A Brief introduction to Bernoulli Numbers

1

2

1

2 3 2

1

3

1

1

( 1) 1 1

2 2 2

( 1)(2 1) 1 1 1

6 3 2 6

n

n

n

n

n

n nr n n

n n nr n n n

r

What about n4 and n5? Can you generalise?

Ada Lovelace

The enchantress

of numbers

Warm up

Write your name in binary Ascii

01101000 01000101 01001100 01001100

0110111/ 01001101 01111001 / 01001110

01000001 01101101 01000101 / 01001001

01110011/

01000001 01000100 01000001

Ada Lovelace

Augusta Ada King-Noel, Countess of

Lovelace (née Byron; 10 December 1815 – 27

November 1852) was an English

mathematician and writer, chiefly known for her

work on Charles Babbage's Analytical Engine.

Her notes on the engine include what is

recognised as the first algorithm intended to be

carried out by a machine. As a result, she is often

regarded as the first computer programmer

Source :Wikipedia

Childhood • Ada Lovelace, born

Augusta Ada Byron,

was the only legitimate

child of the famous

poet Lord George

Gordon Noel Byron.

• Lord Byron's marriage

to Ada's mother,

Lady Anne Isabella

Milbanke Byron,

wasn’t a happy one.

Lady Caroline

Lamb

Claire

Claremont

Teresa

Guiccioli

Lady Oxford

Augusta

Byron

Childhood

• Lady Byron left her husband when

their daughter was eight weeks old.

• A few months later, Lord Byron left England,

and Ada never saw her father again. He died in

Greece when Ada was 8 years old.

Childe Harold's Pilgrimage

Is thy face like thy mother's, my fair child!

ADA! sole daughter of my house and heart? When last I

saw thy young blue eyes they smiled,

And then we parted, -- not as now we part, But with a hope.

-- Awaking with a start,

The waters heave around me; and on high The winds lift up

their voices: I depart,

Whither I know not; but the hour's gone by, When Albion's

lessening shores could grieve or glad mine eye.

Childhood

• Isabella was determined to prevent her

daughter from developing poetic tendencies

and focussed her education on mathematics

• Ada turned out to have an aptitude for maths

Ada aged 5 years

Childhood • Ada had an unusual upbringing for an

aristocratic girl in the mid-1800s.

• Ada’s daily timetable when she was 8 years old

10am Music

11.15 French

11.30 Arithmetic

1.30 Work

3.15 Music

4.30 French exercise

Ada aged 5 years

Flyology

From a young age Ada was often ill

Even during periods of convalescence Ada

never stopped learning and developing her

mathematical skills.

In June 1829 she contracted measles and was

paralysed, only regaining the ability to walk with

crutches in 1831. It may have been due to this

long period of reduced mobility that a 12-year-

old Ada thus decided that she wanted to fly.

“Since last night I have been thinking more about

flying. I can find no difficulty in the motion or the

dimensions of the wings…”

“as soon as I have got flying to perfection I have a

scheme about a steamengine which, if I ever effect

it, will be more wonderful then either steampacket

or steamcarriages”

Ada writing to her mother in 1828

• Ada had several tutors and when she was 17

she tried unsuccessfully to elope with one of

them.

• Her most notable mentor was the scientist,

mathematician and social reformer Mary

Somerville whose book On the

Connexion of the Physical Sciences

was published in 1834.

Growing up

• At age eighteen Ada was presented in court and

made a positive impression, being able to dance

well and having a ‘dainty’ appearance.

• In July 1835 aged 21 she married William King (8th

Baron King) gaining the title baroness King.

• They had three children Byron, Annabella and

Ralph and spent their time between their three

homes, one in Ockham, Surrey, one in London and

one in Loch Torridon.

• In 1838 her husband was created Earl of Lovelace

and she became The Right Honourable Countess

of Lovelace.

Charles Babbage

• Ada first met Charles Babbage shortly after her

coming out party in 1833 when she went with her

mother to see what she called his “thinking machine” a

portion of his difference engine on display in his

drawing room.

• An onlooker reported of the event “While other visitors

gazed on the workings of this beautiful instrument with

the sort of expression, I dare say the sort feeling, that

some savages are said to have shown on first seeing a

looking glass or hearing a gun, Miss Byron, young as she

was, understood its working, and saw the great beauty

of the invention.”

Charles Babbage (the Brian Cox of his day)

• Born December 26, 1791 in London, the son of

a banker

• Went to Cambridge University in 1810

• After graduation was hired by the Royal

Institution to lecture on calculus.

• Within two years he had been elected a

member of the Royal Society

• Was Lucasian Professor of Mathematics at

Cambridge. From 1828 to 1839,

Charles Babbage

• In 1819 Babbage visited France, and learned

about the large-scale government project to

make logarithm and trigonometry tables.

• Mathematical tables were of considerable

military and commercial significance in those

days, being used across science, engineering

and finance, as well as in areas like navigation.

“I wish to God these tables had been made by

steam!”

• In the 1820s he developed

his 'Difference Engine', a machine

which could perform mathematical

calculations.

• A six-wheeled model was built and

demonstrated to audiences in his

London home

• He then developed plans for a bigger,

better, machine - Difference Engine 2.

The Difference Engine

• In 1822 (when Ada was 7) the government

began funding Babbage to produce the

difference engine.

• They funded him for almost nineteen years but

he never produced a finished machine

• The government, who were interested in

receiving accurate tables and not an ever

improving device eventually pulled the plug in

1840.

The Difference Engine

• Babbage had moved onto the complex

Analytical Engine, a revolutionary device

intended to be able to perform any arithmetical

calculation using punched cards

• It would follow the instructions on the punch

card, as well as having a memory unit to store

numbers

• Neither the Analytical Engine nor

Difference Engine 2 were built in B his

lifetime but in 1991, a functioning

Difference Engine was constructed

from Babbage's original plans

Interlude: Joseph-Marie Jacquard

• Invented a mechanical loom in 1801that

simplifies the process of

manufacturing textiles with such

complex patterns.

• The loom was controlled by a chain of

cards of punched cards, laced together

into a continuous sequence.

• Multiple rows of holes were punched

on each card, with one complete card

corresponding to one row of the

design.

Portrait of Jacquard woven

in silk on a Jacquard loom

It required 24,000 punched

cards to create and was only

produced to order.

Charles Babbage owned one

of these portraits; it inspired

him in using perforated cards

in his analytical engine.

In her youth Ada was

fascinated by Jacquard looms

Back to Ada

• Within a few months of the birth of her third

child in 1839, Ada decided to get more serious

about mathematics again.

• She had stayed in touch with Babbage, they

communicated often by letter and with visits

• She turned to Babbage when she wanted to

find a “mathematical Instructor”

• He suggested his friend Augustus de Morgan,

first professor of mathematics at University

College London and noted logician.

What de Morgan said about Ada

“Had any young beginner, about to go

to Cambridge, shown the same power,

I would have prophesied first that his

aptitude at grasping the string points

and real difficulties of first principles

would have very much lowered his chance of

being senior wrangler, secondly that they would

have certainly made him and original

mathematical investigator, perhaps of first rate

eminence”

The first computer programme

• In 1840 Babbage spoke about his Analytical

Engine at a conference in Turin in 1840

• Luigi Menabrea, professor of mechanics and

construction at the military academy at the

university of Turin (and later Prime Minister of

Italy) attended his talk

• In 1842 he published a paper “Sketch of

the Analytical Engine Invented by Charles

Babbage, Esq.”

The first computer programme

• To illustrate the machines capabilities,

Menabrea presented tables of the steps the

machine would go though in performing

calculations and finding numerical solutions to

algebraic equations.

• These steps were designed as instructions that

the engine’s operator would punch in coded

form on cards to be fed into the machine.

The first computer programme • Menabrea realised that the Analytical Engine

was a major step up from the Difference Engine;

it was designed from the start to be

programmable.

• the device had two main parts;

– the store, (equivalent to memory) which could

hold 1000 numbers each with 40 decimal

places, making a total memory of just over 16k

– The mill was the mechanical central

processing unit.

The first computer programme

• In 1842 Ada was asked to translate the paper

into English by Charles Wheatstone

• Babbage asked her to add something of her

own, her notes ended up being

three times longer than the

original article.

• The notes contain instructions

on how to calculate the Bernoulli

Bernoulli numbers.

Interlude Bernoulli numbers

• Jacob Bernoulli (1655 -1705) was one of

the famous Bernoulli family whose members

featured many notable mathematicians and

scientists.

• in 1683 Bernoulli was studying a question

about compound interest which required him

to find the value of

1lim 1n

n n e

So he is now

credited with

having

discovered e

2 23

1

( 1)

4

n n nr

24

1

2 2 25

1

( 1)(2 1)(3 3 1)

30

(2 2 1)( 1)

12

n

n

n n n n nr

n n n nr

1

2

1

2 3 2

1

3 4 3 2

1

4 5 4 3 2

1

5 6 5 4 3 2

1

1

1 1

2 2

1 1 2

3 2 12

1 1 3

3 2 12

1 1 4 10

5 2 12 30

1 1 5 10

6 2 12 12

n

n

n

n

n

n

n

r n n

r n n n

r n n n

r n n n n n

r n n n n n

0

1

2

3

4

5

6

7

8

1

1

2

1

6

0

1

30

0

1

42

0

1

30

B

B

B

B

B

B

B

B

B

Generalise

1 1 2

1

1 10 ...

1 2 12

np p p p pp

r n n n np

11 20 31 2

1

( 1) ...0! 1 1! 2! 3! 1!

pnpp p p p

BB BB Bnr n pn p p n n

p

1

1 0

!

! ( 1 )!

pnp p kkB p

r nk p k

Where Bk are

the Bernoulli

numbers

Bernoulli numbers

• Bk is the coefficient of in the Taylor series

expansion of

2 4 2

1 2 21 ....1 2 2 4! (2 )!

n

nx

x x x x xB B B

e n

!

kx

k

2

11 2 !

k

xk

x x x

e k

1x

x

e

Bernoulli numbers

• The exponential function is

substituting this into the denominator of

gives a strategy for calculating the numbers

recursively:

So the machine could compute the sequence as

long as it could store all the previous numbers.

2 3

1 ....2! 3!

x x xe x

1x

x

e

1

10, n 1

n

i

i

nB

i

Bernoulli numbers

• The generating function is the exponential

function

• They can also be defined by a contour integral

• Ramanujan wrote about them in his first paper

for the Indian Mathematical Society

• The Bernoulli numbers also appear in

the Taylor series expansions of tan and tanh, in

the Euler–Maclaurin formula, and in

expressions for certain values of the Riemann

zeta function.

For a better explanation

• Have a look at

https://www.youtube.com/watch?v=yGpkB2Oo

Qjk

Back to Ada

• In her notes on Ada points out that both the

Jacquard loom, and the Analytical Engine had

the ability to automatically back up the card

sequence and thereby repeat a series of

instructions in what would now be called a

"loop“

The first programme

• The Notes included the first ever published

description of a step by step sequence of

operations for solving mathematical problems

• Since programming languages had not been

invented, Lovelace had to express this in terms

of the way the Jacquard loom worked.

“The Analytical engine weaves algebraic patterns

just as the Jacquard loom weaves flowers and

leaves”

The first programme

• In note G she explains the repetition: cards 1-

12 are processed once (we would call this

initialisation), cards 13-23 are repeated a

number of times depending on which Bernoulli

number is desired (this is the loop) and cards

24-25 are processed once at the end.

(though the calculation of B1 is a special case

and does not involve cards 8-12.)

Back to Ada

• As Ada said in a letter to Babbage while she

was working on debugging her computation of

Bernoulli numbers:

• “My Dear Babbage, I have worked incessantly,

& most successfully, all day. You will admire the

Table & Diagram extremely. They have been

made out with extreme care, & all the indices

most minutely & scrupulously attended to.”

• Then she added that “Lord L (her husband) is at

this moment kindly inking it all over for me. I

had to do it in pencil…”

The first programme

• In her notes on Menbrea’s paper, Ada also

described how codes could be created for the

device to handle letters and symbols along

with numbers.

• She speculated that the Engine 'might act upon

other things besides number... the Engine might

compose elaborate and scientific pieces of

music of any degree of complexity or extent'.

The first programme

• She was also well aware of the limitations of

the machine

“The Analytical engine has no pretentions

whatever to originate anything. It can do whatever

we know how to order it to perform. It can follow

analysis; but it has no power of anticipating any

analytical relations or truths.”

The first programme

• Her work was published in 1843, in an English

science journal. Ada used only the initials

"A.A.L.," for Augusta Ada Lovelace, in the

publication.

• Ada's article attracted little attention when she

was alive.

What Ada did next

• She continued to communicate with Babbage

about mathematics

• She wrote the notes for a paper on using

scientific method to improve agricultural

production published by husband

• In her later years, she tried to develop

mathematical schemes for winning at

gambling. Unfortunately, her schemes failed

and she became heavily in debt.

What Ada did next

• In the 1840 she was very unwell with, what

they later discovered was uterine cancer.

• She was prescribed Laudnum and morphine

which made it hard for her to concentrate.

• Due to the effects that the drugs had on her

system, she also became interested in

mesmerism and impact of chemicals on the

mind

Death • Ada died from uterine cancer in London on

November 27, 1852.

• She was buried next to her father in at the Church

of St. Mary Magdalene in Hucknall

This Daguerreotype is a photograph of a

small portrait of Ada Lovelace, frail and

thin, painted by Henry Wyndham Phillips

in the last months of her life, when she was

in great pain from uterine cancer.

Legacy

• She resurfaces and comes to prominence in

the 20th Century when Alan Turing referred to

her in several contexts including a radio

broadcast

“Let us reconsider Lady Lovelace’s dictat ‘It can

do whatever we know how to order it to perform’

…”

Legacy

• The collaboration with Babbage was close and

biographers debate the extent and originality

of Ada's contribution.

• Whatever you think, Ada had the brain to see

the potential of computers over a century

before one was even built.

• Her foresight was so extraordinary that it

would take another hundred years and Alan

Turing to recognise the significance of her

work.

Legacy

• ADA, named in Ada Lovelace's honour, is a

computer programming language originally

designed for the U.S. Department of Defense

for real-time embedded systems.

• The aim was to find one high-level language to

be used for all DoD software, replacing the

hundreds of languages then in use.

• ADA is the most commonly used language in

U.S. weapons systems

Why the Ascii?

• Ascii is short for American Standard Code for

Information Interchange

• It is a character encoding standard

• Ascii codes are used to represent text in

computers, telecommunications equipment,

and other devices.

Just as Ada predicted

Ada Lovelace

1815 - 1852

references

• Ada’s Algorithm: James Essinger

• The Thrilling Adventures of Lovelace and Babbage: Sydney

Padua

• Childe Harold's Pilgrimage: Lord Byron

• Calculating Ada: Channel 4

• Oxford University:

http://people.maths.ox.ac.uk/kar/AdaLovelace.html

https://blogs.bodleian.ox.ac.uk/adalovelace/2015/10/14/only-

known-photographs-of-ada-lovelace-in-bodleian-display/

• Finding Ada: http://findingada.com/

• The Scientific Life of Ada Lovelace - Professor Ursula Martin

• And Wikipedia (of course)

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