Blaise Pascal

1623 - 1662

Blaise Pascal – background

and early life • Born 1623, Clermont-Ferrand, France

• Mother died when he was 3

• Father was a senior government

administrator, which meant he was a

minor member of the nobility

• Two sisters, one older and one younger

• Father interested in maths and science,

and in education

Blaise Pascal – background

and early life • Family moved to Paris in 1631

• Father educated Blaise and his sisters at

home – intention was to teach Blaise

languages first, then move on to maths

and science from age 14

• Blaise soon showed amazing abilities –

especially in maths and science – family

legend is that he had derived most of

Euclid from scratch by age 12

Blaise Pascal – background

and early life • Family lost its money in 1631 when

Cardinal Richelieu defaulted on the

government’s bonds to pay for the 30

years’ war

• Father forced to flee Paris, having fallen

out with Richelieu – left his children in the

care of a aristocratic society lady

• Father eventually pardoned and made

King’s commissioner of taxes in Rouen

Blaise Pascal – background

and early life • First serious work on mathematics when

he was 16 – essay on conics, sent to Pere

Mersenne – Pascal’s theorem in projective

geometry

• Descartes didn’t believe a 16 year old boy

could have done it

Pascal’s Theorem

• If six arbitrary points are chosen on a conic and

joined by line segments in any order to form

a hexagon, then the three pairs of opposite sides of

the hexagon (extended if necessary) meet in three

points which lie on a straight line, called the Pascal

line of the hexagon.

Pascal’s Theorem

The Pascaline

• A calculating machine designed and built by Pascal

between 1642 and 1644 (when Pascal was between

19 and 21years old!)

• His motivation for building it was to try to help his

father do his tax calculations more efficiently.

• The ‘Pascal’ computer language is named after him

The Pascaline

• Besides being the first calculating machine

made public during its time, the Pascaline

is also:

• the only operational mechanical calculator

in the 17th century

• the first calculator to have a controlled

carry mechanism that allowed for an

effective propagation of multiple carries

• the first calculator to be used in an office

(his father's to compute taxes)

• the first calculator sold commercially

(around twenty machines were built)

• the first calculator to be patented (royal

privilege of 1649 [Louise XIV])

The Pascaline

• Pascal invented the hydraulic press and the

syringe

• He proved the existence of the vacuum

(disagreeing with Descartes)

• Pascal’s law:

any change in pressure applied at any given

point of a fluid is transmitted undiminished

throughout the fluid

This is the basis of hydraulics

Scientific work

• The SI unit of pressure is called the Pascal

(1 Pascal = 1Nm-2)

• Pascal was a pioneer of the scientific

method: “In order to show that a

hypothesis is evident, it does not suffice

that all the phenomena follow from it;

instead, if it leads to something contrary to

a single one of the phenomena, that

suffices to establish its falsity.”

Scientific work

• In 1646 he became interested in

Jansenism – a type of Catholicism

• He then lost interest in religion until 1654

when he had a near death experience in a

carriage accident that was followed by a

religious vision

Theological work

• His religious vision prompted him to give

up maths and science and concentrate on

theology instead

• The Pensees, a theological work

published posthumously, is considered

one of the greatest works of French

literature – it contains Pascal’s wager

Theological work

• Pascal died in 1662, aged just 39

• If he hadn’t died so young, or got

distracted from maths and science by

religion, how much more might he have

achieved?

Blaise Pascal

The problem:

• Two people each put £10 into a pot. They

then choose who will play ‘heads’ and who

will play ‘tails’ when tossing a coin. They

get a point each time the coin shows their

choice. The first one to get 10 points wins.

• If the game is interrupted before the end,

depending on the number of points each

has at that time, how should the pot be

divided?

The problem of points

• Pascal’s friend, the Chevalier de Méré,

asked him to consider this problem.

• Pascal wrote to Pierre de Fermat about it

• Each produced the same answer, but

using different methods

• Pascal’s method introduced the notion of

the expected value and Pascal’s and

Fermat’s work on this problem became the

foundation of probability theory

The problem of points

• Imagine the game is between Jeremy and

Theresa

• An unexpected interruption means the

game stops when Theresa (playing heads)

is leading Jeremy (playing tails) by 8 to 7

The problem of points

• Theresa says she is ahead and so should

get all the money

• Jeremy says the game is incomplete so all

bets are off and they should each get back

their stake

• Tim suggests that since Theresa has won

8/15 of the games, she should get

8/15 X £20 = £10.67

• What do you think?

The problem of points

• When the score is 8 to 7, what is the

maximum number of coin tosses need to

complete the game?

The problem of points

• To solve the problem:

– Fermat used and approach based on

listing all possible outcomes to find the

probability of each winning

– Pascal used an approach based on the

probability of each outcome and the

expected return

• Try it!

The problem of points

Pascal’s wager God exists God does not

exist

You are a believer + Infinity

(eternal bliss in

heaven)

Something finite

You are a non-

believer

- Infinity

(eternal misery in

hell)

Something finite

Pascal is suggesting a way to make a

rational decision, based upon expectation

• Using Pascal’s triangle to determine

binomial coefficients features in AS Maths,

both to expand brackets and to calculate

binomial probabilities

Pascal’s triangle

• Why do the numbers in Pascal’s triangle

give the binomial coefficients?

Pascal’s triangle

If the initial 1 in the triangle forms row 0,

investigate and, where possible, prove these

statements (at least convince your neighbour!):

a.The sum of the numbers in the nth row is 2n

b.The number formed by the digits in the nth row

of Pascal’s triangle is 11n

c.The numbers in the 3rd diagonal are the

triangular numbers

Pascal’s triangle

Discuss how the binomial coefficients relate to

combinations: Explain why the r th entry in the

nth row is equal to

Pascal’s triangle

!

!( )!n r

nC

r n r

Why does the way that Pascal’s triangle is

constructed suggest

?

Prove it!

Pascal’s triangle

1 1 1n r n r n rC C C