounded there's an infinite number of things to be added up and we just can't do that arithmetically so it's an important point

4:32

that i'm going to have to reiterate over and over again but this is in the advice section it's really important that modern students and aspiring researchers adopt this more sensible computational

4:43

approach to the fundamental nature of these these objects which are at the heart of analysis

4:52

okay so how do the Bernoulli numbers arise from this exponential series well if you

4:58

look at this exponential series okay it starts one plus t plus one half t squared and so on

5:04

if you take one from both sides then you get x of t minus 1 equals t plus one half t squared plus 1

5:12

6t cubed so that obviously has a factor of t so let me just do this up here so if we consider

5:22

x of t minus one okay and then we divide that by t

5:31

well then i'll calculate that with a power series in t

5:38

okay so we're going to get this series here and

5:43

that's 1 plus 1 half t plus 1 6 t squared plus 1 24 t cube so it's a lot like the exponential

5:50

except that things are sort of shifted so there's still factorials down here but now the powers are sort of one less than you would expect them to be that's the the effect of things so what we're

5:59

doing now is we're we're taking this series here which is a you could think of it as a variant of

6:05

the exponential series and we're we're inverting it or maybe it's better to say we're taking its

6:11

reciprocal okay because inversion actually has sort of two different kinds of meanings in the

6:16

world of series so we're looking at 1 over this so that'd be t over x plus t minus 1. and we're

6:24

expanding that out so we're we're taking for example let me just uh sort of do it naively

6:30

suppose we take one over and we just uh i'm just gonna grab say six of these terms all right

6:36

again and i think about that as another series if i ask my computer to take a power series for that

6:45

then i'm gonna get one minus one half t plus 1 12 t squared minus 1 over 720 t to the fourth etc but

6:54

you know at some point there will be a divergence between what what's here and what's there because i've only gone up to the sixth power and i've disregarded the rest okay so if we sort of

7:05

expanded the denominator here to include more and more of these terms then we would get more and

7:11

more terms that would be in agreement with this series here okay so this is the series that we get

7:17

by taking one over this series here and and for forming a power series for that and the

7:28

the the brew newly numbers are appearing as coefficients here but this is um what's called

7:33

an exponential generating function which is a little bit of a variance in that there's a k

7:38

factorial in the bottom here in the denominator so we're writing this as the summation of b k over k

7:45

factorial t to the k all right and that's the series that you get by taking the reciprocal

7:52

basically of this series here form from the exponential by subtracting one and dividing by t

8:00

so when we compare the coefficients here to these b sub k's over k factorials

8:07

then we can evaluate what these these b's are and b sub zero is one okay b sub one is minus a half

8:15

b sub two well it's like this one twelfth but you have to multiply by two factorial so that's

8:21

one sixth b sub three is going to be zero because there's no t cubed term in fact there's no t to

8:27

the fifth term no t to the seventh term in fact all the subsequent powers here interestingly um

8:35

are gone okay there's only coefficients of even powers of t after after this place here however

8:41

there's a difference right here's the only sort of odd power of t this minus a half t

8:48

that's responsible for b sub one okay so then the the pattern carries on so

8:53

we're really only interested in the even coefficients b sub 4 is minus 1 over 30.

8:58

b sub 6 is 1 over 42 b sub 8 is minus 1 over 30. b sub 10 is 5 over 66 b sub 12 is minus 691 over 200

9:11

over 2730 b sub 14 is 7 over 6. b sub 16 is minus 3 617 over 510 b sub 18 is 43 867 over 798 b sub

9:29

20 is minus 174 611 over 330. okay so this is a highly unpredictable sequence of rational numbers

9:44

highly unpredictable and in fact uh it's it's not known there's no closed formula for these things

9:53

one can't just write down a formula for that the nth one okay um but i will say something about

9:59

computing these things so the first thing that we should observe here as we already done is that the

10:04

odd Bernoulli numbers after the first one are zero and it's worth noting that this b sub one

10:11

which for us is a minus a half because that's how we obtained it here um this is sometimes

10:18

well there's some variant here okay so sometimes there's a variant used where b sub 1 is actually

10:25

a half okay and so sometimes to distinguish these two different possibilities we talk about b sub 1

10:31

plus being a half and b sub 1 minus being minus a half there's actually good reasons for considering

10:38

both of these cases so it sort of depends on the application certainly if we start like this then

10:44

we get a minus a half to be consistent with this formula but in other cases as we'll see

10:49

this other one also figures so why are these numbers sort of interesting why do they appear

10:56

in analysis okay and then you sum the numbers 1 to the k 2 to the k 3 to the k up to n to the k

11:05

so that's going to be then an expression that depends on n which turns out to be a polynomial

11:11

in n okay this is a polynomial s sub k and uh so maybe we could say this is uh to be a polynomial

11:20

s s sub k in n of degree k plus one and the Bernoulli numbers arise from

11:30

Jacob Bernoulli's formula for foul harbor's sums of power so foul hopper created these these

11:36

polynomials discovered them um up to up to about k equals 17 and then many decades later Bernoulli

11:44

came along and expressed them sort of more succinctly involving these uh Bernoulli numbers

11:51

so here's his formula for for the foul hopper polynomials so i sub 1 to n i sub k the sum is 1

12:02

over k plus 1 and then the bernoulli number b sub j and multiplied by a power of n n to the k plus

12:10

one minus j so this sort of starts with n to the k plus one and then sort of reduces one at a time as

12:18

j is increasing from zero to k so let's have a look at some examples when k equals 1 okay then

12:26

s sub 1 of n is the sum just of the sum of the natural numbers i equals 1 to n of i and okay

12:34

the formula that we all know is that it's one half n times one plus n that's like a triangular number

12:40

but bernoulli's formula says that we should take one over one plus one sum from j equals zero to

12:46

one minus one to the j and then the binomial coefficient one plus 1 choose j times b sub j

12:54

times n to the 2 minus j so when j is 0 well then we get a 2 choose 0 which is just 1

13:04

times b sub zero times n squared and when j is one well we get a minus sign and we get

13:11

two choose one which is two so two times b sub one times n to the one so all right so then you plug

13:20

in b sub 0 is 1 so that's just a single n squared and b sub 1 is minus a half so this becomes just

13:27

plus 1 we get one half n plus n squared which is the usual thing that we know

13:32

when k equals 2 s sub 2 of n equals the sum of the squares the formula is that it's 1 over 2 plus 1

13:40

some j equals 0 to 2 now minus 1 to the j now the coefficient 2 plus i choose j b to the j still

13:50

and multiplied by n to the two plus one minus j okay so we get three terms here's j equals zero

13:58

b zero n cubed here's j equals one we get minus three b one n squared and here's

14:05

j equals two three b two n you plug the values that we know b zero

14:12

b one and b two so b zero is one b one is minus a half of b two is one sixth you plug them in and uh

14:19

you get this expression here which factors as one sixth n times one plus n times one plus two n

14:26

sort of familiar sum of um squares formula and when k equals 3 then s 3 of n equals well you

14:34

get 1 over 3 plus 1 sum from j equals 0 to 3. the same expression so then you end up getting

14:41

this combination involving the essentially the binomial coefficients from the fourth

14:47

row of the pascal triangle so one four six four no final one and uh so decreasing powers of n

14:57

then you have these other factors b0 b1 b2 and b3 so those factors are always appearing first

15:04

b0 then b1 then b2 b3 but now multiplied by the you know the appropriate row of the pascal

15:10

triangle and we have to remember that there's some alternating sign when k equals four against

15:16

the same kind of story now we're talking about the the fifth row of the pascal triangle one five ten

15:21

ten five one with the last one is missing and each of these coefficients is multiplied by b0 then b1

15:29

b2 b3 then b4 there's an alternating sign then we plug in the various values of the b's

15:36

okay and we get this series here and that turns out to have a nice factorization like this

15:46

so this these are the formulas for s1 s2 s3 and s4 and these formulas here were were found by

15:55

foul hopper uh in 1631. small harbor was from ulm was known as the the great calculator of um

16:02

he was quite a famous fellow and and probably deserves a lot more fame than he actually

16:08

gets he calculated these sums all the way up to s 17. so all the way up to 17th powers of natural

16:15

numbers and got these these big polynomials and he discovered a very interesting and important

16:22

aspects of them but he didn't realize like Jacob Bernoulli did or maybe there's a k here but anyway

16:30

um he didn't realize as Jacob Bernoulli did that that there's a kind of a uniform way of writing these polynomials in involving these these sort of curious Bernoulli numbers

16:43

okay so the formula is quite attractive at this stage here but all the complexity in some sense

16:49

is in the actual Bernoulli numbers themselves and as we've seen they are rather unpredictable

16:57

okay so in terms of uh closed formulas for these things um there aren't any okay uh at least

17:04

none that are sort of effective but uh there are recursive formulas and the simplest one is

17:11

is this one here that the sum from k equals 0 to n of n plus 1 choose k b to the k equals the

17:18

direct delta function n zero so most of the time this will be zero unless n is equal to zero

17:26

and what this formula allows you to do is to calculate the top bn in terms of the previous ones so if you know b0 b1 all the way up to bn minus one you can use this

17:38

series to evaluate b sub n so let me illustrate that with say n equals four so i'll show you how to calculate b sub 4 in terms of b0 b1 b2 and b3

17:51

and the the series that we're using is this one right here that uh the sum k equals 0 to 4 or 4

17:57

plus 1 choose k b sub k so that's b 0 plus 5 b 1 plus 10 b 2 plus 10 b 3 plus 5 b 4 equals 0.

18:08

so that that actually you know is is true and so if you know b0 b1 b2 and b3 then you

18:16

can calculate b4 from it now why is such a formula true well this thing here is

18:22

very much like what we saw before right if you look at this expression here

18:28

this is very much what's appearing in in this series for s sub 4 of n okay it's very much

18:34

like this except that the alternating signs have disappeared so we could sort of get at that by

18:40

by putting in n equals minus one if we put in n equals minus one here then we're going to sort

18:46

of re remove the the alternating aspect and if we put n equals one uh minus one in here

18:54

well the result is going to be 0 because of this factor see there's this factor 1 plus n

19:02

in fact you see this vector 1 plus n appearing there in fact it appears in in all of these

19:08

okay so there's this always this factor one plus n so when we plug n equals minus one in here we're

19:14

going to get zero okay and that's essentially the reason why this um this recursive relation is is

19:22

is there it's uh it's a kind of a consequence of one way of thinking about as a consequence of the

19:29

the form of the foul haber polynomials and and basically Bernoulli's formula

19:38

now uh in terms of calculating these b sub n so you know you can use this recursion but

19:43

recursion can be onerous especially since you're using more and more terms so it turns out that

19:49

the world record for calcutta b seven is held by a colleague of mine that's David Harvey at UNSW

19:56

who in 2008 set the world record he calculated the 100 millionth Bernoulli number okay and that turns

20:07

out to be a number with uh 676 million uh digits so you'd express it sort of in scientific notation

20:16

it's actually a rational number but you you you write it out then this is how many digits it has

20:22

in scientific notation and his method involved a modular arithmetic sort of looking at the

20:28

Bernoulli numbers mod p for various primes p and then sort of putting that information together

20:34

using the Chinese remainder theorem and ideas in that direction and the previous record was also established earlier in 2008 and that was for 10 million so that's quite a jump and that's i guess

20:45

still the current record so that you know shows us that you know modern computers are pretty powerful

20:51

and you know these numbers are big but they're not that big and so it's quite a lot of work calculating these Bernoulli numbers and it's quite an interesting topic okay so now let's try to move

21:04

to a two dimensional story we've identified these foul hover polynomials these s sub k's

21:09

as interesting family polynomials and we know that their degrees go up one by one so here's a list of

21:15

them there's s1 s2 s3 etc that we can obtain from say the Bernoulli formula so uh here by the way is

21:27

the original page from Bernoulli okay so we can sort of compare these coefficients here

21:35

and i've just basically recorded them right here okay so for example this number the sum of the

21:43

eighth powers is over here one-ninth n to the nine plus one-half n to the eight you see exactly the

21:49

same two-thirds n to the seven minus seven over fifteen n to the fifth plus two over nine n cubed

21:56

now minus 1 over 30 n okay so also by the way note curiously that they don't write n squared

22:05

okay there's n to the fourth that's the way we write things but n squared is just written as nn

22:12

a little bit interesting okay so what's our our plan our idea is that we should try to

22:18

incorporate these one-dimensional things into a two-dimensional picture okay so we can record this

22:25

family in a two-dimensional array where i've just basically written down the coefficients

22:32

uh in increasing um powers of n so starting with n and then n squared n cubed and the fourth etc

22:40

and i've gone down to the uh i guess the ninth row here okay so let's call this matrix f and we can

22:48

observe that the Bernoulli numbers themselves are appearing down this first column of this array f

22:56

okay and if we want to check the actual foul hopper polynomials are what we get when we

23:03

introduce powers of n so we can do that just by taking our matrix f and multiplying by this column

23:09

vector of powers of n and then we get this thing here and we can actually factor things just as

23:16

as we're used to just factor with one command and we get these um these formulas for the factored

23:22

forms of the file however polynomials okay so when we do this kind of thing what we're aiming to do

23:31

is to try to utilize the matrix algebra from linear algebra to provide new insight okay so

23:40

we're going to take our matrix f right here and we're going to take its inverse this matrix is

23:46

sort of highly unpredictable because in particular there's the highly unpredictable Bernoulli numbers

23:53

going down that first column okay if you look along here you know it's not at all clear what

23:58

the pattern is if there is any pattern that's easily stateable in these entries here but

24:04

that's what we do right that's our our strategy is to at least look to see if there's some

24:10

something that happens when we take the inverse okay so here's f inverse okay here's f just

24:15

repeat it again right here and i'm taking its inverse and lo and behold magic appears

24:24

exceedingly lovely magic the inverse of this complicated unpredictable Bernoulli number

24:32

type matrix is this super regular pascal Harriet like maxel in fact it's exactly the pascal Harriet

24:45

maxel except that the final sort of diagonal of ones has sort of been stripped off okay

24:52

so we're not getting the full rows of the pascal triangle one four six four one the one is in all

24:59

cases dropped and the diagonal just now becomes the increasing sequence one two three four five

25:06

and also there's a checkerboard pattern of minus signs as we've uh you'll come to almost expect

25:12

in a lot of these situations so this is more or less the pascal Harriet maxel and it's really

25:19

an astounding thing i mean we should be hugely surprised that by taking this irregular object

25:27

looking at it from this two-dimensional point of view taking in inverse suddenly the irregularity

25:33

the unpredictability completely disappears so we can think about this now is that we really

25:39

gained some deep understanding that the unpredictable aspect of this can be viewed as

25:45

a result of this complicated process of taking an inverse we're starting with something that's

25:51

very structured very predictable we're taking its inverse and we're getting something that's highly

25:58

unpredictable so this inverse transformation we see its power and we see its power in

26:05

in clarifying what's really going on in particular suppose that we wanted to extend

26:13

you know the the foul harbor polynomials or maybe the Bernoulli numbers a very easy way of doing this now is to just expand this very predictable matrix this pair

26:25

pascal Harriet matrix here okay expand that to a larger size and then take the inverse of

26:30

that because of course taking the inverse of this is going to give you this thing so um we're going to do that but before i do that i just want to to mention that this this

26:40

correspondence here also can run the other way so in one of my algebraic calculus videos one

26:48

called foul habers formula and Bernoulli numbers this in the algebraic calculus one

26:54

series of videos on wild egg in this video i explain uh sort of a derivation of the foul hopper

27:03

polynomials starting with an observation of pascal and using some linear algebra so basically what

27:10

i do is i is i show that you know you can you can create this by by first understanding that

27:19

you can rearrange the the sums of of powers problem into a basically a sum of Harriet

27:30

summations okay so sort of exploiting the the the Harriet basis in in a nice way uh that was

27:38

essentially understood by pascal so if you if you watch this video and if you're interested

27:44

in this i actually just suggest that you do that you'll see that you know we can derive the file

27:50

harbor story from this linear algebra point of view and this is a crucial step we we first get

27:56

to a system of linear equations that has this as a coefficient matrix and then inverting that it amounts to inverting this matrix and so we get this thing here

28:05

so that's a nice complement to what i'm what i'm explaining here today

28:11

okay great so now um let's uh let's increase the size of things because we can generate

28:18

these pascal heritage matrices pretty easily right here's just a very simple formula just introduce a

28:24

matrix function k i j say equals minus 1 to the i plus j and then maybe i minus 1 choose j minus 1.

28:31

that's going to generate a pascal heritage matrix and then you can easily sort of strip off that that diagonal once to get in this case so here's a bigger version all right

28:42

so the same thing as i've shown you before but instead of going just down to the the ninth row here we're going to go down to the 17th arrow and now i'm going to take the inverse of this

28:52

and that's going to then generate uh the Bernoulli story further okay so let's see what we get

29:02

so this is the inverse matrix and so if we just look down that first column which is most

29:08

interesting is that we you know we see the the Bernoulli numbers it gets expanded you know so

29:15

there's the next few okay and in particular were now able to read off further or higher degree file

29:23

hover polynomials pretty easily now a very curious thing happens here that's why i was looking at

29:31

this and comparing just with the values on on Bernoulli's table let me grab Bernoulli's table

29:40

and i will show you okay so there's there's uh the table copy that let's go back down here

29:47

compare the values okay let's look at this thing here so i want you to have a look at the sum of the ninth powers okay this row that starts with n to the ten okay uh so there was a one tenth

29:59

coefficient the one half a three quarters a minus seven tenths of one half and a minus one twelfth

30:05

okay so let's go to the corresponding uh row over here okay so that's there's

30:12

the one-tenth the one-half three-quarters minus seven-tenths one-half and minus three twentieths

30:23

one-tenth one-half three-quarters minus seven-tenths one-half minus one-twelfth hey wait a minute that's not exactly the same so let me actually grab this thing all right so

30:33

let's grab that row okay and i'm going to put it directly adjacent to this thing here okay there's

30:42

there so compare this row that we've obtained now with Bernoulli's row right here starting with

30:51

the the one-tenth one-tenth the one-half yes the three-quarters yes the minus seven-tenths yes the

30:57

one-half yes but the minus one-twelfth no we've got minus three over 20. there's a discrepancy

31:06

so there there appears to be a well let's not say discrepancy

31:15

let's be honest it's an error on Bernoulli's table in the row for the sums

31:29

k equals zero to n of um well maybe i we were saying before

31:39

i equals zero to n of i to the nine okay so let's actually check this so g

31:49

let's call this something like uh okay f

31:55

of n equals this all right

32:01

so uh so let's just check f of one is one yes f of two is five thirteen f of three equals twenty

32:12

29 16. so these are the sums of the of the ninth powers and now let's uh take the corresponding row

32:23

uh which one is it so it's this last one here right this is the one with the minus

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three twentieths so i'm going to let's say find another function say g of n equals this

32:36

all right and i'm going to calculate g of one and g of two that's five thirteen and g g of three

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that's twenty thousand one ninety six this agrees with the actual summations so that's

32:55

telling me that the formula that i've obtained from this linear algebra point of view with the

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minus 3 over 20 here is actually correct and if we replace this minus 3 over 20 as Bernoulli has here

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with the minus 1 12 we're going to get the wrong formula so we see that we we benefit not just in

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you know in augmenting our understanding but this linear algebra very systematic point of view

33:21

i think it has a potential for sort of reducing errors because where we're obtaining these large

33:27

scale structures you know sort of at a single blow with just a single inverse operation and

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we're starting from something which is really very structured right this this matrix that we're

33:41

starting with is given by this very simple formula so there's there's not so much room for error

33:50

so i hope you've enjoyed this introduction to the Bernoulli numbers and the foul hopper

33:56

polynomials which are very important and they they really should be in in every calculus course in my view that's why it's important for you to take the algebraic calculus one course

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and then moving to the algebraic calculus two course where we're gonna be doing a lot more of these sort of on series um kind of activities where we're looking at

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a power series from different points of view the next one we're going to be looking at the

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Bernoulli polynomials and trying to adopt a two-dimensional point of view towards them.