Ada Lovelace

A Brief introduction to Bernoulli Numbers
n
1  n
1
n(n  1) 1 2 1
1 r  2  2 n  2 n
n
n(n  1)(2n  1) 1 3 1 2 1
2
 n  n  n
1 r 
6
3
2
6
n
n
3
r
 
1
What about n4 and n5?
Can you generalise?
Ada Lovelace
The enchantress
of numbers
Warm up
01101000 01000101 01001100 01001100
0110111/ 01001101 01111001 / 01001110
01000001 01101101 01000101 / 01001001
01110011/
01000001 01000100 01000001
Write your name in binary Ascii
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.
Claire
Claremont
Lady Oxford
Lady Caroline
Lamb
Augusta
Byron
Teresa
Guiccioli
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-yearold 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
Growing up
• 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.
• 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!”
The Difference Engine
• 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.
• 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

lim 1  1
n 
 e
n
n
So he is now
credited with
having
discovered e
2
2
n
(
n

1)
3
r
1  4
n
2
n
(
n

1)(2
n

1)(3
n
 3n  1)
4
1 r 
30
n
2
2
2
n
(2
n

2
n

1)(
n

1)
5
r
1 
12
n
n
1  n
1
n
1 2 1
1 r  2 n  2 n
n
1 3 1 2 2
2
1 r  3 n  2 n  12 n
n
1 4 1 3 3 2
3
1 r  3 n  2 n  12 n
n
1 5 1 4 4 3
1
4
2
1 r  5 n  2 n  12 n  0n  30 n
n
1 6 1 5 5 4
1 2
5
3
1 r  6 n  2 n  12 n  0n  12 n
B0  1
1
B1  
2
1
B2 
6
B3  0
1
B4  
30
B5  0
1
B6 
42
B7  0
1
B8  
30
Generalise
n
r
1
n
r
1
p
p
1
1 p
p p 1
p 1

n
 n 
n
 0n p  2  ...
p 1
2
12
Bp
B0 n p 1
B3
B1 p B2
p 1
p 2


n 
pn

p ( p  1)n
 ... 
n
0! p  1 1!
2!
3!
1!
p
n
Bk
p!
p 1 k
r

n
1
0 k ! ( p  1  k )!
p
Where Bk are
the Bernoulli
numbers
Bernoulli numbers
• Bk is the coefficient of
expansion of
x
xk
k!
in the Taylor series
e 1
x
2
4
2n
x
x
x
x
x
 1   B1  B2  ....  B2 n
x
e 1
2
2
4!
(2n)!
x
x  xk
 1  
x
e 1
2 k 2 k !
Bernoulli numbers
• The exponential function is
2
3
x
x
e x  1  x    ....
2! 3!
x
ex 1
substituting this into the denominator of
gives a strategy for calculating the numbers
n
recursively:
 n  1

Bi  0, n  1
i 1  i 
So the machine could compute the sequence as
long as it could store all the previous numbers.
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 112 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/onlyknown-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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